Note 1: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What does it mean for two formulas \(F\) and \(G\) to be logically equivalent (\(F \equiv G\))?
Back
What does it mean for two formulas \(F\) and \(G\) to be logically equivalent (\(F \equiv G\))?
\(F \equiv G\) means they correspond to the same function, i.e., their truth values are equal for all truth assignments to the propositional symbols appearing in \(F\) or \(G\).
\(F \equiv G\) means they correspond to the same function, i.e., their truth values are equal for all truth assignments to the propositional symbols appearing in \(F\) or \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does it mean for two formulas \(F\) and \(G\) to be logically equivalent (\(F \equiv G\))? | |
| Back | \(F \equiv G\) means they correspond to the same function, i.e., their truth values are equal for <strong>all</strong> truth assignments to the propositional symbols appearing in \(F\) or \(G\). |
Note 2: ETH::DiskMat
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Front
What is a tautology in propositional logic?
Back
What is a tautology in propositional logic?
A formula \(F\) is a tautology (or valid) if it is true for all truth assignments of the involved propositional symbols. Denoted as \(\models F\) or \(\top\).
A formula \(F\) is a tautology (or valid) if it is true for all truth assignments of the involved propositional symbols. Denoted as \(\models F\) or \(\top\).
Field-by-field Comparison
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|---|---|---|
| Front | What is a tautology in propositional logic? | |
| Back | A formula \(F\) is a tautology (or valid) if it is true for <strong>all</strong> truth assignments of the involved propositional symbols. Denoted as \(\models F\) or \(\top\). |
Note 3: ETH::DiskMat
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Front
What does it mean for a formula to be satisfiable?
Back
What does it mean for a formula to be satisfiable?
A formula \(F\) is satisfiable if it is true for at least one truth assignment of the involved propositional symbols.
A formula \(F\) is satisfiable if it is true for at least one truth assignment of the involved propositional symbols.
Field-by-field Comparison
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|---|---|---|
| Front | What does it mean for a formula to be satisfiable? | |
| Back | A formula \(F\) is satisfiable if it is true for <strong>at least one</strong> truth assignment of the involved propositional symbols. |
Note 4: ETH::DiskMat
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Front
What does it mean for a formula to be unsatisfiable?
Back
What does it mean for a formula to be unsatisfiable?
A formula is unsatisfiable if it is never true under any truth assignment. Denoted as \(\perp\).
A formula is unsatisfiable if it is never true under any truth assignment. Denoted as \(\perp\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does it mean for a formula to be unsatisfiable? | |
| Back | A formula is unsatisfiable if it is <strong>never</strong> true under any truth assignment. Denoted as \(\perp\). |
Note 5: ETH::DiskMat
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Front
What does \(F \models G\) mean (logical consequence)?
Back
What does \(F \models G\) mean (logical consequence)?
\(G\) is a logical consequence of \(F\) if for all truth assignments where \(F\) evaluates to \(1\), \(G\) also evaluates to \(1\).
\(G\) is a logical consequence of \(F\) if for all truth assignments where \(F\) evaluates to \(1\), \(G\) also evaluates to \(1\).
Field-by-field Comparison
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|---|---|---|
| Front | What does \(F \models G\) mean (logical consequence)? | |
| Back | \(G\) is a logical consequence of \(F\) if for all truth assignments where \(F\) evaluates to \(1\), \(G\) also evaluates to \(1\). |
Note 6: ETH::DiskMat
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Front
What's the difference between \(\equiv\), \(\leftrightarrow\), and \(\Leftrightarrow\)?
Back
What's the difference between \(\equiv\), \(\leftrightarrow\), and \(\Leftrightarrow\)?
- \(\equiv\): links formulas to statements (not part of PL itself)
- \(\leftrightarrow\): formula → formula (part of PL)
- \(\Leftrightarrow\): statement → statement
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What's the difference between \(\equiv\), \(\leftrightarrow\), and \(\Leftrightarrow\)? | |
| Back | <ul> <li>\(\equiv\): links formulas to statements (not part of PL itself)</li> <li>\(\leftrightarrow\): formula → formula (part of PL)</li> <li>\(\Leftrightarrow\): statement → statement</li> </ul> |
Note 7: ETH::DiskMat
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Front
What is a \(k\)-ary predicate on universe \(U\)?
Back
What is a \(k\)-ary predicate on universe \(U\)?
A function \(U^k \rightarrow \{0, 1\}\) that assigns each element of \(U^k\) to a truth value.
A function \(U^k \rightarrow \{0, 1\}\) that assigns each element of \(U^k\) to a truth value.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a \(k\)-ary predicate on universe \(U\)? | |
| Back | A function \(U^k \rightarrow \{0, 1\}\) that assigns each element of \(U^k\) to a truth value. |
Note 8: ETH::DiskMat
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Front
What is \(\lnot \forall x P(x)\) equivalent to?
Back
What is \(\lnot \forall x P(x)\) equivalent to?
\(\lnot \forall x P(x) \equiv \exists x \lnot P(x)\)
\(\lnot \forall x P(x) \equiv \exists x \lnot P(x)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is \(\lnot \forall x P(x)\) equivalent to? | |
| Back | \(\lnot \forall x P(x) \equiv \exists x \lnot P(x)\) |
Note 9: ETH::DiskMat
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Front
What is \(\lnot \exists x P(x)\) equivalent to?
Back
What is \(\lnot \exists x P(x)\) equivalent to?
\(\lnot \exists x P(x) \equiv \forall x \lnot P(x)\)
\(\lnot \exists x P(x) \equiv \forall x \lnot P(x)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is \(\lnot \exists x P(x)\) equivalent to? | |
| Back | \(\lnot \exists x P(x) \equiv \forall x \lnot P(x)\) |
Note 10: ETH::DiskMat
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Front
How does \(\forall\) distribute over \(\land\)?
Back
How does \(\forall\) distribute over \(\land\)?
\(\forall x P(x) \land \forall x Q(x) \equiv \forall x (P(x) \land Q(x))\)
\(\forall x P(x) \land \forall x Q(x) \equiv \forall x (P(x) \land Q(x))\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does \(\forall\) distribute over \(\land\)? | |
| Back | \(\forall x P(x) \land \forall x Q(x) \equiv \forall x (P(x) \land Q(x))\) |
Note 11: ETH::DiskMat
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Front
What is the relationship between \(\exists x (P(x) \land Q(x))\) and \(\exists x P(x) \land \exists x Q(x)\)?
Back
What is the relationship between \(\exists x (P(x) \land Q(x))\) and \(\exists x P(x) \land \exists x Q(x)\)?
\(\exists x (P(x) \land Q(x)) \models \exists x P(x) \land \exists x Q(x)\)
(Note: This is logical consequence, NOT equivalence. The reverse doesn't hold!)
\(\exists x (P(x) \land Q(x)) \models \exists x P(x) \land \exists x Q(x)\)
(Note: This is logical consequence, NOT equivalence. The reverse doesn't hold!)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the relationship between \(\exists x (P(x) \land Q(x))\) and \(\exists x P(x) \land \exists x Q(x)\)? | |
| Back | \(\exists x (P(x) \land Q(x)) \models \exists x P(x) \land \exists x Q(x)\) <br> (Note: This is logical consequence, NOT equivalence. The reverse doesn't hold!) |
Note 12: ETH::DiskMat
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Front
What are the idempotence laws in propositional logic?
Back
What are the idempotence laws in propositional logic?
- \(A \land A \equiv A\)
- \(A \lor A \equiv A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the idempotence laws in propositional logic? | |
| Back | <ul> <li>\(A \land A \equiv A\)</li> <li>\(A \lor A \equiv A\)</li> </ul> |
Note 13: ETH::DiskMat
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Front
What are the commutativity laws for \(\land\) and \(\lor\)?
Back
What are the commutativity laws for \(\land\) and \(\lor\)?
- \(A \land B \equiv B \land A\)
- \(A \lor B \equiv B \lor A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the commutativity laws for \(\land\) and \(\lor\)? | |
| Back | <ul> <li>\(A \land B \equiv B \land A\)</li> <li>\(A \lor B \equiv B \lor A\)</li> </ul> |
Note 14: ETH::DiskMat
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Front
What are the associativity laws for \(\land\) and \(\lor\)?
Back
What are the associativity laws for \(\land\) and \(\lor\)?
- \((A \land B) \land C \equiv A \land (B \land C)\)
- \((A \lor B) \lor C \equiv A \lor (B \lor C)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the associativity laws for \(\land\) and \(\lor\)? | |
| Back | <ul> <li>\((A \land B) \land C \equiv A \land (B \land C)\)</li> <li>\((A \lor B) \lor C \equiv A \lor (B \lor C)\)</li> </ul> |
Note 15: ETH::DiskMat
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Front
What are the absorption laws in propositional logic?
Back
What are the absorption laws in propositional logic?
- \(A \land (A \lor B) \equiv A\)
- \(A \lor (A \land B) \equiv A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the absorption laws in propositional logic? | |
| Back | <ul> <li>\(A \land (A \lor B) \equiv A\)</li> <li>\(A \lor (A \land B) \equiv A\)</li> </ul> |
Note 16: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What are both distributive laws in propositional logic?
Back
What are both distributive laws in propositional logic?
- \(A \land (B \lor C) \equiv (A \land B) \lor (A \land C)\) (distributing \(\land\) over \(\lor\))
- \(A \lor (B \land C) \equiv (A \lor B) \land (A \lor C)\) (distributing \(\lor\) over \(\land\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are both distributive laws in propositional logic? | |
| Back | <ul> <li>\(A \land (B \lor C) \equiv (A \land B) \lor (A \land C)\) (distributing \(\land\) over \(\lor\))</li> <li>\(A \lor (B \land C) \equiv (A \lor B) \land (A \lor C)\) (distributing \(\lor\) over \(\land\))</li> </ul> |
Note 17: ETH::DiskMat
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Front
What are De Morgan's laws?
Back
What are De Morgan's laws?
- \(\lnot(A \land B) \equiv \lnot A \lor \lnot B\)
- \(\lnot(A \lor B) \equiv \lnot A \land \lnot B\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are De Morgan's laws? | |
| Back | <ul> <li>\(\lnot(A \land B) \equiv \lnot A \lor \lnot B\)</li> <li>\(\lnot(A \lor B) \equiv \lnot A \land \lnot B\)</li> </ul> |
Note 18: ETH::DiskMat
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Front
What is the double negation law?
Back
What is the double negation law?
\(\lnot \lnot A \equiv A\)
\(\lnot \lnot A \equiv A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the double negation law? | |
| Back | \(\lnot \lnot A \equiv A\) |
Note 19: ETH::DiskMat
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Front
What is the relationship between tautologies and unsatisfiable formulas?
Back
What is the relationship between tautologies and unsatisfiable formulas?
A formula \(F\) is a tautology if and only if \(\lnot F\) is unsatisfiable.
A formula \(F\) is a tautology if and only if \(\lnot F\) is unsatisfiable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the relationship between tautologies and unsatisfiable formulas? | |
| Back | A formula \(F\) is a tautology <strong>if and only if</strong> \(\lnot F\) is unsatisfiable. |
Note 20: ETH::DiskMat
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Front
How are tautologies related to logical consequence (implication)?
Back
How are tautologies related to logical consequence (implication)?
For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology if and only if \(F \models G\).
For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology if and only if \(F \models G\).
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|---|---|---|
| Front | How are tautologies related to logical consequence (implication)? | |
| Back | For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology <strong>if and only if</strong> \(F \models G\). |
Note 21: ETH::DiskMat
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Front
If \(F \models G\) in predicate logic, what can we conclude about validity?
Back
If \(F \models G\) in predicate logic, what can we conclude about validity?
If \(F\) is valid, then \(G\) is also valid. (Logical consequence preserves validity)
If \(F\) is valid, then \(G\) is also valid. (Logical consequence preserves validity)
Field-by-field Comparison
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|---|---|---|
| Front | If \(F \models G\) in predicate logic, what can we conclude about validity? | |
| Back | If \(F\) is valid, then \(G\) is also valid. (Logical consequence preserves validity) |
Note 22: ETH::DiskMat
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Front
What is the transitivity property of implication?
Back
What is the transitivity property of implication?
\((A \rightarrow B) \land (B \rightarrow C) \models A \rightarrow C\)
\((A \rightarrow B) \land (B \rightarrow C) \models A \rightarrow C\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the transitivity property of implication? | |
| Back | \((A \rightarrow B) \land (B \rightarrow C) \models A \rightarrow C\) |
Note 23: ETH::DiskMat
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Front
What is the modus ponens logical rule?
Back
What is the modus ponens logical rule?
\(A \land (A \rightarrow B) \models B\)
(If \(A\) is true and \(A\) implies \(B\), then \(B\) is true)
\(A \land (A \rightarrow B) \models B\)
(If \(A\) is true and \(A\) implies \(B\), then \(B\) is true)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the modus ponens logical rule? | |
| Back | \(A \land (A \rightarrow B) \models B\) <br> (If \(A\) is true and \(A\) implies \(B\), then \(B\) is true) |
Note 24: ETH::DiskMat
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Front
What is the logical rule for case distinction?
Back
What is the logical rule for case distinction?
For every \(k\) we have: \[(A_1 \lor \dots \lor A_k) \land (A_1 \rightarrow B) \land \dots \land (A_k \rightarrow B) \models B\]
(If at least one case occurs, and all cases imply \(B\), then \(B\) holds)
For every \(k\) we have: \[(A_1 \lor \dots \lor A_k) \land (A_1 \rightarrow B) \land \dots \land (A_k \rightarrow B) \models B\]
(If at least one case occurs, and all cases imply \(B\), then \(B\) holds)
Field-by-field Comparison
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|---|---|---|
| Front | What is the logical rule for case distinction? | |
| Back | For every \(k\) we have: \[(A_1 \lor \dots \lor A_k) \land (A_1 \rightarrow B) \land \dots \land (A_k \rightarrow B) \models B\] <br> (If at least one case occurs, and all cases imply \(B\), then \(B\) holds) |
Note 25: ETH::DiskMat
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Front
What is the logical rule for proof by contradiction?
Back
What is the logical rule for proof by contradiction?
(If assuming \(\lnot A\) leads to something false, then \(A\) must be true)
- \((\lnot A \rightarrow B) \land \lnot B \models A\)
- Alternative: \((A \lor B) \land \lnot B \models A\)
(If assuming \(\lnot A\) leads to something false, then \(A\) must be true)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the logical rule for proof by contradiction? | |
| Back | <ul> <li>\((\lnot A \rightarrow B) \land \lnot B \models A\)</li> <li>Alternative: \((A \lor B) \land \lnot B \models A\)</li> </ul> <br> (If assuming \(\lnot A\) leads to something false, then \(A\) must be true) |
Note 26: ETH::DiskMat
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Front
What is the Pigeonhole Principle?
Back
What is the Pigeonhole Principle?
If a set of \(n\) objects is partitioned into \(k < n\) sets, then at least one of those sets contains at least \(\lceil \frac{n}{k} \rceil\) objects.
(If you have more pigeons than holes, at least one hole must contain multiple pigeons)
If a set of \(n\) objects is partitioned into \(k < n\) sets, then at least one of those sets contains at least \(\lceil \frac{n}{k} \rceil\) objects.
(If you have more pigeons than holes, at least one hole must contain multiple pigeons)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the Pigeonhole Principle? | |
| Back | If a set of \(n\) objects is partitioned into \(k < n\) sets, then at least one of those sets contains at least \(\lceil \frac{n}{k} \rceil\) objects. <br> (If you have more pigeons than holes, at least one hole must contain multiple pigeons) |
Note 27: ETH::DiskMat
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Front
What is the Principle of Mathematical Induction?
Back
What is the Principle of Mathematical Induction?
For the universe \(\mathbb{N}\) and an arbitrary unary predicate \(P\): \[P(0) \land \forall n (P(n) \rightarrow P(n+1)) \Rightarrow \forall n P(n)\]
(If the base case holds and the induction step holds, then the property holds for all natural numbers)
For the universe \(\mathbb{N}\) and an arbitrary unary predicate \(P\): \[P(0) \land \forall n (P(n) \rightarrow P(n+1)) \Rightarrow \forall n P(n)\]
(If the base case holds and the induction step holds, then the property holds for all natural numbers)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the Principle of Mathematical Induction? | |
| Back | For the universe \(\mathbb{N}\) and an arbitrary unary predicate \(P\): \[P(0) \land \forall n (P(n) \rightarrow P(n+1)) \Rightarrow \forall n P(n)\] <br> (If the base case holds and the induction step holds, then the property holds for all natural numbers) |
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Front
How does an indirect proof of \(S \Rightarrow T\) work?
Back
How does an indirect proof of \(S \Rightarrow T\) work?
An indirect proof assumes that \(T\) is false and proves that \(S\) is false under this assumption. This works because \(\lnot B \rightarrow \lnot A \models A \rightarrow B\).
An indirect proof assumes that \(T\) is false and proves that \(S\) is false under this assumption. This works because \(\lnot B \rightarrow \lnot A \models A \rightarrow B\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does an indirect proof of \(S \Rightarrow T\) work? | |
| Back | An indirect proof assumes that \(T\) is <strong>false</strong> and proves that \(S\) is <strong>false</strong> under this assumption. This works because \(\lnot B \rightarrow \lnot A \models A \rightarrow B\). |
Note 29: ETH::DiskMat
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Front
Describe the three steps of a modus ponens proof of statement \(S\).
Back
Describe the three steps of a modus ponens proof of statement \(S\).
1. Find a suitable mathematical statement \(R\)
2. Prove \(R\)
3. Prove \(R \Rightarrow S\)
1. Find a suitable mathematical statement \(R\)
2. Prove \(R\)
3. Prove \(R \Rightarrow S\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Describe the three steps of a modus ponens proof of statement \(S\). | |
| Back | 1. Find a suitable mathematical statement \(R\) <br>2. Prove \(R\) <br>3. Prove \(R \Rightarrow S\) |
Note 30: ETH::DiskMat
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Front
Describe the three steps of a case distinction proof of statement \(S\).
Back
Describe the three steps of a case distinction proof of statement \(S\).
1. Find a finite list \(R_1, \ldots, R_k\) of mathematical statements (the cases)
2. Prove that at least one of \(R_i\) is true (at least one case occurs)
3. Prove that \(R_i \Rightarrow S\) for \(i = 1, \ldots, k\)
1. Find a finite list \(R_1, \ldots, R_k\) of mathematical statements (the cases)
2. Prove that at least one of \(R_i\) is true (at least one case occurs)
3. Prove that \(R_i \Rightarrow S\) for \(i = 1, \ldots, k\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Describe the three steps of a case distinction proof of statement \(S\). | |
| Back | 1. Find a finite list \(R_1, \ldots, R_k\) of mathematical statements (the cases)<br>2. Prove that at least one of \(R_i\) is true (at least one case occurs)<br>3. Prove that \(R_i \Rightarrow S\) for \(i = 1, \ldots, k\) |
Note 31: ETH::DiskMat
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Front
Describe the three steps of a proof by contradiction of statement \(S\).
Back
Describe the three steps of a proof by contradiction of statement \(S\).
1. Find a suitable statement \(T\)
2. Prove that \(T\) is false
3. Assume \(S\) is false and prove that \(T\) is true (a contradiction)
1. Find a suitable statement \(T\)
2. Prove that \(T\) is false
3. Assume \(S\) is false and prove that \(T\) is true (a contradiction)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Describe the three steps of a proof by contradiction of statement \(S\). | |
| Back | 1. Find a suitable statement \(T\) <br>2. Prove that \(T\) is false <br>3. Assume \(S\) is false and prove that \(T\) is true (a contradiction) |
Note 32: ETH::DiskMat
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Front
What is the difference between a constructive and non-constructive existence proof?
Back
What is the difference between a constructive and non-constructive existence proof?
- Constructive: Exhibits an explicit \(a\) for which \(S_a\) is true
- Non-constructive: Proves existence without constructing a specific example
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the difference between a constructive and non-constructive existence proof? | |
| Back | <ul> <li><strong>Constructive</strong>: Exhibits an explicit \(a\) for which \(S_a\) is true</li> <li><strong>Non-constructive</strong>: Proves existence without constructing a specific example</li> </ul> |
Note 33: ETH::DiskMat
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Front
What is a proof by counterexample?
Back
What is a proof by counterexample?
A proof that \(S_x\) is not true for all \(x \in X\) by exhibiting an \(a\) (called a counterexample) such that \(S_a\) is false.
A proof that \(S_x\) is not true for all \(x \in X\) by exhibiting an \(a\) (called a counterexample) such that \(S_a\) is false.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a proof by counterexample? | |
| Back | A proof that \(S_x\) is <strong>not</strong> true for all \(x \in X\) by exhibiting an \(a\) (called a counterexample) such that \(S_a\) is <strong>false</strong>. |
Note 34: ETH::DiskMat
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Front
What are the two steps of a proof by induction?
Back
What are the two steps of a proof by induction?
1. Basis Step: Prove \(P(0)\) (or \(P(1)\) or \(P(k)\) depending on universe) 2. Induction Step: Prove that for any arbitrary \(n\), \(P(n) \Rightarrow P(n+1)\) (assuming \(P(n)\) as the induction hypothesis)
1. Basis Step: Prove \(P(0)\) (or \(P(1)\) or \(P(k)\) depending on universe) 2. Induction Step: Prove that for any arbitrary \(n\), \(P(n) \Rightarrow P(n+1)\) (assuming \(P(n)\) as the induction hypothesis)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the two steps of a proof by induction? | |
| Back | 1. <strong>Basis Step</strong>: Prove \(P(0)\) (or \(P(1)\) or \(P(k)\) depending on universe) 2. <strong>Induction Step</strong>: Prove that for any arbitrary \(n\), \(P(n) \Rightarrow P(n+1)\) (assuming \(P(n)\) as the induction hypothesis) |
Note 35: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the cardinality of a finite set \(A\)?
Back
What is the cardinality of a finite set \(A\)?
The number of elements of \(A\), denoted \(|A|\).
The number of elements of \(A\), denoted \(|A|\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the cardinality of a finite set \(A\)? | |
| Back | The number of elements of \(A\), denoted \(|A|\). |
Note 36: ETH::DiskMat
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Front
Give the formal definition of set equality.
Back
Give the formal definition of set equality.
\[A = B \overset{\text{def}}{\Longleftrightarrow} \forall x (x \in A \leftrightarrow x \in B)\]
\[A = B \overset{\text{def}}{\Longleftrightarrow} \forall x (x \in A \leftrightarrow x \in B)\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of set equality. | |
| Back | \[A = B \overset{\text{def}}{\Longleftrightarrow} \forall x (x \in A \leftrightarrow x \in B)\] |
Note 37: ETH::DiskMat
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Front
How are ordered pairs \((a, b)\) formally defined in set theory?
Back
How are ordered pairs \((a, b)\) formally defined in set theory?
\[(a, b) \overset{\text{def}}{=} \{\{a\}, \{a, b\}\}\]
\[(a, b) \overset{\text{def}}{=} \{\{a\}, \{a, b\}\}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How are ordered pairs \((a, b)\) formally defined in set theory? | |
| Back | \[(a, b) \overset{\text{def}}{=} \{\{a\}, \{a, b\}\}\] |
Note 38: ETH::DiskMat
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Front
Give the formal definition of subset (\(A \subseteq B\)).
Back
Give the formal definition of subset (\(A \subseteq B\)).
\[A \subseteq B \overset{\text{def}}{\Longleftrightarrow} \forall x (x \in A \rightarrow x \in B)\]
\[A \subseteq B \overset{\text{def}}{\Longleftrightarrow} \forall x (x \in A \rightarrow x \in B)\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of subset (\(A \subseteq B\)). | |
| Back | \[A \subseteq B \overset{\text{def}}{\Longleftrightarrow} \forall x (x \in A \rightarrow x \in B)\] |
Note 39: ETH::DiskMat
Deck: ETH::DiskMat
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Front
If two singleton sets are equal, what can we conclude about their elements?
Back
If two singleton sets are equal, what can we conclude about their elements?
For any sets \(a\) and \(b\), \(\{a\} = \{b\} \Longrightarrow a = b\)
For any sets \(a\) and \(b\), \(\{a\} = \{b\} \Longrightarrow a = b\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If two singleton sets are equal, what can we conclude about their elements? | |
| Back | For any sets \(a\) and \(b\), \(\{a\} = \{b\} \Longrightarrow a = b\) |
Note 40: ETH::DiskMat
Deck: ETH::DiskMat
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Front
How can we prove two sets are equal using subsets?
Back
How can we prove two sets are equal using subsets?
\[A = B \Longleftrightarrow (A \subseteq B) \land (B \subseteq A)\]
(To prove equality, show mutual subset inclusion)
\[A = B \Longleftrightarrow (A \subseteq B) \land (B \subseteq A)\]
(To prove equality, show mutual subset inclusion)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How can we prove two sets are equal using subsets? | |
| Back | \[A = B \Longleftrightarrow (A \subseteq B) \land (B \subseteq A)\] <br> (To prove equality, show mutual subset inclusion) |
Note 41: ETH::DiskMat
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Front
Is the subset relation transitive?
Back
Is the subset relation transitive?
Yes: \[(A \subseteq B) \land (B \subseteq C) \Rightarrow A \subseteq C\]
Yes: \[(A \subseteq B) \land (B \subseteq C) \Rightarrow A \subseteq C\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the subset relation transitive? | |
| Back | Yes: \[(A \subseteq B) \land (B \subseteq C) \Rightarrow A \subseteq C\] |
Note 42: ETH::DiskMat
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Front
Give the formal definitions of union and intersection.
Back
Give the formal definitions of union and intersection.
- \(A \cup B \overset{\text{def}}{=} \{x \ | \ x \in A \lor x \in B\}\)
- \(A \cap B \overset{\text{def}}{=} \{x \ | \ x \in A \land x \in B\}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definitions of union and intersection. | |
| Back | <ul> <li>\(A \cup B \overset{\text{def}}{=} \{x \ | \ x \in A \lor x \in B\}\)</li> <li>\(A \cap B \overset{\text{def}}{=} \{x \ | \ x \in A \land x \in B\}\)</li> </ul> |
Note 43: ETH::DiskMat
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Front
Give the formal definition of set difference \(B \setminus A\).
Back
Give the formal definition of set difference \(B \setminus A\).
\[B \setminus A \overset{\text{def}}{=} \{x \in B \ | \ x \not\in A\}\] (Elements in \(B\) that are not in \(A\))
\[B \setminus A \overset{\text{def}}{=} \{x \in B \ | \ x \not\in A\}\] (Elements in \(B\) that are not in \(A\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of set difference \(B \setminus A\). | |
| Back | \[B \setminus A \overset{\text{def}}{=} \{x \in B \ | \ x \not\in A\}\] (Elements in \(B\) that are not in \(A\)) |
Note 44: ETH::DiskMat
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o[h]hYy%u}
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Front
What is the relationship between the empty set and all other sets?
Back
What is the relationship between the empty set and all other sets?
\(\forall A (\emptyset \subseteq A)\) - The empty set is a subset of every set.
\(\forall A (\emptyset \subseteq A)\) - The empty set is a subset of every set.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the relationship between the empty set and all other sets? | |
| Back | \(\forall A (\emptyset \subseteq A)\) - The empty set is a subset of every set. |
Note 45: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the power set \(\mathcal{P}(A)\) of a set \(A\)?
Back
What is the power set \(\mathcal{P}(A)\) of a set \(A\)?
\[\mathcal{P}(A) \overset{\text{def}}{=} \{S \ | \ S \subseteq A\}\] The set of all subsets of \(A\).
\[\mathcal{P}(A) \overset{\text{def}}{=} \{S \ | \ S \subseteq A\}\] The set of all subsets of \(A\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the power set \(\mathcal{P}(A)\) of a set \(A\)? | |
| Back | \[\mathcal{P}(A) \overset{\text{def}}{=} \{S \ | \ S \subseteq A\}\] The set of all subsets of \(A\). |
Note 46: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the cardinality of the power set of a finite set with cardinality \(k\)?
Back
What is the cardinality of the power set of a finite set with cardinality \(k\)?
\(|\mathcal{P}(A)| = 2^k\) (hence the alternative notation \(2^A\))
\(|\mathcal{P}(A)| = 2^k\) (hence the alternative notation \(2^A\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the cardinality of the power set of a finite set with cardinality \(k\)? | |
| Back | \(|\mathcal{P}(A)| = 2^k\) (hence the alternative notation \(2^A\)) |
Note 47: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Give the formal definition of Cartesian product \(A \times B\).
Back
Give the formal definition of Cartesian product \(A \times B\).
\[A \times B = \{(a, b) \ | \ a \in A \land b \in B\}\] The set of all ordered pairs with first component from \(A\) and second from \(B\).
\[A \times B = \{(a, b) \ | \ a \in A \land b \in B\}\] The set of all ordered pairs with first component from \(A\) and second from \(B\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of Cartesian product \(A \times B\). | |
| Back | \[A \times B = \{(a, b) \ | \ a \in A \land b \in B\}\] The set of all ordered pairs with first component from \(A\) and second from \(B\). |
Note 48: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the cardinality of \(A \times B\) for finite sets?
Back
What is the cardinality of \(A \times B\) for finite sets?
\(|A \times B| = |A| \cdot |B|\)
\(|A \times B| = |A| \cdot |B|\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the cardinality of \(A \times B\) for finite sets? | |
| Back | \(|A \times B| = |A| \cdot |B|\) |
Note 49: ETH::DiskMat
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Front
Is the Cartesian product associative? Give an example.
Back
Is the Cartesian product associative? Give an example.
No, it's NOT associative.
No, it's NOT associative.
- \(\bigtimes_{i=1}^3 A_i\) gives \((a_1, a_2, a_3)\)
- \((A_1 \times A_2) \times A_3\) gives \(((a_1, a_2), a_3)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the Cartesian product associative? Give an example. | |
| Back | <strong>No</strong>, it's NOT associative. <ul> <li>\(\bigtimes_{i=1}^3 A_i\) gives \((a_1, a_2, a_3)\)</li> <li>\((A_1 \times A_2) \times A_3\) gives \(((a_1, a_2), a_3)\)</li> </ul> |
Note 50: ETH::DiskMat
Deck: ETH::DiskMat
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M
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Front
What are the idempotence laws for sets?
Back
What are the idempotence laws for sets?
- \(A \cap A = A\)
- \(A \cup A = A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the idempotence laws for sets? | |
| Back | <ul> <li>\(A \cap A = A\)</li> <li>\(A \cup A = A\)</li> </ul> |
Note 51: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What are the commutativity laws for sets?
Back
What are the commutativity laws for sets?
- \(A \cap B = B \cap A\)
- \(A \cup B = B \cup A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the commutativity laws for sets? | |
| Back | <ul> <li>\(A \cap B = B \cap A\)</li> <li>\(A \cup B = B \cup A\)</li> </ul> |
Note 52: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What are the associativity laws for sets?
Back
What are the associativity laws for sets?
- \(A \cap (B \cap C) = (A \cap B) \cap C\)
- \(A \cup (B \cup C) = (A \cup B) \cup C\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the associativity laws for sets? | |
| Back | <ul> <li>\(A \cap (B \cap C) = (A \cap B) \cap C\)</li> <li>\(A \cup (B \cup C) = (A \cup B) \cup C\)</li> </ul> |
Note 53: ETH::DiskMat
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Front
What are the absorption laws for sets?
Back
What are the absorption laws for sets?
- \(A \cap (A \cup B) = A\)
- \(A \cup (A \cap B) = A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the absorption laws for sets? | |
| Back | <ul> <li>\(A \cap (A \cup B) = A\)</li> <li>\(A \cup (A \cap B) = A\)</li> </ul> |
Note 54: ETH::DiskMat
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Front
What are the distributive laws for sets?
Back
What are the distributive laws for sets?
- \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
- \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the distributive laws for sets? | |
| Back | <ul> <li>\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)</li> <li>\(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)</li> </ul> |
Note 55: ETH::DiskMat
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eMixId]]vy
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Front
How can we characterize the subset relation using union and intersection?
Back
How can we characterize the subset relation using union and intersection?
\[A \subseteq B \iff A \cap B = A \iff A \cup B = B\]
\[A \subseteq B \iff A \cap B = A \iff A \cup B = B\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How can we characterize the subset relation using union and intersection? | |
| Back | \[A \subseteq B \iff A \cap B = A \iff A \cup B = B\] |
Note 56: ETH::DiskMat
Deck: ETH::DiskMat
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yl]2udZDYh
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Front
What is a binary relation from set \(A\) to set \(B\)?
Back
What is a binary relation from set \(A\) to set \(B\)?
A subset of \(A \times B\). If \(A = B\), then \(\rho\) is called a relation on \(A\).
A subset of \(A \times B\). If \(A = B\), then \(\rho\) is called a relation on \(A\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a binary relation from set \(A\) to set \(B\)? | |
| Back | A subset of \(A \times B\). If \(A = B\), then \(\rho\) is called a relation <strong>on</strong> \(A\). |
Note 57: ETH::DiskMat
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Front
What is the identity relation \(\text{id}_A\) on set \(A\)?
Back
What is the identity relation \(\text{id}_A\) on set \(A\)?
\[\text{id}_A = \{(a, a) \ | \ a \in A\}\]
\[\text{id}_A = \{(a, a) \ | \ a \in A\}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the identity relation \(\text{id}_A\) on set \(A\)? | |
| Back | \[\text{id}_A = \{(a, a) \ | \ a \in A\}\] |
Note 58: ETH::DiskMat
Deck: ETH::DiskMat
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Front
How many distinct relations are possible on a finite set \(A\) with \(|A|\) elements?
Back
How many distinct relations are possible on a finite set \(A\) with \(|A|\) elements?
\(2^{|A \times A|} = 2^{|A|^2}\) (because \(\rho \subseteq A \times A\))
\(2^{|A \times A|} = 2^{|A|^2}\) (because \(\rho \subseteq A \times A\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How many distinct relations are possible on a finite set \(A\) with \(|A|\) elements? | |
| Back | \(2^{|A \times A|} = 2^{|A|^2}\) (because \(\rho \subseteq A \times A\)) |
Note 59: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Give the formal definition of the inverse \(\hat{\rho}\) of a relation \(\rho \subseteq A \times B\).
Back
Give the formal definition of the inverse \(\hat{\rho}\) of a relation \(\rho \subseteq A \times B\).
\[\hat{\rho} \overset{\text{def}}{=} \{(b,a) \ | \ (a, b) \in \rho\} \subseteq B \times A\]
\[\hat{\rho} \overset{\text{def}}{=} \{(b,a) \ | \ (a, b) \in \rho\} \subseteq B \times A\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of the inverse \(\hat{\rho}\) of a relation \(\rho \subseteq A \times B\). | |
| Back | \[\hat{\rho} \overset{\text{def}}{=} \{(b,a) \ | \ (a, b) \in \rho\} \subseteq B \times A\] |
Note 60: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
C:,kb=SDJG
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Front
How does the inverse of a relation appear in matrix and graph representations?
Back
How does the inverse of a relation appear in matrix and graph representations?
- Matrix: The transpose of the matrix
- Graph: Reversing the direction of all edges
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does the inverse of a relation appear in matrix and graph representations? | |
| Back | <ul> <li><strong>Matrix</strong>: The transpose of the matrix</li> <li><strong>Graph</strong>: Reversing the direction of all edges</li> </ul> |
Note 61: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
zRN1V|E{mK
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Front
Give the formal definition of composition \(\rho \circ \sigma\) where \(\rho: A \to B\) and \(\sigma: B \to C\).
Back
Give the formal definition of composition \(\rho \circ \sigma\) where \(\rho: A \to B\) and \(\sigma: B \to C\).
\[\rho \circ \sigma \overset{\text{def}}{=} \{(a, c) \ | \ \exists b ((a, b) \in \rho \land (b, c) \in \sigma)\}\]
\[\rho \circ \sigma \overset{\text{def}}{=} \{(a, c) \ | \ \exists b ((a, b) \in \rho \land (b, c) \in \sigma)\}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of composition \(\rho \circ \sigma\) where \(\rho: A \to B\) and \(\sigma: B \to C\). | |
| Back | \[\rho \circ \sigma \overset{\text{def}}{=} \{(a, c) \ | \ \exists b ((a, b) \in \rho \land (b, c) \in \sigma)\}\] |
Note 62: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
QHq8d__[K&
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Front
How is composition of relations represented in matrix and graph form?
Back
How is composition of relations represented in matrix and graph form?
- Matrix: Matrix multiplication
- Graph: Natural composition - there's a path from \(a\) to \(c\) if there's a path \(a \to b\) in graph 1 and \(b \to c\) in graph 2
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How is composition of relations represented in matrix and graph form? | |
| Back | <ul> <li><strong>Matrix</strong>: Matrix multiplication</li> <li><strong>Graph</strong>: Natural composition - there's a path from \(a\) to \(c\) if there's a path \(a \to b\) in graph 1 and \(b \to c\) in graph 2</li> </ul> |
Note 63: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
gK5yW[0/~7
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Front
Is composition of relations associative?
Back
Is composition of relations associative?
Yes: \(\rho \circ (\sigma \circ \phi) = (\rho \circ \sigma) \circ \phi\)
Yes: \(\rho \circ (\sigma \circ \phi) = (\rho \circ \sigma) \circ \phi\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is composition of relations associative? | |
| Back | Yes: \(\rho \circ (\sigma \circ \phi) = (\rho \circ \sigma) \circ \phi\) |
Note 64: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
fll?FK2HQW
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Front
How does the inverse of a composition of relations behave?
Back
How does the inverse of a composition of relations behave?
Let \(\rho: A \to B\) and \(\sigma: B \to C\). Then: \[\widehat{\rho \sigma} = \hat{\sigma}\hat{\rho}\] (The inverse of a composition reverses the order)
Let \(\rho: A \to B\) and \(\sigma: B \to C\). Then: \[\widehat{\rho \sigma} = \hat{\sigma}\hat{\rho}\] (The inverse of a composition reverses the order)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does the inverse of a composition of relations behave? | |
| Back | Let \(\rho: A \to B\) and \(\sigma: B \to C\). Then: \[\widehat{\rho \sigma} = \hat{\sigma}\hat{\rho}\] (The inverse of a composition reverses the order) |
Note 65: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
GUID:
cMp-bYX->s
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Front
When is a relation \(\rho\) on set \(A\) reflexive?
Back
When is a relation \(\rho\) on set \(A\) reflexive?
When \(a \ \rho \ a\) is true for all \(a \in A\), i.e., \(\text{id} \subseteq \rho\)
When \(a \ \rho \ a\) is true for all \(a \in A\), i.e., \(\text{id} \subseteq \rho\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a relation \(\rho\) on set \(A\) reflexive? | |
| Back | When \(a \ \rho \ a\) is true for all \(a \in A\), i.e., \(\text{id} \subseteq \rho\) |
Note 66: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
l&wZTwi1Sq
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Front
When is a relation \(\rho\) on set \(A\) irreflexive?
Back
When is a relation \(\rho\) on set \(A\) irreflexive?
When \(a \ \not\rho \ a\) is true for all \(a \in A\), i.e., \(\rho \cap \text{id} = \emptyset\)
When \(a \ \not\rho \ a\) is true for all \(a \in A\), i.e., \(\rho \cap \text{id} = \emptyset\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a relation \(\rho\) on set \(A\) irreflexive? | |
| Back | When \(a \ \not\rho \ a\) is true for all \(a \in A\), i.e., \(\rho \cap \text{id} = \emptyset\) |
Note 67: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
Qzp().;[`~
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Front
When is a relation \(\rho\) on set \(A\) symmetric?
Back
When is a relation \(\rho\) on set \(A\) symmetric?
When \(a \ \rho \ b \Longleftrightarrow b \ \rho \ a\) for all \(a, b \in A\), i.e., \(\rho = \hat{\rho}\)
When \(a \ \rho \ b \Longleftrightarrow b \ \rho \ a\) for all \(a, b \in A\), i.e., \(\rho = \hat{\rho}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a relation \(\rho\) on set \(A\) symmetric? | |
| Back | When \(a \ \rho \ b \Longleftrightarrow b \ \rho \ a\) for all \(a, b \in A\), i.e., \(\rho = \hat{\rho}\) |
Note 68: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
yvfeO9PXGk
Previous
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Front
How does symmetry of a relation appear in matrix representation?
Back
How does symmetry of a relation appear in matrix representation?
The matrix is symmetric (equals its own transpose).
The matrix is symmetric (equals its own transpose).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does symmetry of a relation appear in matrix representation? | |
| Back | The matrix is symmetric (equals its own transpose). |
Note 69: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
GUID:
n{BL(`.1n2
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Front
When is a relation \(\rho\) on set \(A\) antisymmetric?
Back
When is a relation \(\rho\) on set \(A\) antisymmetric?
When \(a \ \rho \ b \land b \ \rho \ a \Longleftrightarrow a = b\) for all \(a, b \in A\), i.e., \(\rho \cap \hat{\rho} \subseteq \text{id}\)
When \(a \ \rho \ b \land b \ \rho \ a \Longleftrightarrow a = b\) for all \(a, b \in A\), i.e., \(\rho \cap \hat{\rho} \subseteq \text{id}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a relation \(\rho\) on set \(A\) antisymmetric? | |
| Back | When \(a \ \rho \ b \land b \ \rho \ a \Longleftrightarrow a = b\) for all \(a, b \in A\), i.e., \(\rho \cap \hat{\rho} \subseteq \text{id}\) |
Note 70: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
tmXe!J(6@%
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Front
Is antisymmetric the negation of symmetric?
Back
Is antisymmetric the negation of symmetric?
NO! Antisymmetry is NOT the negation of symmetry. They are independent properties.
NO! Antisymmetry is NOT the negation of symmetry. They are independent properties.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is antisymmetric the negation of symmetric? | |
| Back | <strong>NO!</strong> Antisymmetry is NOT the negation of symmetry. They are independent properties. |
Note 71: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
D0bP}6KO$]
Previous
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Front
How does antisymmetry appear in graph representation?
Back
How does antisymmetry appear in graph representation?
There is not a single cycle of length 2 (no edge from \(a\) to \(b\) AND from \(b\) to \(a\)).
There is not a single cycle of length 2 (no edge from \(a\) to \(b\) AND from \(b\) to \(a\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does antisymmetry appear in graph representation? | |
| Back | There is not a single cycle of length 2 (no edge from \(a\) to \(b\) AND from \(b\) to \(a\)). |
Note 72: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
p|78(#x<7
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Front
When is a relation \(\rho\) on set \(A\) transitive?
Back
When is a relation \(\rho\) on set \(A\) transitive?
When \(a \ \rho \ b \land b \ \rho \ c \Rightarrow a \ \rho \ c\) for all \(a, b, c \in A\).
When \(a \ \rho \ b \land b \ \rho \ c \Rightarrow a \ \rho \ c\) for all \(a, b, c \in A\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a relation \(\rho\) on set \(A\) transitive? | |
| Back | When \(a \ \rho \ b \land b \ \rho \ c \Rightarrow a \ \rho \ c\) for all \(a, b, c \in A\). |
Note 73: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
GoF!ZP7[i!
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Front
How can we test if a relation is transitive using composition?
Back
How can we test if a relation is transitive using composition?
A relation \(\rho\) is transitive if and only if \(\rho^2 \subseteq \rho\).
(If all two-step paths are already direct edges, the relation is transitive)
A relation \(\rho\) is transitive if and only if \(\rho^2 \subseteq \rho\).
(If all two-step paths are already direct edges, the relation is transitive)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How can we test if a relation is transitive using composition? | |
| Back | A relation \(\rho\) is transitive <strong>if and only if</strong> \(\rho^2 \subseteq \rho\). <br> (If all two-step paths are already direct edges, the relation is transitive) |
Note 74: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
vHNwBT2PnJ
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Front
What is the transitive closure \(\rho^*\) of a relation \(\rho\)?
Back
What is the transitive closure \(\rho^*\) of a relation \(\rho\)?
\[\rho^{*} = \bigcup_{n \in \mathbb{N} \setminus \{0\}} \rho^n\] The "reachability relation" - \((a,b) \in \rho^*\) iff there's a path of finite length from \(a\) to \(b\) in \(\rho\).
\[\rho^{*} = \bigcup_{n \in \mathbb{N} \setminus \{0\}} \rho^n\] The "reachability relation" - \((a,b) \in \rho^*\) iff there's a path of finite length from \(a\) to \(b\) in \(\rho\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the transitive closure \(\rho^*\) of a relation \(\rho\)? | |
| Back | \[\rho^{*} = \bigcup_{n \in \mathbb{N} \setminus \{0\}} \rho^n\] The "reachability relation" - \((a,b) \in \rho^*\) iff there's a path of finite length from \(a\) to \(b\) in \(\rho\). |
Note 75: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
LWf)m2..vK
Previous
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Front
What three properties must a relation have to be an equivalence relation?
Back
What three properties must a relation have to be an equivalence relation?
1. Reflexive 2. Symmetric 3. Transitive
1. Reflexive 2. Symmetric 3. Transitive
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What three properties must a relation have to be an equivalence relation? | |
| Back | 1. <span style="color: rgb(255, 255, 255);"><b>Reflexive</b></span> 2. <b>Symmetric</b> 3. <b>Transitive</b> |
Note 76: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
x^Wv3;n[%Q
Previous
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Front
What is the equivalence class \([a]_\theta\) of element \(a\) under equivalence relation \(\theta\)?
Back
What is the equivalence class \([a]_\theta\) of element \(a\) under equivalence relation \(\theta\)?
\[[a]_{\theta} \overset{\text{def}}{=} \{b \in A \ | \ b \ \theta \ a\}\] The set of all elements equivalent to \(a\).
\[[a]_{\theta} \overset{\text{def}}{=} \{b \in A \ | \ b \ \theta \ a\}\] The set of all elements equivalent to \(a\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the equivalence class \([a]_\theta\) of element \(a\) under equivalence relation \(\theta\)? | |
| Back | \[[a]_{\theta} \overset{\text{def}}{=} \{b \in A \ | \ b \ \theta \ a\}\] The set of all elements equivalent to \(a\). |
Note 77: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
HTIS
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Front
What are the two trivial equivalence relations on a set \(A\)?
Back
What are the two trivial equivalence relations on a set \(A\)?
1. Complete relation \(A \times A\) → single equivalence class \(A\) 2. Identity relation → equivalence classes are all singletons \(\{a\}\)
1. Complete relation \(A \times A\) → single equivalence class \(A\) 2. Identity relation → equivalence classes are all singletons \(\{a\}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the two trivial equivalence relations on a set \(A\)? | |
| Back | 1. <strong>Complete relation</strong> \(A \times A\) → single equivalence class \(A\) 2. <strong>Identity relation</strong> → equivalence classes are all singletons \(\{a\}\) |
Note 78: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
GUID:
y+%Du@ss=x
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Front
If we intersect two equivalence relations, what do we get?
Back
If we intersect two equivalence relations, what do we get?
The intersection of two equivalence relations (on the same set) is also an equivalence relation.
The intersection of two equivalence relations (on the same set) is also an equivalence relation.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If we intersect two equivalence relations, what do we get? | |
| Back | The intersection of two equivalence relations (on the same set) is also an equivalence relation. |
Note 79: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
op#z.)
Previous
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Front
What is a partition of a set \(A\)?
Back
What is a partition of a set \(A\)?
A set \(\{S_i \ | \ i \in \mathcal{I}\}\) of mutually disjoint subsets that cover \(A\):
A set \(\{S_i \ | \ i \in \mathcal{I}\}\) of mutually disjoint subsets that cover \(A\):
- \(S_i \cap S_j = \emptyset\) for \(i \neq j\)
- \(\bigcup_{i \in \mathcal{I}} S_i = A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a partition of a set \(A\)? | |
| Back | A set \(\{S_i \ | \ i \in \mathcal{I}\}\) of mutually disjoint subsets that cover \(A\): <ul> <li>\(S_i \cap S_j = \emptyset\) for \(i \neq j\)</li> <li>\(\bigcup_{i \in \mathcal{I}} S_i = A\)</li> </ul> |
Note 80: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
GV.6~{1l[p
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Front
What is the quotient set \(A / \theta\)?
Back
What is the quotient set \(A / \theta\)?
\[A / \theta \overset{\text{def}}{=} \{[a]_{\theta} \ | \ a \in A\}\] The set of all equivalence classes of \(\theta\) on \(A\) (also called "\(A\) modulo \(\theta\)" or "\(A\) mod \(\theta\)").
\[A / \theta \overset{\text{def}}{=} \{[a]_{\theta} \ | \ a \in A\}\] The set of all equivalence classes of \(\theta\) on \(A\) (also called "\(A\) modulo \(\theta\)" or "\(A\) mod \(\theta\)").
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the quotient set \(A / \theta\)? | |
| Back | \[A / \theta \overset{\text{def}}{=} \{[a]_{\theta} \ | \ a \in A\}\] The set of all equivalence classes of \(\theta\) on \(A\) (also called "\(A\) modulo \(\theta\)" or "\(A\) mod \(\theta\)"). |
Note 81: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
LTXK//3RaH
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Front
What important property do equivalence classes have?
Back
What important property do equivalence classes have?
The set \(A / \theta\) of equivalence classes of an equivalence relation \(\theta\) on \(A\) is a partition of \(A\).
(Equivalence classes are disjoint and cover the entire set)
The set \(A / \theta\) of equivalence classes of an equivalence relation \(\theta\) on \(A\) is a partition of \(A\).
(Equivalence classes are disjoint and cover the entire set)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What important property do equivalence classes have? | |
| Back | The set \(A / \theta\) of equivalence classes of an equivalence relation \(\theta\) on \(A\) is a partition of \(A\). <br> (Equivalence classes are disjoint and cover the entire set) |
Note 82: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
F2
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Front
How are the rational numbers \(\mathbb{Q}\) defined using equivalence relations?
Back
How are the rational numbers \(\mathbb{Q}\) defined using equivalence relations?
Let \(A = \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})\) and \((a, b) \sim (c,d) \overset{\text{def}}{\Longleftrightarrow} ad = bc\)
Then: \(\mathbb{Q} \overset{\text{def}}{=} (\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})) / \sim\)
Let \(A = \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})\) and \((a, b) \sim (c,d) \overset{\text{def}}{\Longleftrightarrow} ad = bc\)
Then: \(\mathbb{Q} \overset{\text{def}}{=} (\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})) / \sim\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How are the rational numbers \(\mathbb{Q}\) defined using equivalence relations? | |
| Back | Let \(A = \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})\) and \((a, b) \sim (c,d) \overset{\text{def}}{\Longleftrightarrow} ad = bc\) <br> Then: \(\mathbb{Q} \overset{\text{def}}{=} (\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})) / \sim\) |
Note 83: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
GUID:
qnpI?yoaky
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Front
What three properties must a relation have to be a partial order?
Back
What three properties must a relation have to be a partial order?
1. Reflexive 2. Antisymmetric 3. Transitive
1. Reflexive 2. Antisymmetric 3. Transitive
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What three properties must a relation have to be a partial order? | |
| Back | 1. <strong>Reflexive</strong> 2. <strong>Antisymmetric</strong> 3. <strong>Transitive</strong> |
Note 84: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
dQBgAq4%4R
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Front
What is the key difference between a partial order and an equivalence relation?
Back
What is the key difference between a partial order and an equivalence relation?
Replace the symmetry condition with an antisymmetry condition.
Replace the symmetry condition with an antisymmetry condition.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the key difference between a partial order and an equivalence relation? | |
| Back | Replace the <strong>symmetry</strong> condition with an <strong>antisymmetry</strong> condition. |
Note 85: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
v2<,m(`1YY
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Front
When are two elements \(a\) and \(b\) comparable in a poset \((A; \preceq)\)?
Back
When are two elements \(a\) and \(b\) comparable in a poset \((A; \preceq)\)?
When \(a \preceq b\) or \(b \preceq a\). Otherwise they are incomparable.
When \(a \preceq b\) or \(b \preceq a\). Otherwise they are incomparable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When are two elements \(a\) and \(b\) comparable in a poset \((A; \preceq)\)? | |
| Back | When \(a \preceq b\) <strong>or</strong> \(b \preceq a\). Otherwise they are <strong>incomparable</strong>. |
Note 86: ETH::DiskMat
Deck: ETH::DiskMat
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Note Type: Basic (with MathJax)
GUID:
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Front
When is a poset \((A; \preceq)\) totally ordered (linearly ordered)?
Back
When is a poset \((A; \preceq)\) totally ordered (linearly ordered)?
When any two elements of \(A\) are comparable.
When any two elements of \(A\) are comparable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a poset \((A; \preceq)\) totally ordered (linearly ordered)? | |
| Back | When <strong>any two elements</strong> of \(A\) are comparable. |
Note 87: ETH::DiskMat
Deck: ETH::DiskMat
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Note Type: Basic (with MathJax)
GUID:
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Front
When does element \(b\) cover element \(a\) in a poset \((A; \preceq)\)?
Back
When does element \(b\) cover element \(a\) in a poset \((A; \preceq)\)?
When \(a \prec b\) and there exists no \(c\) with \(a \prec c \prec b\) (i.e., \(b\) is the direct successor of \(a\)).
When \(a \prec b\) and there exists no \(c\) with \(a \prec c \prec b\) (i.e., \(b\) is the direct successor of \(a\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When does element \(b\) <strong>cover</strong> element \(a\) in a poset \((A; \preceq)\)? | |
| Back | When \(a \prec b\) and there exists <strong>no</strong> \(c\) with \(a \prec c \prec b\) (i.e., \(b\) is the direct successor of \(a\)). |
Note 88: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
What is a Hasse diagram of a poset \((A; \preceq)\)?
Back
What is a Hasse diagram of a poset \((A; \preceq)\)?
A directed graph whose vertices are labeled with elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).
A directed graph whose vertices are labeled with elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a Hasse diagram of a poset \((A; \preceq)\)? | |
| Back | A directed graph whose vertices are labeled with elements of \(A\) and where there is an edge from \(a\) to \(b\) <strong>if and only if</strong> \(b\) <strong>covers</strong> \(a\). |
Note 89: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
How is the direct product \((A; \preceq) \times (B; \sqsubseteq)\) defined?
Back
How is the direct product \((A; \preceq) \times (B; \sqsubseteq)\) defined?
The set \(A \times B\) with relation \(\leq\) defined by: \[(a_1, b_1) \leq (a_2, b_2) \overset{\text{def}}{\Longleftrightarrow} a_1 \preceq a_2 \land b_1 \sqsubseteq b_2\]
The set \(A \times B\) with relation \(\leq\) defined by: \[(a_1, b_1) \leq (a_2, b_2) \overset{\text{def}}{\Longleftrightarrow} a_1 \preceq a_2 \land b_1 \sqsubseteq b_2\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How is the direct product \((A; \preceq) \times (B; \sqsubseteq)\) defined? | |
| Back | The set \(A \times B\) with relation \(\leq\) defined by: \[(a_1, b_1) \leq (a_2, b_2) \overset{\text{def}}{\Longleftrightarrow} a_1 \preceq a_2 \land b_1 \sqsubseteq b_2\] |
Note 90: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
If we take the direct product of two posets, what do we get?
Back
If we take the direct product of two posets, what do we get?
\((A; \preceq) \times (B;\sqsubseteq)\) is also a poset.
(The direct product preserves the poset structure)
\((A; \preceq) \times (B;\sqsubseteq)\) is also a poset.
(The direct product preserves the poset structure)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If we take the direct product of two posets, what do we get? | |
| Back | \((A; \preceq) \times (B;\sqsubseteq)\) is also a poset. <br> (The direct product preserves the poset structure) |
Note 91: ETH::DiskMat
Deck: ETH::DiskMat
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Front
How is the lexicographic order \(\leq_{\text{lex}}\) on \(A \times B\) defined?
Back
How is the lexicographic order \(\leq_{\text{lex}}\) on \(A \times B\) defined?
\[(a_1, b_1) \leq_{\text{lex}} (a_2, b_2) \overset{\text{def}}{\Longleftrightarrow} a_1 \preceq a_2 \lor (a_1 = a_2 \land b_1 \sqsubseteq b_2)\]
\[(a_1, b_1) \leq_{\text{lex}} (a_2, b_2) \overset{\text{def}}{\Longleftrightarrow} a_1 \preceq a_2 \lor (a_1 = a_2 \land b_1 \sqsubseteq b_2)\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How is the lexicographic order \(\leq_{\text{lex}}\) on \(A \times B\) defined? | |
| Back | \[(a_1, b_1) \leq_{\text{lex}} (a_2, b_2) \overset{\text{def}}{\Longleftrightarrow} a_1 \preceq a_2 \lor (a_1 = a_2 \land b_1 \sqsubseteq b_2)\] |
Note 92: ETH::DiskMat
Deck: ETH::DiskMat
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Front
When is the lexicographic order on \(A \times B\) totally ordered?
Back
When is the lexicographic order on \(A \times B\) totally ordered?
When both \((A; \preceq)\) and \((B; \sqsubseteq)\) are totally ordered.
When both \((A; \preceq)\) and \((B; \sqsubseteq)\) are totally ordered.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is the lexicographic order on \(A \times B\) totally ordered? | |
| Back | When <strong>both</strong> \((A; \preceq)\) and \((B; \sqsubseteq)\) are totally ordered. |
Note 93: ETH::DiskMat
Deck: ETH::DiskMat
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cm^dke)Enb
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Front
What's the difference between a minimal element and the least element in a poset?
Back
What's the difference between a minimal element and the least element in a poset?
- Minimal: No \(b\) exists with \(b \prec a\) (could be multiple minimal elements)
- Least: \(a \preceq b\) for all \(b \in A\) (unique if it exists)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What's the difference between a minimal element and the least element in a poset? | |
| Back | <ul> <li><strong>Minimal</strong>: No \(b\) exists with \(b \prec a\) (could be multiple minimal elements)</li> <li><strong>Least</strong>: \(a \preceq b\) for <strong>all</strong> \(b \in A\) (unique if it exists)</li> </ul> |
Note 94: ETH::DiskMat
Deck: ETH::DiskMat
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Front
When is \(a \in A\) a lower bound of subset \(S \subseteq A\) in poset \((A; \preceq)\)?
Back
When is \(a \in A\) a lower bound of subset \(S \subseteq A\) in poset \((A; \preceq)\)?
When \(a \preceq b\) for all \(b \in S\).
When \(a \preceq b\) for all \(b \in S\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is \(a \in A\) a lower bound of subset \(S \subseteq A\) in poset \((A; \preceq)\)? | |
| Back | When \(a \preceq b\) for <strong>all</strong> \(b \in S\). |
Note 95: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the greatest lower bound (glb) of a subset \(S\) in a poset?
Back
What is the greatest lower bound (glb) of a subset \(S\) in a poset?
The greatest element (by the relation, not just integer ordering) of the set of all lower bounds of \(S\). Also called the infimum.
The greatest element (by the relation, not just integer ordering) of the set of all lower bounds of \(S\). Also called the infimum.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the greatest lower bound (glb) of a subset \(S\) in a poset? | |
| Back | The <strong>greatest element</strong> (by the relation, not just integer ordering) of the set of all lower bounds of \(S\). Also called the <strong>infimum</strong>. |
Note 96: ETH::DiskMat
Deck: ETH::DiskMat
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Front
When is a poset \((A; \preceq)\) well-ordered?
Back
When is a poset \((A; \preceq)\) well-ordered?
When it is totally ordered AND every non-empty subset of \(A\) has a least element.
When it is totally ordered AND every non-empty subset of \(A\) has a least element.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is a poset \((A; \preceq)\) well-ordered? | |
| Back | When it is <strong>totally ordered</strong> AND every non-empty subset of \(A\) has a <strong>least element</strong>. |
Note 97: ETH::DiskMat
Deck: ETH::DiskMat
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Front
For what types of posets is well-ordering primarily of interest?
Back
For what types of posets is well-ordering primarily of interest?
Infinite posets. (Every totally ordered finite poset is automatically well-ordered)
Infinite posets. (Every totally ordered finite poset is automatically well-ordered)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | For what types of posets is well-ordering primarily of interest? | |
| Back | <strong>Infinite posets</strong>. (Every totally ordered finite poset is automatically well-ordered) |
Note 98: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What are the meet and join of elements \(a\) and \(b\) in a poset?
Back
What are the meet and join of elements \(a\) and \(b\) in a poset?
- Meet (\(a \land b\)): The greatest lower bound of \(\{a, b\}\)
- Join (\(a \lor b\)): The least upper bound of \(\{a, b\}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the meet and join of elements \(a\) and \(b\) in a poset? | |
| Back | <ul> <li><strong>Meet</strong> (\(a \land b\)): The greatest lower bound of \(\{a, b\}\)</li> <li><strong>Join</strong> (\(a \lor b\)): The least upper bound of \(\{a, b\}\)</li> </ul> |
Note 99: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is a lattice?
Back
What is a lattice?
A poset \((A; \preceq)\) in which every pair of elements has a meet and join.
A poset \((A; \preceq)\) in which every pair of elements has a meet and join.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a lattice? | |
| Back | A poset \((A; \preceq)\) in which <strong>every pair</strong> of elements has a meet and join. |
Note 100: ETH::DiskMat
Deck: ETH::DiskMat
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Front
In the lattice \((\{1,2,3,4,5,6,8,12,24\}; |)\), what are the meet and join for two examples of pairs of numbers?
Back
In the lattice \((\{1,2,3,4,5,6,8,12,24\}; |)\), what are the meet and join for two examples of pairs of numbers?
Example: \(6 \land 8 = 2\), \(6 \lor 8 = 24\)
- Meet: The gcd (greatest common divisor)
- Join: The lcm (least common multiple)
Example: \(6 \land 8 = 2\), \(6 \lor 8 = 24\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | In the lattice \((\{1,2,3,4,5,6,8,12,24\}; |)\), what are the meet and join for two examples of pairs of numbers? | |
| Back | <ul> <li><strong>Meet</strong>: The gcd (greatest common divisor)</li> <li><strong>Join</strong>: The lcm (least common multiple)</li> </ul> <br> Example: \(6 \land 8 = 2\), \(6 \lor 8 = 24\) |
Note 101: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What two properties must a relation \(f: A \to B\) have to be a function?
Back
What two properties must a relation \(f: A \to B\) have to be a function?
1. Totally defined: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 2. Well-defined: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
1. Totally defined: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 2. Well-defined: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What two properties must a relation \(f: A \to B\) have to be a function? | |
| Back | 1. <strong>Totally defined</strong>: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 2. <strong>Well-defined</strong>: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\) |
Note 102: ETH::DiskMat
Deck: ETH::DiskMat
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mIeYx_[A>*
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Front
What notation denotes the set of all functions \(A \to B\)?
Back
What notation denotes the set of all functions \(A \to B\)?
\(B^A\)
\(B^A\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What notation denotes the set of all functions \(A \to B\)? | |
| Back | \(B^A\) |
Note 103: ETH::DiskMat
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Front
What is a partial function \(A \to B\)?
Back
What is a partial function \(A \to B\)?
A relation from \(A\) to \(B\) that satisfies only the well-defined property (condition 2), NOT necessarily totally defined.
A relation from \(A\) to \(B\) that satisfies only the well-defined property (condition 2), NOT necessarily totally defined.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a partial function \(A \to B\)? | |
| Back | A relation from \(A\) to \(B\) that satisfies only the <strong>well-defined</strong> property (condition 2), NOT necessarily totally defined. |
Note 104: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the image \(f(S)\) of a subset \(S \subseteq A\) under function \(f: A \to B\)?
Back
What is the image \(f(S)\) of a subset \(S \subseteq A\) under function \(f: A \to B\)?
\[f(S) \overset{\text{def}}{=} \{f(a) \ | \ a \in S\}\]
\[f(S) \overset{\text{def}}{=} \{f(a) \ | \ a \in S\}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the image \(f(S)\) of a subset \(S \subseteq A\) under function \(f: A \to B\)? | |
| Back | \[f(S) \overset{\text{def}}{=} \{f(a) \ | \ a \in S\}\] |
Note 105: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the image (or range) of a function \(f: A \to B\)?
Back
What is the image (or range) of a function \(f: A \to B\)?
The subset \(f(A)\) of \(B\), also denoted \(\text{Im}(f)\).
The subset \(f(A)\) of \(B\), also denoted \(\text{Im}(f)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the image (or range) of a function \(f: A \to B\)? | |
| Back | The subset \(f(A)\) of \(B\), also denoted \(\text{Im}(f)\). |
Note 106: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the preimage \(f^{-1}(T)\) of a subset \(T \subseteq B\)?
Back
What is the preimage \(f^{-1}(T)\) of a subset \(T \subseteq B\)?
\[f^{-1}(T) \overset{\text{def}}{=} \{a \in A \ | \ f(a) \in T\}\] The set of values in \(A\) that map into \(T\).
\[f^{-1}(T) \overset{\text{def}}{=} \{a \in A \ | \ f(a) \in T\}\] The set of values in \(A\) that map into \(T\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the preimage \(f^{-1}(T)\) of a subset \(T \subseteq B\)? | |
| Back | \[f^{-1}(T) \overset{\text{def}}{=} \{a \in A \ | \ f(a) \in T\}\] The set of values in \(A\) that map into \(T\). |
Note 107: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What does it mean for a function \(f: A \to B\) to be injective (one-to-one)?
Back
What does it mean for a function \(f: A \to B\) to be injective (one-to-one)?
For \(a \neq a'\) we have \(f(a) \neq f(a')\). No two distinct values are mapped to the same function value (no "collisions").
For \(a \neq a'\) we have \(f(a) \neq f(a')\). No two distinct values are mapped to the same function value (no "collisions").
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does it mean for a function \(f: A \to B\) to be injective (one-to-one)? | |
| Back | For \(a \neq a'\) we have \(f(a) \neq f(a')\). No two distinct values are mapped to the same function value (no "collisions"). |
Note 108: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What does it mean for a function \(f: A \to B\) to be surjective (onto)?
Back
What does it mean for a function \(f: A \to B\) to be surjective (onto)?
\(f(A) = B\), i.e., for every \(b \in B\), \(b = f(a)\) for some \(a \in A\). Every value in the codomain is taken on.
\(f(A) = B\), i.e., for every \(b \in B\), \(b = f(a)\) for some \(a \in A\). Every value in the codomain is taken on.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does it mean for a function \(f: A \to B\) to be surjective (onto)? | |
| Back | \(f(A) = B\), i.e., for every \(b \in B\), \(b = f(a)\) for some \(a \in A\). Every value in the codomain is taken on. |
Note 109: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What does it mean for a function to be bijective?
Back
What does it mean for a function to be bijective?
It is both injective and surjective.
It is both injective and surjective.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does it mean for a function to be bijective? | |
| Back | It is both <strong>injective</strong> and <strong>surjective</strong>. |
Note 110: ETH::DiskMat
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Front
When does a function have an inverse function?
Back
When does a function have an inverse function?
When the function is bijective. The inverse (as a relation) is called the inverse function, denoted \(f^{-1}\).
When the function is bijective. The inverse (as a relation) is called the inverse function, denoted \(f^{-1}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When does a function have an inverse function? | |
| Back | When the function is <strong>bijective</strong>. The inverse (as a relation) is called the inverse function, denoted \(f^{-1}\). |
Note 111: ETH::DiskMat
Deck: ETH::DiskMat
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Front
How is the composition \(g \circ f\) of functions \(f: A \to B\) and \(g: B \to C\) defined?
Back
How is the composition \(g \circ f\) of functions \(f: A \to B\) and \(g: B \to C\) defined?
\[(g \circ f)(a) = g(f(a))\] Critical: \(f\) is applied FIRST, then \(g\) (read right to left!)
\[(g \circ f)(a) = g(f(a))\] Critical: \(f\) is applied FIRST, then \(g\) (read right to left!)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How is the composition \(g \circ f\) of functions \(f: A \to B\) and \(g: B \to C\) defined? | |
| Back | \[(g \circ f)(a) = g(f(a))\] <strong>Critical</strong>: \(f\) is applied <strong>FIRST</strong>, then \(g\) (read right to left!) |
Note 112: ETH::DiskMat
Deck: ETH::DiskMat
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Front
In the composition \(g \circ f\), which function is applied first?
Back
In the composition \(g \circ f\), which function is applied first?
\(f\) is applied FIRST, then \(g\). The order of letters (left to right) is OPPOSITE to the order of application (right to left).
\(f\) is applied FIRST, then \(g\). The order of letters (left to right) is OPPOSITE to the order of application (right to left).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | In the composition \(g \circ f\), which function is applied first? | |
| Back | \(f\) is applied FIRST, then \(g\). The order of letters (left to right) is <strong>OPPOSITE</strong> to the order of application (right to left). |
Note 113: ETH::DiskMat
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Front
Is function composition associative?
Back
Is function composition associative?
Yes: \((h \circ g) \circ f = h \circ (g \circ f)\)
Yes: \((h \circ g) \circ f = h \circ (g \circ f)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is function composition associative? | |
| Back | Yes: \((h \circ g) \circ f = h \circ (g \circ f)\) |
Note 114: ETH::DiskMat
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Front
When are two sets \(A\) and \(B\) equinumerous (\(A \sim B\))?
Back
When are two sets \(A\) and \(B\) equinumerous (\(A \sim B\))?
When there exists a bijection \(A \to B\).
When there exists a bijection \(A \to B\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When are two sets \(A\) and \(B\) equinumerous (\(A \sim B\))? | |
| Back | When there exists a <strong>bijection</strong> \(A \to B\). |
Note 115: ETH::DiskMat
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Front
When does set \(B\) dominate set \(A\) (denoted \(A \preceq B\))?
Back
When does set \(B\) dominate set \(A\) (denoted \(A \preceq B\))?
When \(A \sim C\) for some subset \(C \subseteq B\), or equivalently, when there exists an injection \(A \to B\).
When \(A \sim C\) for some subset \(C \subseteq B\), or equivalently, when there exists an injection \(A \to B\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When does set \(B\) dominate set \(A\) (denoted \(A \preceq B\))? | |
| Back | When \(A \sim C\) for some subset \(C \subseteq B\), or equivalently, when there exists an <strong>injection</strong> \(A \to B\). |
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Front
What does it mean for a set \(A\) to be countable?
Back
What does it mean for a set \(A\) to be countable?
\(A \preceq \mathbb{N}\) (i.e., there exists an injection \(A \to \mathbb{N}\))
\(A \preceq \mathbb{N}\) (i.e., there exists an injection \(A \to \mathbb{N}\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does it mean for a set \(A\) to be countable? | |
| Back | \(A \preceq \mathbb{N}\) (i.e., there exists an injection \(A \to \mathbb{N}\)) |
Note 117: ETH::DiskMat
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Front
What kind of relation is equinumerosity (\(\sim\))?
Back
What kind of relation is equinumerosity (\(\sim\))?
The relation \(\sim\) (equinumerous) is an equivalence relation.
(It is reflexive, symmetric, and transitive)
The relation \(\sim\) (equinumerous) is an equivalence relation.
(It is reflexive, symmetric, and transitive)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What kind of relation is equinumerosity (\(\sim\))? | |
| Back | The relation \(\sim\) (equinumerous) is an <strong>equivalence relation</strong>. <br> (It is reflexive, symmetric, and transitive) |
Note 118: ETH::DiskMat
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Front
Is the "dominates" relation (\(\preceq\)) transitive?
Back
Is the "dominates" relation (\(\preceq\)) transitive?
Yes: \(A \preceq B \land B \preceq C \Rightarrow A \preceq C\)
(If \(A\) injects into \(B\) and \(B\) injects into \(C\), then \(A\) injects into \(C\))
Yes: \(A \preceq B \land B \preceq C \Rightarrow A \preceq C\)
(If \(A\) injects into \(B\) and \(B\) injects into \(C\), then \(A\) injects into \(C\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the "dominates" relation (\(\preceq\)) transitive? | |
| Back | Yes: \(A \preceq B \land B \preceq C \Rightarrow A \preceq C\) <br> (If \(A\) injects into \(B\) and \(B\) injects into \(C\), then \(A\) injects into \(C\)) |
Note 119: ETH::DiskMat
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Front
If two sets each dominate the other, what can we conclude?
Back
If two sets each dominate the other, what can we conclude?
For sets \(A\) and \(B\): \[A \preceq B \land B \preceq A \quad \Rightarrow \quad A \sim B\] If there's an injection \(f: A \to B\) and an injection \(g: B \to A\), then there's a bijection between \(A\) and \(B\).
For sets \(A\) and \(B\): \[A \preceq B \land B \preceq A \quad \Rightarrow \quad A \sim B\] If there's an injection \(f: A \to B\) and an injection \(g: B \to A\), then there's a bijection between \(A\) and \(B\).
Bernstein-Schröder Theorem
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If two sets each dominate the other, what can we conclude? | |
| Back | For sets \(A\) and \(B\): \[A \preceq B \land B \preceq A \quad \Rightarrow \quad A \sim B\] If there's an injection \(f: A \to B\) and an injection \(g: B \to A\), then there's a bijection between \(A\) and \(B\).<div><br></div><div>Bernstein-Schröder Theorem</div> |
Note 120: ETH::DiskMat
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Front
What are the two types of countable sets?
Back
What are the two types of countable sets?
\(A\) is countable if and only if \(A \sim \mathbb{N}\) or \(A \sim \mathbf{n}\) for some \(n \in \mathbb{N}\) (i.e., \(A\) is finite or equinumerous with \(\mathbb{N}\)).
Conclusion: No cardinality level exists between finite and countably infinite.
\(A\) is countable if and only if \(A \sim \mathbb{N}\) or \(A \sim \mathbf{n}\) for some \(n \in \mathbb{N}\) (i.e., \(A\) is finite or equinumerous with \(\mathbb{N}\)).
Conclusion: No cardinality level exists between finite and countably infinite.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the two types of countable sets? | |
| Back | \(A\) is countable <strong>if and only if</strong> \(A \sim \mathbb{N}\) or \(A \sim \mathbf{n}\) for some \(n \in \mathbb{N}\) (i.e., \(A\) is finite or equinumerous with \(\mathbb{N}\)). <br> <strong>Conclusion</strong>: No cardinality level exists between finite and countably infinite. |
Note 121: ETH::DiskMat
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Front
Is the set of all finite binary sequences countable?
Back
Is the set of all finite binary sequences countable?
Yes, the set \(\{0, 1\}^* = \{\epsilon, 0, 1, 00, 01, 10, 11, 000, \ldots\}\) of finite binary sequences is countable.
Yes, the set \(\{0, 1\}^* = \{\epsilon, 0, 1, 00, 01, 10, 11, 000, \ldots\}\) of finite binary sequences is countable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the set of all finite binary sequences countable? | |
| Back | Yes, the set \(\{0, 1\}^* = \{\epsilon, 0, 1, 00, 01, 10, 11, 000, \ldots\}\) of finite binary sequences is <strong>countable</strong>. |
Note 122: ETH::DiskMat
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Front
Is \(\mathbb{N} \times \mathbb{N}\) countable?
Back
Is \(\mathbb{N} \times \mathbb{N}\) countable?
Yes, the set \(\mathbb{N} \times \mathbb{N}\) (= \(\mathbb{N}^2\)) of ordered pairs of natural numbers is countable.
Yes, the set \(\mathbb{N} \times \mathbb{N}\) (= \(\mathbb{N}^2\)) of ordered pairs of natural numbers is countable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is \(\mathbb{N} \times \mathbb{N}\) countable? | |
| Back | Yes, the set \(\mathbb{N} \times \mathbb{N}\) (= \(\mathbb{N}^2\)) of ordered pairs of natural numbers is <strong>countable</strong>. |
Note 123: ETH::DiskMat
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Front
If two sets are countable, what about their Cartesian product?
Back
If two sets are countable, what about their Cartesian product?
The Cartesian product \(A \times B\) of two countable sets is also countable: \[A \preceq \mathbb{N} \land B \preceq \mathbb{N} \Rightarrow A \times B \preceq \mathbb{N}\]
The Cartesian product \(A \times B\) of two countable sets is also countable: \[A \preceq \mathbb{N} \land B \preceq \mathbb{N} \Rightarrow A \times B \preceq \mathbb{N}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If two sets are countable, what about their Cartesian product? | |
| Back | The Cartesian product \(A \times B\) of two countable sets is also countable: \[A \preceq \mathbb{N} \land B \preceq \mathbb{N} \Rightarrow A \times B \preceq \mathbb{N}\] |
Note 124: ETH::DiskMat
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Front
Are the rational numbers \(\mathbb{Q}\) countable?
Back
Are the rational numbers \(\mathbb{Q}\) countable?
Yes, the rational numbers \(\mathbb{Q}\) are countable. They correspond to (a, b) tuples which can be mapped bijectively to the natural numbers.
Yes, the rational numbers \(\mathbb{Q}\) are countable. They correspond to (a, b) tuples which can be mapped bijectively to the natural numbers.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Are the rational numbers \(\mathbb{Q}\) countable? | |
| Back | Yes, the rational numbers \(\mathbb{Q}\) are <strong>countable</strong>. They correspond to (a, b) tuples which can be mapped bijectively to the natural numbers. |
Note 125: ETH::DiskMat
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Front
What operations preserve countability?
Back
What operations preserve countability?
Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then:
Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then:
- (i) \(A^n\) (\(n\)-tuples) is countable
- (ii) \(\bigcup_{i\in \mathbb{N}} A_i\) (countable union) is countable
- (iii) \(A^*\) (finite sequences) is countable
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What operations preserve countability? | |
| Back | Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then: <div> - (i) \(A^n\) (\(n\)-tuples) is countable </div><div> - (ii) \(\bigcup_{i\in \mathbb{N}} A_i\) (countable union) is countable </div><div> - (iii) \(A^*\) (finite sequences) is countable</div> |
Note 126: ETH::DiskMat
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Front
What is the set \(\{0, 1\}^{\infty}\)?
Back
What is the set \(\{0, 1\}^{\infty}\)?
The set of semi-infinite binary sequences, or equivalently, the set of functions \(\mathbb{N} \to \{0,1\}\).
The set of semi-infinite binary sequences, or equivalently, the set of functions \(\mathbb{N} \to \{0,1\}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the set \(\{0, 1\}^{\infty}\)? | |
| Back | The set of semi-infinite binary sequences, or equivalently, the set of functions \(\mathbb{N} \to \{0,1\}\). |
Note 127: ETH::DiskMat
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Front
Is the set of infinite binary sequences countable?
Back
Is the set of infinite binary sequences countable?
No, the set \(\{0,1\}^{\infty}\) is uncountable.
(Proven by Cantor's diagonalization argument)
No, the set \(\{0,1\}^{\infty}\) is uncountable.
(Proven by Cantor's diagonalization argument)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the set of infinite binary sequences countable? | |
| Back | No, the set \(\{0,1\}^{\infty}\) is <strong>uncountable</strong>. <br> (Proven by Cantor's diagonalization argument) |
Note 128: ETH::DiskMat
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Front
What is a computable function \(f: \mathbb{N} \to \{0, 1\}\)?
Back
What is a computable function \(f: \mathbb{N} \to \{0, 1\}\)?
A function for which there exists a program that, for every \(n \in \mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).
A function for which there exists a program that, for every \(n \in \mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a computable function \(f: \mathbb{N} \to \{0, 1\}\)? | |
| Back | A function for which there exists a program that, for every \(n \in \mathbb{N}\), when given \(n\) as input, outputs \(f(n)\). |
Note 129: ETH::DiskMat
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Front
Do uncomputable functions exist?
Back
Do uncomputable functions exist?
Yes, there exist uncomputable functions \(\mathbb{N} \to \{0, 1\}\).
Proof idea: Programs are finite bitstrings (\(\{0,1\}^*\) is countable), but functions \(\mathbb{N} \to \{0,1\}\) are uncountable.
Yes, there exist uncomputable functions \(\mathbb{N} \to \{0, 1\}\).
Proof idea: Programs are finite bitstrings (\(\{0,1\}^*\) is countable), but functions \(\mathbb{N} \to \{0,1\}\) are uncountable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Do uncomputable functions exist? | |
| Back | Yes, there exist <strong>uncomputable</strong> functions \(\mathbb{N} \to \{0, 1\}\). <br> <strong>Proof idea</strong>: Programs are finite bitstrings (\(\{0,1\}^*\) is countable), but functions \(\mathbb{N} \to \{0,1\}\) are uncountable. |
Note 130: ETH::DiskMat
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Front
Show that \(\mathbb{Z}\) is countable by exhibiting a bijection with \(\mathbb{N}\).
Back
Show that \(\mathbb{Z}\) is countable by exhibiting a bijection with \(\mathbb{N}\).
The function \(f(n) = (-1)^n \lceil n/2 \rceil\) is a bijection \(\mathbb{N} \to \mathbb{Z}\).
Maps: \(0 \mapsto 0, 1 \mapsto 1, 2 \mapsto -1, 3 \mapsto 2, 4 \mapsto -2, \ldots\)
The function \(f(n) = (-1)^n \lceil n/2 \rceil\) is a bijection \(\mathbb{N} \to \mathbb{Z}\).
Maps: \(0 \mapsto 0, 1 \mapsto 1, 2 \mapsto -1, 3 \mapsto 2, 4 \mapsto -2, \ldots\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Show that \(\mathbb{Z}\) is countable by exhibiting a bijection with \(\mathbb{N}\). | |
| Back | The function \(f(n) = (-1)^n \lceil n/2 \rceil\) is a bijection \(\mathbb{N} \to \mathbb{Z}\). <br> Maps: \(0 \mapsto 0, 1 \mapsto 1, 2 \mapsto -1, 3 \mapsto 2, 4 \mapsto -2, \ldots\) |
Note 131: ETH::DiskMat
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Front
What are the equivalence classes of \(\equiv_3\) on \(\mathbb{Z}\)?
Back
What are the equivalence classes of \(\equiv_3\) on \(\mathbb{Z}\)?
- \([0] = \{\ldots, -6, -3, 0, 3, 6, \ldots\}\)
- \([1] = \{\ldots, -5, -2, 1, 4, 7, \ldots\}\)
- \([2] = \{\ldots, -4, -1, 2, 5, 8, \ldots\}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the equivalence classes of \(\equiv_3\) on \(\mathbb{Z}\)? | |
| Back | <ul> <li>\([0] = \{\ldots, -6, -3, 0, 3, 6, \ldots\}\)</li> <li>\([1] = \{\ldots, -5, -2, 1, 4, 7, \ldots\}\)</li> <li>\([2] = \{\ldots, -4, -1, 2, 5, 8, \ldots\}\)</li> </ul> |
Note 132: ETH::DiskMat
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Front
What is \(\equiv_5 \cap \equiv_3\) (as equivalence relations on \(\mathbb{Z}\))?
Back
What is \(\equiv_5 \cap \equiv_3\) (as equivalence relations on \(\mathbb{Z}\))?
\(\equiv_{15}\) (equivalence modulo 15)
\(\equiv_{15}\) (equivalence modulo 15)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is \(\equiv_5 \cap \equiv_3\) (as equivalence relations on \(\mathbb{Z}\))? | |
| Back | \(\equiv_{15}\) (equivalence modulo 15) |
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Front
Why is \((\mathcal{P}(\{1,2,3\}); \subseteq)\) NOT totally ordered?
Back
Why is \((\mathcal{P}(\{1,2,3\}); \subseteq)\) NOT totally ordered?
Because \(\{2, 3\} \not\subseteq \{3, 4\}\) and \(\{3, 4\} \not\subseteq \{2, 3\}\) (they are incomparable).
Because \(\{2, 3\} \not\subseteq \{3, 4\}\) and \(\{3, 4\} \not\subseteq \{2, 3\}\) (they are incomparable).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why is \((\mathcal{P}(\{1,2,3\}); \subseteq)\) NOT totally ordered? | |
| Back | Because \(\{2, 3\} \not\subseteq \{3, 4\}\) and \(\{3, 4\} \not\subseteq \{2, 3\}\) (they are incomparable). |
Note 134: ETH::DiskMat
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Front
Why is \((\mathbb{N}; |)\) NOT totally ordered?
Back
Why is \((\mathbb{N}; |)\) NOT totally ordered?
Because \(2 \nmid 3\) and \(3 \nmid 2\) (they are incomparable).
Because \(2 \nmid 3\) and \(3 \nmid 2\) (they are incomparable).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why is \((\mathbb{N}; |)\) NOT totally ordered? | |
| Back | Because \(2 \nmid 3\) and \(3 \nmid 2\) (they are incomparable). |
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Front
In the poset \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), identify the minimal and maximal elements.
Back
In the poset \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), identify the minimal and maximal elements.
- Minimal elements: \(2, 3, 5, 7\) (primes)
- Maximal elements: \(5, 6, 7, 8, 9\)
- No least or greatest element (not all elements comparable)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | In the poset \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), identify the minimal and maximal elements. | |
| Back | <ul> <li><strong>Minimal elements</strong>: \(2, 3, 5, 7\) (primes)</li> <li><strong>Maximal elements</strong>: \(5, 6, 7, 8, 9\)</li> <li><strong>No least or greatest element</strong> (not all elements comparable)</li> </ul> |
Note 136: ETH::DiskMat
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Front
In \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), what is the glb of \(\{4, 6, 8\}\)?
Back
In \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), what is the glb of \(\{4, 6, 8\}\)?
\(2\) is a lower bound (and the greatest lower bound/glb) of \(\{4, 6, 8\}\).
\(2\) is a lower bound (and the greatest lower bound/glb) of \(\{4, 6, 8\}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | In \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), what is the glb of \(\{4, 6, 8\}\)? | |
| Back | \(2\) is a lower bound (and the greatest lower bound/glb) of \(\{4, 6, 8\}\). |
Note 137: ETH::DiskMat
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Front
If \(\phi\) is the parenthood relation on the set \(H\) of all humans, what is \(\phi^2\)?
Back
If \(\phi\) is the parenthood relation on the set \(H\) of all humans, what is \(\phi^2\)?
The grand-parenthood relation.
The grand-parenthood relation.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If \(\phi\) is the parenthood relation on the set \(H\) of all humans, what is \(\phi^2\)? | |
| Back | The grand-parenthood relation. |
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Front
What fundamental property distinguishes finite from infinite sets regarding proper subsets?
Back
What fundamental property distinguishes finite from infinite sets regarding proper subsets?
A finite set never has the same cardinality as one of its proper subsets. An infinite set can (e.g., \(\mathbb{N} \sim \mathbb{O}\) where \(\mathbb{O}\) is the set of odd numbers).
A finite set never has the same cardinality as one of its proper subsets. An infinite set can (e.g., \(\mathbb{N} \sim \mathbb{O}\) where \(\mathbb{O}\) is the set of odd numbers).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What fundamental property distinguishes finite from infinite sets regarding proper subsets? | |
| Back | A <strong>finite</strong> set never has the same cardinality as one of its proper subsets. An <strong>infinite</strong> set can (e.g., \(\mathbb{N} \sim \mathbb{O}\) where \(\mathbb{O}\) is the set of odd numbers). |
Note 139: ETH::DiskMat
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Front
For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are:
1. The range of \(f\)
2. The preimage of \([4, 9]\)
1. The range of \(f\)
2. The preimage of \([4, 9]\)
Back
For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are:
1. The range of \(f\)
2. The preimage of \([4, 9]\)
1. Range: \(\mathbb{R}^{\geq 0}\) (non-negative reals)
2. Preimage of \([4, 9]\): \([-3, -2] \cup [2, 3]\)
1. The range of \(f\)
2. The preimage of \([4, 9]\)
1. Range: \(\mathbb{R}^{\geq 0}\) (non-negative reals)
2. Preimage of \([4, 9]\): \([-3, -2] \cup [2, 3]\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are: <br>1. The range of \(f\) <br>2. The preimage of \([4, 9]\) | |
| Back | 1. <strong>Range</strong>: \(\mathbb{R}^{\geq 0}\) (non-negative reals) <br>2. <strong>Preimage of \([4, 9]\)</strong>: \([-3, -2] \cup [2, 3]\) |
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Front
How can we construct the first few natural numbers using only the empty set?
Back
How can we construct the first few natural numbers using only the empty set?
- \(\mathbf{0} = \emptyset\)
- \(\mathbf{1} = \{\emptyset\}\)
- \(\mathbf{2} = \{\emptyset, \{\emptyset\}\}\)
- Successor: \(s(\mathbf{n}) = \mathbf{n} \cup \{\mathbf{n}\}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How can we construct the first few natural numbers using only the empty set? | |
| Back | <ul> <li>\(\mathbf{0} = \emptyset\)</li> <li>\(\mathbf{1} = \{\emptyset\}\)</li> <li>\(\mathbf{2} = \{\emptyset, \{\emptyset\}\}\)</li> <li>Successor: \(s(\mathbf{n}) = \mathbf{n} \cup \{\mathbf{n}\}\)</li> </ul> |
Note 141: ETH::DiskMat
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Front
What happens when a formula in predicate logic has a free variable (no quantifier)?
Back
What happens when a formula in predicate logic has a free variable (no quantifier)?
The variable must be replaced by a specific constant from the universe for any interpretation. Without a quantifier, \(x\) is not bound and requires a concrete value.
The variable must be replaced by a specific constant from the universe for any interpretation. Without a quantifier, \(x\) is not bound and requires a concrete value.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What happens when a formula in predicate logic has a free variable (no quantifier)? | |
| Back | The variable must be replaced by a <strong>specific constant</strong> from the universe for any interpretation. Without a quantifier, \(x\) is not bound and requires a concrete value. |
Note 142: ETH::DiskMat
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Front
How does satisfiability differ between propositional logic and predicate logic?
Back
How does satisfiability differ between propositional logic and predicate logic?
- Propositional Logic: About truth assignments to symbols
- Predicate Logic: About interpretations (universe, predicates, and constants)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does satisfiability differ between propositional logic and predicate logic? | |
| Back | <ul> <li><strong>Propositional Logic</strong>: About truth assignments to symbols</li> <li><strong>Predicate Logic</strong>: About interpretations (universe, predicates, and constants)</li> </ul> |
Note 143: ETH::DiskMat
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Front
What is the principle behind composing proofs (Definition 2.12)?
Back
What is the principle behind composing proofs (Definition 2.12)?
If \(S \Rightarrow T\) and \(T \Rightarrow U\) are both true, then \(S \Rightarrow U\) is also true (transitivity of implication).
If \(S \Rightarrow T\) and \(T \Rightarrow U\) are both true, then \(S \Rightarrow U\) is also true (transitivity of implication).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the principle behind composing proofs (Definition 2.12)? | |
| Back | If \(S \Rightarrow T\) and \(T \Rightarrow U\) are both true, then \(S \Rightarrow U\) is also true (transitivity of implication). |
Note 144: ETH::DiskMat
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Front
What are the three ways to represent a relation on finite sets?
Back
What are the three ways to represent a relation on finite sets?
1. Set notation (subset of \(A \times B\)) 2. Boolean matrix (1 if \((a,b) \in \rho\), 0 otherwise) 3. Directed graph (nodes are elements, edges are relations)
1. Set notation (subset of \(A \times B\)) 2. Boolean matrix (1 if \((a,b) \in \rho\), 0 otherwise) 3. Directed graph (nodes are elements, edges are relations)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the three ways to represent a relation on finite sets? | |
| Back | 1. <strong>Set notation</strong> (subset of \(A \times B\)) 2. <strong>Boolean matrix</strong> (1 if \((a,b) \in \rho\), 0 otherwise) 3. <strong>Directed graph</strong> (nodes are elements, edges are relations) |
Note 145: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Is the divisibility relation \(|\) antisymmetric on \(\mathbb{N}\)? On \(\mathbb{Z}\)?
Back
Is the divisibility relation \(|\) antisymmetric on \(\mathbb{N}\)? On \(\mathbb{Z}\)?
- On \(\mathbb{N}\): YES (if \(a | b\) and \(b | a\), then \(a = b\))
- On \(\mathbb{Z}\): NO (e.g., \(2 | -2\) and \(-2 | 2\) but \(2 \neq -2\))
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the divisibility relation \(|\) antisymmetric on \(\mathbb{N}\)? On \(\mathbb{Z}\)? | |
| Back | <ul> <li><strong>On \(\mathbb{N}\)</strong>: YES (if \(a | b\) and \(b | a\), then \(a = b\))</li> <li><strong>On \(\mathbb{Z}\)</strong>: NO (e.g., \(2 | -2\) and \(-2 | 2\) but \(2 \neq -2\))</li> </ul> |
Note 146: ETH::DiskMat
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Front
Is \(\mathbb{N}\) well-ordered by \(\leq\)? What about \(\mathbb{Z}\)?
Back
Is \(\mathbb{N}\) well-ordered by \(\leq\)? What about \(\mathbb{Z}\)?
- \(\mathbb{N}\): YES (every non-empty subset has a least element)
- \(\mathbb{Z}\): NO (e.g., \(\mathbb{Z}\) itself has no least element)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is \(\mathbb{N}\) well-ordered by \(\leq\)? What about \(\mathbb{Z}\)? | |
| Back | <ul> <li><strong>\(\mathbb{N}\)</strong>: YES (every non-empty subset has a least element)</li> <li><strong>\(\mathbb{Z}\)</strong>: NO (e.g., \(\mathbb{Z}\) itself has no least element)</li> </ul> |
Note 147: ETH::DiskMat
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Front
If \((A; \preceq)\) is well-ordered, is every subset of \(A\) also well-ordered?
Back
If \((A; \preceq)\) is well-ordered, is every subset of \(A\) also well-ordered?
YES, any subset of a well-ordered set is well-ordered (by the same relation).
YES, any subset of a well-ordered set is well-ordered (by the same relation).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If \((A; \preceq)\) is well-ordered, is every subset of \(A\) also well-ordered? | |
| Back | <strong>YES</strong>, any subset of a well-ordered set is well-ordered (by the same relation). |
Note 148: ETH::DiskMat
Deck: ETH::DiskMat
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Front
List all three pairs of related but distinct logical symbols and their types.
Back
List all three pairs of related but distinct logical symbols and their types.
1. \(\equiv\) (formula→statement), \(\leftrightarrow\) (formula→formula), \(\Leftrightarrow\) (statement→statement) 2. \(\models\) (formula→statement), \(\rightarrow\) (formula→formula), \(\Rightarrow\) (statement→statement)
1. \(\equiv\) (formula→statement), \(\leftrightarrow\) (formula→formula), \(\Leftrightarrow\) (statement→statement) 2. \(\models\) (formula→statement), \(\rightarrow\) (formula→formula), \(\Rightarrow\) (statement→statement)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | List all three pairs of related but distinct logical symbols and their types. | |
| Back | 1. \(\equiv\) (formula→statement), \(\leftrightarrow\) (formula→formula), \(\Leftrightarrow\) (statement→statement) 2. \(\models\) (formula→statement), \(\rightarrow\) (formula→formula), \(\Rightarrow\) (statement→statement) |
Note 149: ETH::DiskMat
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Front
What are the 7 main proof patterns covered in the course?
Back
What are the 7 main proof patterns covered in the course?
1. Direct proof 2. Indirect proof (contraposition) 3. Modus ponens 4. Case distinction 5. Contradiction 6. Existence proof 7. Induction
1. Direct proof 2. Indirect proof (contraposition) 3. Modus ponens 4. Case distinction 5. Contradiction 6. Existence proof 7. Induction
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the 7 main proof patterns covered in the course? | |
| Back | 1. Direct proof 2. Indirect proof (contraposition) 3. Modus ponens 4. Case distinction 5. Contradiction 6. Existence proof 7. Induction |
Note 150: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Can a relation be both symmetric and antisymmetric?
Back
Can a relation be both symmetric and antisymmetric?
YES - the identity relation is both symmetric and antisymmetric. The properties are independent, not mutually exclusive.
YES - the identity relation is both symmetric and antisymmetric. The properties are independent, not mutually exclusive.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Can a relation be both symmetric and antisymmetric? | |
| Back | <strong>YES</strong> - the identity relation is both symmetric and antisymmetric. The properties are <strong>independent</strong>, not mutually exclusive. |
Note 151: ETH::DiskMat
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Front
Which of the following are countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), \(\{0,1\}^*\), \(\{0,1\}^{\infty}\)?
Back
Which of the following are countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), \(\{0,1\}^*\), \(\{0,1\}^{\infty}\)?
Countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\{0,1\}^*\) Uncountable: \(\mathbb{R}\), \(\{0,1\}^{\infty}\)
Countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\{0,1\}^*\) Uncountable: \(\mathbb{R}\), \(\{0,1\}^{\infty}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Which of the following are countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), \(\{0,1\}^*\), \(\{0,1\}^{\infty}\)? | |
| Back | <strong>Countable</strong>: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\{0,1\}^*\) <strong>Uncountable</strong>: \(\mathbb{R}\), \(\{0,1\}^{\infty}\) |
Note 152: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Is the interval \([0, 1]\) countable or uncountable? What does this imply for \(\mathbb{R}\)?
Back
Is the interval \([0, 1]\) countable or uncountable? What does this imply for \(\mathbb{R}\)?
The interval is uncountable by Cantor's diagonal argument, thus \(\mathbb{R}\) is too.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Is the interval \([0, 1]\) countable or uncountable? What does this imply for \(\mathbb{R}\)? | |
| Back | The interval is uncountable by Cantor's diagonal argument, thus \(\mathbb{R}\) is too. |
Note 153: ETH::DiskMat
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Front
How is the countability of the power set of any set related to the countability of that set?
Back
How is the countability of the power set of any set related to the countability of that set?
\[\mathbb{N} \prec \mathcal{P}(\mathbb{N}) \prec \mathcal{P}(\mathcal{P}(\mathbb{N}))\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How is the countability of the power set of any set related to the countability of that set? | |
| Back | \[\mathbb{N} \prec \mathcal{P}(\mathbb{N}) \prec \mathcal{P}(\mathcal{P}(\mathbb{N}))\] |
Note 154: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What properties does the relation \(=\) satisfy?
Back
What properties does the relation \(=\) satisfy?
- Equivalence relation
- Partial order relation
As it's reflexive, transitive, symmetric and antisymmetric.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What properties does the relation \(=\) satisfy? | |
| Back | <ul><li>Equivalence relation</li><li>Partial order relation</li></ul><div>As it's reflexive, transitive, symmetric and antisymmetric.</div> |
Note 155: ETH::DiskMat
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Front
Name the binding strengths of PL tokens in order.
Back
Name the binding strengths of PL tokens in order.
- unary operators (NOT)
- quantifiers (for all and exists)
- operators (AND, OR)
- Implication
- quantifiers (for all and exists)
- operators (AND, OR)
- Implication
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Name the binding strengths of PL tokens in order. | |
| Back | - unary operators (NOT)<br> - quantifiers (for all and exists)<br> - operators (AND, OR)<br> - Implication | |
| Tags | #ch2 |
Note 156: ETH::DiskMat
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Front
Give the formal definition of "\(a\) divides \(b\)" (denoted \(a | b\)).
Back
Give the formal definition of "\(a\) divides \(b\)" (denoted \(a | b\)).
\[a | b \overset{\text{def}}{\Longleftrightarrow} \exists c \in \mathbb{Z} \ b = ac\] \(a\) is a divisor of \(b\), \(b\) is a multiple of \(a\), and \(c\) is the quotient.
\[a | b \overset{\text{def}}{\Longleftrightarrow} \exists c \in \mathbb{Z} \ b = ac\] \(a\) is a divisor of \(b\), \(b\) is a multiple of \(a\), and \(c\) is the quotient.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of "\(a\) divides \(b\)" (denoted \(a | b\)). | |
| Back | \[a | b \overset{\text{def}}{\Longleftrightarrow} \exists c \in \mathbb{Z} \ b = ac\] \(a\) is a divisor of \(b\), \(b\) is a multiple of \(a\), and \(c\) is the quotient. |
Note 157: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What are the trivial divisors that apply to all integers?
Back
What are the trivial divisors that apply to all integers?
- Every non-zero integer is a divisor of \(0\)
- \(1\) and \(-1\) are divisors of every integer
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the trivial divisors that apply to all integers? | |
| Back | <ul> <li>Every non-zero integer is a divisor of \(0\)</li> <li>\(1\) and \(-1\) are divisors of every integer</li> </ul> |
Note 158: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Give the formal definition of a prime number \(p\).
Back
Give the formal definition of a prime number \(p\).
\[p \ \text{prime} \overset{\text{def}}{\Longleftrightarrow} p > 1 \land \forall d \ ((d > 1) \land (d | p) \rightarrow d = p)\] A prime is greater than 1 and its only positive divisors are 1 and itself.
\[p \ \text{prime} \overset{\text{def}}{\Longleftrightarrow} p > 1 \land \forall d \ ((d > 1) \land (d | p) \rightarrow d = p)\] A prime is greater than 1 and its only positive divisors are 1 and itself.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of a prime number \(p\). | |
| Back | \[p \ \text{prime} \overset{\text{def}}{\Longleftrightarrow} p > 1 \land \forall d \ ((d > 1) \land (d | p) \rightarrow d = p)\] A prime is greater than 1 and its only positive divisors are 1 and itself. |
Note 159: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is a composite number?
Back
What is a composite number?
An integer greater than 1 that is not prime (i.e., it has divisors other than 1 and itself).
An integer greater than 1 that is not prime (i.e., it has divisors other than 1 and itself).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a composite number? | |
| Back | An integer greater than 1 that is <strong>not prime</strong> (i.e., it has divisors other than 1 and itself). |
Note 160: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Give the formal definition of a greatest common divisor \(d\) of integers \(a\) and \(b\) (not both 0).
Back
Give the formal definition of a greatest common divisor \(d\) of integers \(a\) and \(b\) (not both 0).
\[d | a \land d | b \land \forall c \ ((c | a \land c | b) \rightarrow c | d)\] \(d\) divides both \(a\) and \(b\), AND every common divisor of \(a\) and \(b\) divides \(d\).
\[d | a \land d | b \land \forall c \ ((c | a \land c | b) \rightarrow c | d)\] \(d\) divides both \(a\) and \(b\), AND every common divisor of \(a\) and \(b\) divides \(d\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of a greatest common divisor \(d\) of integers \(a\) and \(b\) (not both 0). | |
| Back | \[d | a \land d | b \land \forall c \ ((c | a \land c | b) \rightarrow c | d)\] \(d\) divides both \(a\) and \(b\), AND every common divisor of \(a\) and \(b\) divides \(d\). |
Note 161: ETH::DiskMat
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Front
What is \(\text{gcd}(a, b)\)?
Back
What is \(\text{gcd}(a, b)\)?
The unique positive greatest common divisor of \(a\) and \(b\).
The unique positive greatest common divisor of \(a\) and \(b\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is \(\text{gcd}(a, b)\)? | |
| Back | The <strong>unique positive</strong> greatest common divisor of \(a\) and \(b\). |
Note 162: ETH::DiskMat
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Front
When are two integers \(a\) and \(b\) called relatively prime (or coprime)?
Back
When are two integers \(a\) and \(b\) called relatively prime (or coprime)?
When \(\text{gcd}(a, b) = 1\).
When \(\text{gcd}(a, b) = 1\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When are two integers \(a\) and \(b\) called relatively prime (or coprime)? | |
| Back | When \(\text{gcd}(a, b) = 1\). |
Note 163: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the ideal \((a, b)\) generated by integers \(a\) and \(b\) and just \(a\)?
Back
What is the ideal \((a, b)\) generated by integers \(a\) and \(b\) and just \(a\)?
\[(a, b) \overset{\text{def}}{=} \{ ua + vb \ | \ u, v \in \mathbb{Z}\}\] and \[(a) \overset{\text{def}}{=} \{ ua \ | \ u \in \mathbb{Z}\}\] The set of all integer linear combinations of \(a\) and \(b\).
\[(a, b) \overset{\text{def}}{=} \{ ua + vb \ | \ u, v \in \mathbb{Z}\}\] and \[(a) \overset{\text{def}}{=} \{ ua \ | \ u \in \mathbb{Z}\}\] The set of all integer linear combinations of \(a\) and \(b\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the ideal \((a, b)\) generated by integers \(a\) and \(b\) and just \(a\)? | |
| Back | \[(a, b) \overset{\text{def}}{=} \{ ua + vb \ | \ u, v \in \mathbb{Z}\}\] and \[(a) \overset{\text{def}}{=} \{ ua \ | \ u \in \mathbb{Z}\}\] The set of all integer linear combinations of \(a\) and \(b\). |
Note 164: ETH::DiskMat
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Front
Give the formal definition of the least common multiple \(\text{lcm}(a, b)\).
Back
Give the formal definition of the least common multiple \(\text{lcm}(a, b)\).
\[a | l \land b | l \land \forall m \ ((a | m \land b | m) \rightarrow l | m)\] \(l\) is a common multiple of \(a\) and \(b\) which divides every common multiple of \(a\) and \(b\).
\[a | l \land b | l \land \forall m \ ((a | m \land b | m) \rightarrow l | m)\] \(l\) is a common multiple of \(a\) and \(b\) which divides every common multiple of \(a\) and \(b\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of the least common multiple \(\text{lcm}(a, b)\). | |
| Back | \[a | l \land b | l \land \forall m \ ((a | m \land b | m) \rightarrow l | m)\] \(l\) is a common multiple of \(a\) and \(b\) which divides every common multiple of \(a\) and \(b\). |
Note 165: ETH::DiskMat
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Front
Give the formal definition of "\(a\) is congruent to \(b\) modulo \(m\)".
Back
Give the formal definition of "\(a\) is congruent to \(b\) modulo \(m\)".
\[a \equiv_m b \overset{\text{def}}{\Longleftrightarrow} m | (a - b)\] Also written as \(a \equiv b \pmod{m}\).
\[a \equiv_m b \overset{\text{def}}{\Longleftrightarrow} m | (a - b)\] Also written as \(a \equiv b \pmod{m}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Give the formal definition of "\(a\) is congruent to \(b\) modulo \(m\)". | |
| Back | \[a \equiv_m b \overset{\text{def}}{\Longleftrightarrow} m | (a - b)\] Also written as \(a \equiv b \pmod{m}\). |
Note 166: ETH::DiskMat
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Front
What is \(R_m(x)\)?
Back
What is \(R_m(x)\)?
The smallest non-negative integer \(n \in \mathbb{N}\) for which \(x \equiv_m n\). Equivalently, the remainder when \(x\) is divided by \(m\) (so \(0 \leq R_m(x) < m\)).
The smallest non-negative integer \(n \in \mathbb{N}\) for which \(x \equiv_m n\). Equivalently, the remainder when \(x\) is divided by \(m\) (so \(0 \leq R_m(x) < m\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is \(R_m(x)\)? | |
| Back | The smallest <strong>non-negative</strong> integer \(n \in \mathbb{N}\) for which \(x \equiv_m n\). Equivalently, the remainder when \(x\) is divided by \(m\) (so \(0 \leq R_m(x) < m\)). |
Note 167: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is the multiplicative inverse of \(a\) modulo \(m\)?
Back
What is the multiplicative inverse of \(a\) modulo \(m\)?
The unique solution \(x \in \mathbb{Z}_m\) to the congruence equation \(ax \equiv_m 1\), where \(\text{gcd}(a, m) = 1\). Denoted \(a^{-1} \pmod{m}\) or \(1/a \pmod{m}\).
The unique solution \(x \in \mathbb{Z}_m\) to the congruence equation \(ax \equiv_m 1\), where \(\text{gcd}(a, m) = 1\). Denoted \(a^{-1} \pmod{m}\) or \(1/a \pmod{m}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the multiplicative inverse of \(a\) modulo \(m\)? | |
| Back | The unique solution \(x \in \mathbb{Z}_m\) to the congruence equation \(ax \equiv_m 1\), where \(\text{gcd}(a, m) = 1\). Denoted \(a^{-1} \pmod{m}\) or \(1/a \pmod{m}\). |
Note 168: ETH::DiskMat
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Front
What is the prime counting function \(\pi(x)\)?
Back
What is the prime counting function \(\pi(x)\)?
\[\pi : \mathbb{R} \rightarrow \mathbb{N}\] For any real \(x\), \(\pi(x)\) is the number of primes \(\leq x\).
\[\pi : \mathbb{R} \rightarrow \mathbb{N}\] For any real \(x\), \(\pi(x)\) is the number of primes \(\leq x\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the prime counting function \(\pi(x)\)? | |
| Back | \[\pi : \mathbb{R} \rightarrow \mathbb{N}\] For any real \(x\), \(\pi(x)\) is the number of primes \(\leq x\). |
Note 169: ETH::DiskMat
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Front
State the Euclidean Division Theorem.
Back
State the Euclidean Division Theorem.
For all integers \(a\) and \(d \neq 0\), there exist unique integers \(q\) and \(r\) satisfying: \[a = dq + r \quad \text{and} \quad 0 \leq r < |d|\] (\(r\) is the remainder, \(q\) is the quotient)
For all integers \(a\) and \(d \neq 0\), there exist unique integers \(q\) and \(r\) satisfying: \[a = dq + r \quad \text{and} \quad 0 \leq r < |d|\] (\(r\) is the remainder, \(q\) is the quotient)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | State the Euclidean Division Theorem. | |
| Back | For all integers \(a\) and \(d \neq 0\), there exist <strong>unique</strong> integers \(q\) and \(r\) satisfying: \[a = dq + r \quad \text{and} \quad 0 \leq r < |d|\] (\(r\) is the remainder, \(q\) is the quotient) |
Note 170: ETH::DiskMat
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Front
How does the GCD change when we subtract a multiple? (Lemma 4.2)
Back
How does the GCD change when we subtract a multiple? (Lemma 4.2)
For any integers \(m, n\) and \(q\): \[\text{gcd}(m, n - qm) = \text{gcd}(m, n)\] This is the key property for Euclid's algorithm: \[\text{gcd}(m, R_m(n)) = \text{gcd}(m, n)\]
For any integers \(m, n\) and \(q\): \[\text{gcd}(m, n - qm) = \text{gcd}(m, n)\] This is the key property for Euclid's algorithm: \[\text{gcd}(m, R_m(n)) = \text{gcd}(m, n)\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does the GCD change when we subtract a multiple? (Lemma 4.2) | |
| Back | For any integers \(m, n\) and \(q\): \[\text{gcd}(m, n - qm) = \text{gcd}(m, n)\] This is the key property for Euclid's algorithm: \[\text{gcd}(m, R_m(n)) = \text{gcd}(m, n)\] |
Note 171: ETH::DiskMat
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Front
What important property do ideals in \(\mathbb{Z}\) have? (Lemma 4.3)
Back
What important property do ideals in \(\mathbb{Z}\) have? (Lemma 4.3)
For \(a, b \in \mathbb{Z}\), there exists \(d \in \mathbb{Z}\) such that \((a, b) = (d)\).
Every ideal can be generated by a single integer.
For \(a, b \in \mathbb{Z}\), there exists \(d \in \mathbb{Z}\) such that \((a, b) = (d)\).
Every ideal can be generated by a single integer.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What important property do ideals in \(\mathbb{Z}\) have? (Lemma 4.3) | |
| Back | For \(a, b \in \mathbb{Z}\), there exists \(d \in \mathbb{Z}\) such that \((a, b) = (d)\). <br> <strong>Every ideal</strong> can be generated by a <strong>single integer</strong>. |
Note 172: ETH::DiskMat
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Front
How is the GCD related to ideals? (Lemma 4.4)
Back
How is the GCD related to ideals? (Lemma 4.4)
Let \(a, b \in \mathbb{Z}\) (not both 0). If \((a, b) = (d)\), then \(d\) is a greatest common divisor of \(a\) and \(b\).
Let \(a, b \in \mathbb{Z}\) (not both 0). If \((a, b) = (d)\), then \(d\) is a greatest common divisor of \(a\) and \(b\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How is the GCD related to ideals? (Lemma 4.4) | |
| Back | Let \(a, b \in \mathbb{Z}\) (not both 0). If \((a, b) = (d)\), then \(d\) is a greatest common divisor of \(a\) and \(b\). |
Note 173: ETH::DiskMat
Deck: ETH::DiskMat
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Front
State Bézout's identity (Corollary 4.5).
Back
State Bézout's identity (Corollary 4.5).
For \(a, b \in \mathbb{Z}\) (not both 0), there exist \(u, v \in \mathbb{Z}\) such that: \[\text{gcd}(a, b) = ua + vb\] The GCD can be expressed as an integer linear combination.
For \(a, b \in \mathbb{Z}\) (not both 0), there exist \(u, v \in \mathbb{Z}\) such that: \[\text{gcd}(a, b) = ua + vb\] The GCD can be expressed as an integer linear combination.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | State Bézout's identity (Corollary 4.5). | |
| Back | For \(a, b \in \mathbb{Z}\) (not both 0), there exist \(u, v \in \mathbb{Z}\) such that: \[\text{gcd}(a, b) = ua + vb\] The GCD can be expressed as an integer linear combination. |
Note 174: ETH::DiskMat
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Front
State the Fundamental Theorem of Arithmetic (Theorem 4.6).
Back
State the Fundamental Theorem of Arithmetic (Theorem 4.6).
Every positive integer can be written uniquely (up to the order of factors) as the product of primes: \[x = \prod p_i^{e_i}\]
Every positive integer can be written uniquely (up to the order of factors) as the product of primes: \[x = \prod p_i^{e_i}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | State the Fundamental Theorem of Arithmetic (Theorem 4.6). | |
| Back | Every positive integer can be written <strong>uniquely</strong> (up to the order of factors) as the product of primes: \[x = \prod p_i^{e_i}\] |
Note 175: ETH::DiskMat
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Front
If a prime divides a product, what can we conclude? (Lemma 4.7)
Back
If a prime divides a product, what can we conclude? (Lemma 4.7)
If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]
If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If a prime divides a product, what can we conclude? (Lemma 4.7) | |
| Back | If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\] |
Note 176: ETH::DiskMat
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Front
How many primes exist? (Theorem 4.9)
Back
How many primes exist? (Theorem 4.9)
There are infinitely many primes.
There are infinitely many primes.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How many primes exist? (Theorem 4.9) | |
| Back | There are <strong>infinitely many</strong> primes. |
Note 177: ETH::DiskMat
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Front
What shortcut exists for testing whether \(n\) is prime? (Lemma 4.12)
Back
What shortcut exists for testing whether \(n\) is prime? (Lemma 4.12)
Every composite integer \(n\) has a prime divisor \(\leq \sqrt{n}\).
Therefore, to test if \(n\) is prime, we only need to check divisibility by primes up to \(\sqrt{n}\).
Every composite integer \(n\) has a prime divisor \(\leq \sqrt{n}\).
Therefore, to test if \(n\) is prime, we only need to check divisibility by primes up to \(\sqrt{n}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What shortcut exists for testing whether \(n\) is prime? (Lemma 4.12) | |
| Back | Every composite integer \(n\) has a prime divisor \(\leq \sqrt{n}\). <br> Therefore, to test if \(n\) is prime, we only need to check divisibility by primes up to \(\sqrt{n}\). |
Note 178: ETH::DiskMat
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Front
What kind of relation is \(\equiv_m\)? (Lemma 4.13)
Back
What kind of relation is \(\equiv_m\)? (Lemma 4.13)
For any \(m > 1\), \(\equiv_m\) is an equivalence relation on \(\mathbb{Z}\) (reflexive, symmetric, and transitive).
For any \(m > 1\), \(\equiv_m\) is an equivalence relation on \(\mathbb{Z}\) (reflexive, symmetric, and transitive).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What kind of relation is \(\equiv_m\)? (Lemma 4.13) | |
| Back | For any \(m > 1\), \(\equiv_m\) is an <strong>equivalence relation</strong> on \(\mathbb{Z}\) (reflexive, symmetric, and transitive). |
Note 179: ETH::DiskMat
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If \(a \equiv_m b\) and \(c \equiv_m d\), what operations are preserved? (Lemma 4.14)
Back
If \(a \equiv_m b\) and \(c \equiv_m d\), what operations are preserved? (Lemma 4.14)
\[a + c \equiv_m b + d \quad \text{and} \quad ac \equiv_m bd\] Both addition and multiplication are preserved under congruence.
\[a + c \equiv_m b + d \quad \text{and} \quad ac \equiv_m bd\] Both addition and multiplication are preserved under congruence.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | If \(a \equiv_m b\) and \(c \equiv_m d\), what operations are preserved? (Lemma 4.14) | |
| Back | \[a + c \equiv_m b + d \quad \text{and} \quad ac \equiv_m bd\] Both addition and multiplication are preserved under congruence. |
Note 180: ETH::DiskMat
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Front
How do polynomials behave under modular reduction? (Corollary 4.15)
Back
How do polynomials behave under modular reduction? (Corollary 4.15)
Let \(f(x_1, \dots, x_k)\) be a polynomial with integer coefficients, and let \(m \geq 1\). If \(a_i \equiv_m b_i\) for \(1 \leq i \leq k\), then: \[f(a_1, \dots, a_k) \equiv_m f(b_1, \dots, b_k)\]
Let \(f(x_1, \dots, x_k)\) be a polynomial with integer coefficients, and let \(m \geq 1\). If \(a_i \equiv_m b_i\) for \(1 \leq i \leq k\), then: \[f(a_1, \dots, a_k) \equiv_m f(b_1, \dots, b_k)\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How do polynomials behave under modular reduction? (Corollary 4.15) | |
| Back | Let \(f(x_1, \dots, x_k)\) be a polynomial with integer coefficients, and let \(m \geq 1\). If \(a_i \equiv_m b_i\) for \(1 \leq i \leq k\), then: \[f(a_1, \dots, a_k) \equiv_m f(b_1, \dots, b_k)\] |
Note 181: ETH::DiskMat
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Front
What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16)
Back
What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16)
(i) \(a \equiv_m R_m(a)\) (the remainder represents the equivalence class) (ii) \(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congruence iff same remainder)
(i) \(a \equiv_m R_m(a)\) (the remainder represents the equivalence class) (ii) \(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congruence iff same remainder)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16) | |
| Back | <strong>(i)</strong> \(a \equiv_m R_m(a)\) (the remainder represents the equivalence class) <strong>(ii)</strong> \(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congruence iff same remainder) |
Note 182: ETH::DiskMat
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How can we simplify \(R_m(f(a_1, \dots, a_k))\) for a polynomial \(f\)? (Corollary 4.17)
Back
How can we simplify \(R_m(f(a_1, \dots, a_k))\) for a polynomial \(f\)? (Corollary 4.17)
\[R_m(f(a_1, \dots, a_k)) = R_m(f(R_m(a_1), \dots, R_m(a_k)))\] We can first reduce each argument modulo \(m\), then evaluate the polynomial.
\[R_m(f(a_1, \dots, a_k)) = R_m(f(R_m(a_1), \dots, R_m(a_k)))\] We can first reduce each argument modulo \(m\), then evaluate the polynomial.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How can we simplify \(R_m(f(a_1, \dots, a_k))\) for a polynomial \(f\)? (Corollary 4.17) | |
| Back | \[R_m(f(a_1, \dots, a_k)) = R_m(f(R_m(a_1), \dots, R_m(a_k)))\] We can first reduce each argument modulo \(m\), then evaluate the polynomial. |
Note 183: ETH::DiskMat
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Front
When does \(ax \equiv_m 1\) have a solution? (Lemma 4.18)
Back
When does \(ax \equiv_m 1\) have a solution? (Lemma 4.18)
The equation \(ax \equiv_m 1\) has a unique solution \(x \in \mathbb{Z}_m\) if and only if \(\text{gcd}(a, m) = 1\).
The equation \(ax \equiv_m 1\) has a unique solution \(x \in \mathbb{Z}_m\) if and only if \(\text{gcd}(a, m) = 1\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When does \(ax \equiv_m 1\) have a solution? (Lemma 4.18) | |
| Back | The equation \(ax \equiv_m 1\) has a <strong>unique</strong> solution \(x \in \mathbb{Z}_m\) <strong>if and only if</strong> \(\text{gcd}(a, m) = 1\). |
Note 184: ETH::DiskMat
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Front
State the Chinese Remainder Theorem (Theorem 4.19).
Back
State the Chinese Remainder Theorem (Theorem 4.19).
Let \(m_1, m_2, \dots, m_r\) be pairwise relatively prime integers and let \(M = \prod_{i=1}^{r} m_i\). For every list \(a_1, \dots, a_r\) with \(0 \leq a_i < m_i\), the system \[\begin{align} x &\equiv_{m_1} a_1 \\ x &\equiv_{m_2} a_2 \\ &\vdots \\ x &\equiv_{m_r} a_r \end{align}\] has a unique solution \(x\) satisfying \(0 \leq x < M\).
Let \(m_1, m_2, \dots, m_r\) be pairwise relatively prime integers and let \(M = \prod_{i=1}^{r} m_i\). For every list \(a_1, \dots, a_r\) with \(0 \leq a_i < m_i\), the system \[\begin{align} x &\equiv_{m_1} a_1 \\ x &\equiv_{m_2} a_2 \\ &\vdots \\ x &\equiv_{m_r} a_r \end{align}\] has a unique solution \(x\) satisfying \(0 \leq x < M\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | State the Chinese Remainder Theorem (Theorem 4.19). | |
| Back | Let \(m_1, m_2, \dots, m_r\) be <strong>pairwise relatively prime</strong> integers and let \(M = \prod_{i=1}^{r} m_i\). For every list \(a_1, \dots, a_r\) with \(0 \leq a_i < m_i\), the system \[\begin{align} x &\equiv_{m_1} a_1 \\ x &\equiv_{m_2} a_2 \\ &\vdots \\ x &\equiv_{m_r} a_r \end{align}\] has a <strong>unique solution</strong> \(x\) satisfying \(0 \leq x < M\). |
Note 185: ETH::DiskMat
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Front
Given prime factorizations \(a = \prod_i p_i^{e_i}\) and \(b = \prod_i p_i^{f_i}\), express \(\text{gcd}(a,b)\) and \(\text{lcm}(a,b)\).
Back
Given prime factorizations \(a = \prod_i p_i^{e_i}\) and \(b = \prod_i p_i^{f_i}\), express \(\text{gcd}(a,b)\) and \(\text{lcm}(a,b)\).
\[\text{gcd}(a,b) = \prod_i p_i^{\min(e_i, f_i)}\] \[\text{lcm}(a,b) = \prod_i p_i^{\max(e_i, f_i)}\]
\[\text{gcd}(a,b) = \prod_i p_i^{\min(e_i, f_i)}\] \[\text{lcm}(a,b) = \prod_i p_i^{\max(e_i, f_i)}\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Given prime factorizations \(a = \prod_i p_i^{e_i}\) and \(b = \prod_i p_i^{f_i}\), express \(\text{gcd}(a,b)\) and \(\text{lcm}(a,b)\). | |
| Back | \[\text{gcd}(a,b) = \prod_i p_i^{\min(e_i, f_i)}\] \[\text{lcm}(a,b) = \prod_i p_i^{\max(e_i, f_i)}\] |
Note 186: ETH::DiskMat
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Front
What is the relationship between \(\text{gcd}(a,b)\), \(\text{lcm}(a,b)\), and \(ab\)?
Back
What is the relationship between \(\text{gcd}(a,b)\), \(\text{lcm}(a,b)\), and \(ab\)?
\[\text{gcd}(a,b) \cdot \text{lcm}(a,b) = ab\] This follows from \(\min(e_i, f_i) + \max(e_i, f_i) = e_i + f_i\).
\[\text{gcd}(a,b) \cdot \text{lcm}(a,b) = ab\] This follows from \(\min(e_i, f_i) + \max(e_i, f_i) = e_i + f_i\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the relationship between \(\text{gcd}(a,b)\), \(\text{lcm}(a,b)\), and \(ab\)? | |
| Back | \[\text{gcd}(a,b) \cdot \text{lcm}(a,b) = ab\] This follows from \(\min(e_i, f_i) + \max(e_i, f_i) = e_i + f_i\). |
Note 187: ETH::DiskMat
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Front
How many equivalence classes does \(\equiv_m\) have, and what are they?
Back
How many equivalence classes does \(\equiv_m\) have, and what are they?
There are \(m\) equivalence classes: \([0], [1], \dots, [m-1]\).
The set \(\mathbb{Z}_m = \{0, 1, \dots, m-1\}\) contains the canonical representative from each class.
There are \(m\) equivalence classes: \([0], [1], \dots, [m-1]\).
The set \(\mathbb{Z}_m = \{0, 1, \dots, m-1\}\) contains the canonical representative from each class.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How many equivalence classes does \(\equiv_m\) have, and what are they? | |
| Back | There are \(m\) equivalence classes: \([0], [1], \dots, [m-1]\). <br> The set \(\mathbb{Z}_m = \{0, 1, \dots, m-1\}\) contains the canonical representative from each class. |
Note 188: ETH::DiskMat
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Front
Why must every common divisor of \(a\) and \(b\) also divide \(\text{gcd}(a,b)\)?
Back
Why must every common divisor of \(a\) and \(b\) also divide \(\text{gcd}(a,b)\)?
This is part of the definition of GCD - it's not just the largest common divisor by magnitude, but also the one that is divisible by all other common divisors. This makes it "greatest" in the divisibility ordering, not just in size.
This is part of the definition of GCD - it's not just the largest common divisor by magnitude, but also the one that is divisible by all other common divisors. This makes it "greatest" in the divisibility ordering, not just in size.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why must every common divisor of \(a\) and \(b\) also divide \(\text{gcd}(a,b)\)? | |
| Back | This is part of the definition of GCD - it's not just the largest common divisor by magnitude, but also the one that is divisible by all other common divisors. This makes it "greatest" in the divisibility ordering, not just in size. |
Note 189: ETH::DiskMat
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Front
Why is Bézout's identity useful for finding modular inverses?
Back
Why is Bézout's identity useful for finding modular inverses?
If \(\text{gcd}(a, m) = 1\), then \(ua + vm = 1\) for some \(u, v\). This means \(ua = 1 - vm\), so \(ua \equiv_m 1\), making \(u\) the multiplicative inverse of \(a\) modulo \(m\).
If \(\text{gcd}(a, m) = 1\), then \(ua + vm = 1\) for some \(u, v\). This means \(ua = 1 - vm\), so \(ua \equiv_m 1\), making \(u\) the multiplicative inverse of \(a\) modulo \(m\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why is Bézout's identity useful for finding modular inverses? | |
| Back | If \(\text{gcd}(a, m) = 1\), then \(ua + vm = 1\) for some \(u, v\). This means \(ua = 1 - vm\), so \(ua \equiv_m 1\), making \(u\) the multiplicative inverse of \(a\) modulo \(m\). |
Note 190: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
Why does \(ax \equiv_m 1\) have no solution when \(\text{gcd}(a, m) = d > 1\)?
Back
Why does \(ax \equiv_m 1\) have no solution when \(\text{gcd}(a, m) = d > 1\)?
If \(d | a\) and \(d | m\), then \(d | ax\) for any \(x\). But \(d \nmid 1\), so \(ax\) can never be congruent to \(1\) modulo \(m\).
See bezouts identity on why there is no solution ax - ym = 1.
If \(d | a\) and \(d | m\), then \(d | ax\) for any \(x\). But \(d \nmid 1\), so \(ax\) can never be congruent to \(1\) modulo \(m\).
See bezouts identity on why there is no solution ax - ym = 1.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why does \(ax \equiv_m 1\) have no solution when \(\text{gcd}(a, m) = d > 1\)? | |
| Back | If \(d | a\) and \(d | m\), then \(d | ax\) for any \(x\). But \(d \nmid 1\), so \(ax\) can never be congruent to \(1\) modulo \(m\).<br><br>See bezouts identity on why there is no solution ax - ym = 1. |
Note 191: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
What does "unique up to order" mean in the Fundamental Theorem of Arithmetic?
Back
What does "unique up to order" mean in the Fundamental Theorem of Arithmetic?
Every integer has exactly one prime factorization if we don't care about the order of factors. For example, \(12 = 2^2 \cdot 3 = 3 \cdot 2 \cdot 2 = 2 \cdot 3 \cdot 2\) are all the same factorization, just written differently.
Every integer has exactly one prime factorization if we don't care about the order of factors. For example, \(12 = 2^2 \cdot 3 = 3 \cdot 2 \cdot 2 = 2 \cdot 3 \cdot 2\) are all the same factorization, just written differently.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What does "unique up to order" mean in the Fundamental Theorem of Arithmetic? | |
| Back | Every integer has exactly one prime factorization if we don't care about the order of factors. For example, \(12 = 2^2 \cdot 3 = 3 \cdot 2 \cdot 2 = 2 \cdot 3 \cdot 2\) are all the same factorization, just written differently. |
Note 192: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
Why does Euclid's algorithm work? (Based on Lemma 4.2)
Back
Why does Euclid's algorithm work? (Based on Lemma 4.2)
Because \(\text{gcd}(m, n) = \text{gcd}(m, R_m(n))\). We repeatedly replace the larger number with its remainder when divided by the smaller, preserving the GCD while reducing the problem size. Eventually we reach \(\text{gcd}(d, 0) = d\).
Because \(\text{gcd}(m, n) = \text{gcd}(m, R_m(n))\). We repeatedly replace the larger number with its remainder when divided by the smaller, preserving the GCD while reducing the problem size. Eventually we reach \(\text{gcd}(d, 0) = d\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why does Euclid's algorithm work? (Based on Lemma 4.2) | |
| Back | Because \(\text{gcd}(m, n) = \text{gcd}(m, R_m(n))\). We repeatedly replace the larger number with its remainder when divided by the smaller, preserving the GCD while reducing the problem size. Eventually we reach \(\text{gcd}(d, 0) = d\). |
Note 193: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
Why does the Chinese Remainder Theorem require \(m_1, \dots, m_r\) to be pairwise relatively prime?
Back
Why does the Chinese Remainder Theorem require \(m_1, \dots, m_r\) to be pairwise relatively prime?
If \(\text{gcd}(m_i, m_j) = d > 1\), then the system could be inconsistent (e.g., \(x \equiv 0 \pmod{6}\) and \(x \equiv 1 \pmod{4}\) has no solution) or have multiple solutions (destroying uniqueness).
If \(\text{gcd}(m_i, m_j) = d > 1\), then the system could be inconsistent (e.g., \(x \equiv 0 \pmod{6}\) and \(x \equiv 1 \pmod{4}\) has no solution) or have multiple solutions (destroying uniqueness).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Why does the Chinese Remainder Theorem require \(m_1, \dots, m_r\) to be <strong>pairwise relatively prime</strong>? | |
| Back | If \(\text{gcd}(m_i, m_j) = d > 1\), then the system could be <strong>inconsistent</strong> (e.g., \(x \equiv 0 \pmod{6}\) and \(x \equiv 1 \pmod{4}\) has no solution) or have <strong>multiple solutions</strong> (destroying uniqueness). |
Note 194: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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mA6Xn.z1Ap
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Front
How does \(\mathbb{Z}_m = \{0, 1, \dots, m-1\}\) relate to equivalence classes of modulo?
Back
How does \(\mathbb{Z}_m = \{0, 1, \dots, m-1\}\) relate to equivalence classes of modulo?
\(\mathbb{Z}_m\) is the set of canonical representatives from \(\mathbb{Z} / \equiv_m\). Each element of \(\mathbb{Z}_m\) represents one of the \(m\) equivalence classes of integers congruent modulo \(m\).
\(\mathbb{Z}_m\) is the set of canonical representatives from \(\mathbb{Z} / \equiv_m\). Each element of \(\mathbb{Z}_m\) represents one of the \(m\) equivalence classes of integers congruent modulo \(m\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How does \(\mathbb{Z}_m = \{0, 1, \dots, m-1\}\) relate to equivalence classes of modulo? | |
| Back | \(\mathbb{Z}_m\) is the set of <strong>canonical representatives</strong> from \(\mathbb{Z} / \equiv_m\). Each element of \(\mathbb{Z}_m\) represents one of the \(m\) equivalence classes of integers congruent modulo \(m\). |
Note 195: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
Verify that \(\equiv_m\) is reflexive, symmetric, and transitive.
Back
Verify that \(\equiv_m\) is reflexive, symmetric, and transitive.
- Reflexive: \(a \equiv_m a\) since \(m | (a - a) = 0\) ✓
- Symmetric: \(a \equiv_m b \Rightarrow m | (a-b) \Rightarrow m | (b-a) \Rightarrow b \equiv_m a\) ✓
- Transitive: If \(m | (a-b)\) and \(m | (b-c)\), then \(m | (a-b+b-c) = (a-c)\) ✓
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Verify that \(\equiv_m\) is reflexive, symmetric, and transitive. | |
| Back | <ul> <li><strong>Reflexive</strong>: \(a \equiv_m a\) since \(m | (a - a) = 0\) ✓</li> <li><strong>Symmetric</strong>: \(a \equiv_m b \Rightarrow m | (a-b) \Rightarrow m | (b-a) \Rightarrow b \equiv_m a\) ✓</li> <li><strong>Transitive</strong>: If \(m | (a-b)\) and \(m | (b-c)\), then \(m | (a-b+b-c) = (a-c)\) ✓</li> </ul> |
Note 196: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Discrete Mathematics Card
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Front
Are we allowed to assume that the inverse of the inverse of a relation is simply the relation itself?
Back
Are we allowed to assume that the inverse of the inverse of a relation is simply the relation itself?
No, we need to prove it every time.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Are we allowed to assume that the inverse of the inverse of a relation is simply the relation itself? | |
| Back | No, we need to prove it every time. |
Note 197: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Discrete Mathematics Card
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Front
When is writing \(\top\) or \(\perp\) allowed in formulas (proof steps for example)?
Back
When is writing \(\top\) or \(\perp\) allowed in formulas (proof steps for example)?
We are not allowed to use $\top$ or $\perp$ in formulas, to replace statement that are `true` or `false` under our interpretation.
It's only allowed when the formula is actually a tautology (or unsatisfiable), i.e. true or false under all interpretations!
For example, in $U = \mathbb{N}$, $x \geq 0 \land x = 5 \implies \top \land x = 5 \implies x = 5$ but this is wrong as $x \geq 0$ is only equivalent to $\top$ in this specific universe. We instead can just write the implication directly.
It's only allowed when the formula is actually a tautology (or unsatisfiable), i.e. true or false under all interpretations!
For example, in $U = \mathbb{N}$, $x \geq 0 \land x = 5 \implies \top \land x = 5 \implies x = 5$ but this is wrong as $x \geq 0$ is only equivalent to $\top$ in this specific universe. We instead can just write the implication directly.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | When is writing \(\top\) or \(\perp\) allowed in formulas (proof steps for example)? | |
| Back | We are not allowed to use $\top$ or $\perp$ in formulas, to replace statement that are `true` or `false` under our interpretation.<br><br>It's only allowed when the formula is actually a tautology (or unsatisfiable), i.e. true or false under <b>all</b> interpretations!<br><br>For example, in $U = \mathbb{N}$, $x \geq 0 \land x = 5 \implies \top \land x = 5 \implies x = 5$ but this is wrong as $x \geq 0$ is only equivalent to $\top$ in this specific universe. We instead can just write the implication directly. |
Note 198: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Discrete Mathematics Card
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Note Type: Discrete Mathematics Card
GUID:
juXB+9W`+)
Previous
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Front
Does \( p | a \land q | a \land \gcd(p, q) = 1 \implies pq | a \) hold?
Back
Does \( p | a \land q | a \land \gcd(p, q) = 1 \implies pq | a \) hold?
Yes, but this has to be reproven before using.
The proof technique is important. Replacing a neutral element by something it's equal is often a smart move.
Proof: This is an important result for the exam:
Since \(p \mid a\) and \(q \mid a\), we have:
The proof technique is important. Replacing a neutral element by something it's equal is often a smart move.
Proof: This is an important result for the exam:
\[p \mid a \land q \mid a \land \gcd(p, q) = 1 \implies pq \mid a\]
Which is the same as saying \(\exists k \in \mathbb{Z}\) such that \(a = pq \cdot k\).
Since \(p \mid a\) and \(q \mid a\), we have:
\[\exists k, k' \in \mathbb{Z} \text{ such that } a = pk \land a = qk'\]
Since \(\gcd(p, q) = 1\), by Bézout's identity:
\[\exists u, v \in \mathbb{Z} \text{ such that } 1 = pu + qv\]
Now we can write:
\[\begin{align}
a &= 1 \cdot a \\
&= a \cdot (pu + qv) \\
&= pua + qva \\
&= pu \cdot qk' + qv \cdot pk \\
&= pq(uk' + vk')
\end{align}\]
Thus \(pq \mid a\). \(\square\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Does \( p | a \land q | a \land \gcd(p, q) = 1 \implies pq | a \) hold? | |
| Back | Yes, but this has to be reproven before using.<br><br>The proof technique is important. Replacing a neutral element by something it's equal is often a smart move.<br> <b>Proof:</b> This is an important result for the exam: <div>\[p \mid a \land q \mid a \land \gcd(p, q) = 1 \implies pq \mid a\]</div> Which is the same as saying \(\exists k \in \mathbb{Z}\) such that \(a = pq \cdot k\). <br> Since \(p \mid a\) and \(q \mid a\), we have: <div>\[\exists k, k' \in \mathbb{Z} \text{ such that } a = pk \land a = qk'\]</div> Since \(\gcd(p, q) = 1\), by Bézout's identity: <div>\[\exists u, v \in \mathbb{Z} \text{ such that } 1 = pu + qv\]</div> Now we can write: <div>\[\begin{align} a &= 1 \cdot a \\ &= a \cdot (pu + qv) \\ &= pua + qva \\ &= pu \cdot qk' + qv \cdot pk \\ &= pq(uk' + vk') \end{align}\]</div> Thus \(pq \mid a\). \(\square\) |
Note 199: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Discrete Mathematics Card
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GUID:
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Front
How can we use the CRT to decompose remainders like \(R_{77}(n)\)?
Back
How can we use the CRT to decompose remainders like \(R_{77}(n)\)?
We can decompose \(77 = 11 \cdot 7\) and then calculate:
- \(R_7(n) = 3\)
- \(R_{11}(n) = 5\)
- Find \(11^{-1} \pmod{7} = 2\) (since \(11 \cdot 2 = 22 \equiv 1 \pmod{7}\))
- Find \(7^{-1} \pmod{11} = 8\) (since \(7 \cdot 8 = 56 \equiv 1 \pmod{11}\))
- Calculate: \(x = 3 \cdot 11 \cdot 2 + 5 \cdot 7 \cdot 8 = 66 + 280 = 346 \equiv 38 \pmod{77}\)
- Therefore \(R_{77}(n) = 38\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How can we use the CRT to decompose remainders like \(R_{77}(n)\)? | |
| Back | We can decompose \(77 = 11 \cdot 7\) and then calculate:<br><ul><li>\(R_7(n) = 3\)</li><li>\(R_{11}(n) = 5\)</li></ul>Then to find the result mod 77, we use the CRT.<br><ol><li>Find \(11^{-1} \pmod{7} = 2\) (since \(11 \cdot 2 = 22 \equiv 1 \pmod{7}\))</li><li>Find \(7^{-1} \pmod{11} = 8\) (since \(7 \cdot 8 = 56 \equiv 1 \pmod{11}\))</li><li>Calculate: \(x = 3 \cdot 11 \cdot 2 + 5 \cdot 7 \cdot 8 = 66 + 280 = 346 \equiv 38 \pmod{77}\)</li><li>Therefore \(R_{77}(n) = 38\)</li></ol> |
Note 200: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Front
\( \neg (A \land B) \) \( \equiv \) \( \neg A \lor \neg B \)
Back
\( \neg (A \land B) \) \( \equiv \) \( \neg A \lor \neg B \)
de Morgan rules
de Morgan rules
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | {{c1::\( \neg (A \land B) \)}} \( \equiv \) {{c2::\( \neg A \lor \neg B \)}}<br> | |
| Extra | de Morgan rules |
Note 201: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Front
\( \neg (A \lor B) \) \( \equiv \) \( \neg A \land \neg B\)
Back
\( \neg (A \lor B) \) \( \equiv \) \( \neg A \land \neg B\)
De Morgan rules
De Morgan rules
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | {{c1::\( \neg (A \lor B) \) }} \( \equiv \) {{c2::\( \neg A \land \neg B\)}}<br> | |
| Extra | De Morgan rules |
Note 202: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
What is the difference between propositional and predicate logic?
Back
What is the difference between propositional and predicate logic?
propositional: only values of \(\{0,1\}\), finite
propositional: only values of \(\{0,1\}\), finite
predicate: any values in our universe, infinite
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the difference between propositional and predicate logic? | |
| Back | propositional: only values of \(\{0,1\}\), finite<div>predicate: any values in our universe, infinite</div> |
Note 203: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
What is a predicate?
Back
What is a predicate?
A k-ary predicate on \( U \) is a function \( U^k \rightarrow \{0,1\}\).
A k-ary predicate on \( U \) is a function \( U^k \rightarrow \{0,1\}\).
It's like a function that takes any number of arguments, but only returns boolean results.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a predicate? | |
| Back | A k-ary predicate on \( U \) is a function \( U^k \rightarrow \{0,1\}\).<div>It's like a function that takes any number of arguments, but only returns boolean results.</div> |
Note 204: ETH::DiskMat
Deck: ETH::DiskMat
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Front
What is a tautology?
Back
What is a tautology?
A formula F (propositional logic) is a tautology/valid if it is true for all truth combinations of the involved symbols, so if it is always true. Symbol: \( \top \)
A formula F (propositional logic) is a tautology/valid if it is true for all truth combinations of the involved symbols, so if it is always true. Symbol: \( \top \)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a tautology? | |
| Back | A formula F (propositional logic) is a tautology/valid if it is true for all truth combinations of the involved symbols, so if it is always true. Symbol: \( \top \) |
Note 205: ETH::DiskMat
Deck: ETH::DiskMat
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Front
G is a logical conseqence of F. What does that mean?
Back
G is a logical conseqence of F. What does that mean?
\( F \models G\) if G is always true if F is true, but not necessarily false when F is false. (No causality!)
\( F \models G\) if G is always true if F is true, but not necessarily false when F is false. (No causality!)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | G is a <i>logical conse</i><i>qence </i>of F. What does that mean? | |
| Back | \( F \models G\) if G is always true if F is true, but not necessarily false when F is false. (No causality!)<br> |
Note 206: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
Predicate logic: universe
Back
Predicate logic: universe
A universe is the non-empty set that we work within. Examples: \( \mathbb{N}, \mathbb{R}, \{ 0,1,2,3,6 \} \)
A universe is the non-empty set that we work within. Examples: \( \mathbb{N}, \mathbb{R}, \{ 0,1,2,3,6 \} \)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Predicate logic: universe | |
| Back | A universe is the non-empty set that we work within. Examples: \( \mathbb{N}, \mathbb{R}, \{ 0,1,2,3,6 \} \) |
Note 207: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Proof method: "Composition of Implications"
Back
Proof method: "Composition of Implications"
Idea: If \( S \implies T \) and \( T \implies U \) are both true, then \( S \implies U \) is also true.
Idea: If \( S \implies T \) and \( T \implies U \) are both true, then \( S \implies U \) is also true.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: "Composition of Implications" | |
| Back | Idea: If \( S \implies T \) and \( T \implies U \) are both true, then \( S \implies U \) is also true. |
Note 208: ETH::DiskMat
Deck: ETH::DiskMat
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y)
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Front
Proof method: "Direct Proof of an Implication"
Back
Proof method: "Direct Proof of an Implication"
Direct proof of \( S \implies T \): assume S and prove T under that assumption
Direct proof of \( S \implies T \): assume S and prove T under that assumption
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: "Direct Proof of an Implication" | |
| Back | Direct proof of \( S \implies T \): assume S and prove T under that assumption |
Note 209: ETH::DiskMat
Deck: ETH::DiskMat
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Front
Proof method: "Indirect Proof of an Implication"
Back
Proof method: "Indirect Proof of an Implication"
Indirect proof of \( S \implies T \): Assume T is false, prove that S is false.
Indirect proof of \( S \implies T \): Assume T is false, prove that S is false.
Follows from \( (\neg B \to \neg A) \models (A \to B) \)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <i>Proof method:</i> "Indirect Proof of an Implication" | |
| Back | Indirect proof of \( S \implies T \): Assume T is false, prove that S is false.<div><br></div><div>Follows from \( (\neg B \to \neg A) \models (A \to B) \)</div> |
Note 210: ETH::DiskMat
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Front
Proof method: "Modus Ponens"
Back
Proof method: "Modus Ponens"
1. Find a suitable statement \(R\)
1. Find a suitable statement \(R\)
2. Prove \(R\)
3. Prove \(R \implies S\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: "Modus Ponens" | |
| Back | 1. Find a suitable statement \(R\)<div>2. Prove \(R\)</div><div>3. Prove \(R \implies S\)</div> |
Note 211: ETH::DiskMat
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Front
Proof method: "Case Distinction"
Back
Proof method: "Case Distinction"
1. Find a finite list \( R_1, \ldots, R_k\) of statements (cases)
1. Find a finite list \( R_1, \ldots, R_k\) of statements (cases)
2. Prove that one case applies for the situation (prove one \(R_i\))
3. Prove \( R_i \implies S\) for \(i = 1, \ldots, k\)
Basically, show for all cases that they are correct.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: "Case Distinction" | |
| Back | 1. Find a finite list \( R_1, \ldots, R_k\) of statements (cases)<div>2. Prove that one case applies for the situation (prove one \(R_i\))</div><div>3. Prove \( R_i \implies S\) for \(i = 1, \ldots, k\)</div><div><br></div><div>Basically, show for all cases that they are correct.</div> |
Note 212: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
Cardinality of a set
Back
Cardinality of a set
The number of elements in the set, written as \( |A| \).
The number of elements in the set, written as \( |A| \).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Cardinality of a set | |
| Back | The number of elements in the set, written as \( |A| \). |
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Front
Proof method: Proof by Contradiction
Back
Proof method: Proof by Contradiction
1. Find a suitable statement \( T\)
1. Find a suitable statement \( T\)
2. Prove that \( T \) is false
3. Assume that \( S \) is false and prove that \( T \) is true (-> contradiction)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: Proof by Contradiction | |
| Back | 1. Find a suitable statement \( T\)<div>2. Prove that \( T \) is false</div><div>3. Assume that \( S \) is false and prove that \( T \) is true (-> contradiction)</div> |
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Front
Proof method: Existence Proof
Back
Proof method: Existence Proof
We just want to prove that there exists a \( x \) such that a statement \( S_x \) is true. (e.g. There exists a prime number such that \( n - 10\) and \( n + 10\) are also prime.)
We just want to prove that there exists a \( x \) such that a statement \( S_x \) is true. (e.g. There exists a prime number such that \( n - 10\) and \( n + 10\) are also prime.)
constructive: We know the x.
non-constructive: We know that x has to exist, but we don't know its value.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: Existence Proof | |
| Back | We just want to prove that there exists a \( x \) such that a statement \( S_x \) is true. (e.g. There exists a prime number such that \( n - 10\) and \( n + 10\) are also prime.) <div><br></div><div><i>constructive: </i>We know the x.</div><div><i>non-constructive: </i>We know that x has to exist, but we don't know its value.</div> |
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Front
Proof method: Pigeonhole Principle
Back
Proof method: Pigeonhole Principle
If a set of \( n \) objects is divided into \( k < n\) sets, then at least one of the sets contains at least \( \left \lceil{\frac{n}{k}}\right \rceil\) objects.
If a set of \( n \) objects is divided into \( k < n\) sets, then at least one of the sets contains at least \( \left \lceil{\frac{n}{k}}\right \rceil\) objects.
Informally: If there are more objects than sets, there is a set with more than one object in it.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: Pigeonhole Principle | |
| Back | If a set of \( n \) objects is divided into \( k < n\) sets, then at least one of the sets contains at least \( \left \lceil{\frac{n}{k}}\right \rceil\) objects.<div><br></div><div>Informally: If there are more objects than sets, there is a set with more than one object in it.</div> |
Note 216: ETH::DiskMat
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Front
Proof method: Proofs by counterexample
Back
Proof method: Proofs by counterexample
Special case of constructive existence proofs. By finding a counter example \( x\) such that \(S_x\) is not true, we can prove that \( S_i \) isn't always true.
Special case of constructive existence proofs. By finding a counter example \( x\) such that \(S_x\) is not true, we can prove that \( S_i \) isn't always true.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Proof method: Proofs by counterexample | |
| Back | Special case of constructive existence proofs. By finding a counter example \( x\) such that \(S_x\) is not true, we can prove that \( S_i \) isn't always true. |
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A relation ρ on a set A is called reflexive if {{c2::\( a \ \rho \ a\) is true for all \( a \in A\), i.e. if \( \text{id} \subseteq \rho\).}}
Back
A relation ρ on a set A is called reflexive if {{c2::\( a \ \rho \ a\) is true for all \( a \in A\), i.e. if \( \text{id} \subseteq \rho\).}}
Example: \( \ge, \le \) are reflexive, while \( <, > \) are not.
Example: \( \ge, \le \) are reflexive, while \( <, > \) are not.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A relation ρ on a set A is called {{c1::reflexive}} if {{c2::\( a \ \rho \ a\) is true for all \( a \in A\), i.e. if \( \text{id} \subseteq \rho\).}} | |
| Extra | Example: \( \ge, \le \) are reflexive, while \( <, > \) are not. |
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A relation ρ on a set A is called symmetric if {{c2::\( a \ \rho \ b \iff b \ \rho \ a\) is true, i.e. if \( \rho = \hat{\rho}\)}}
Back
A relation ρ on a set A is called symmetric if {{c2::\( a \ \rho \ b \iff b \ \rho \ a\) is true, i.e. if \( \rho = \hat{\rho}\)}}
Examples: \( \equiv_m\), marriage
Examples: \( \equiv_m\), marriage
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A relation ρ on a set A is called {{c1::symmetric}} if {{c2::\( a \ \rho \ b \iff b \ \rho \ a\) is true, i.e. if \( \rho = \hat{\rho}\)}} | |
| Extra | Examples: \( \equiv_m\), marriage |
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A relation ρ on a set A is called antisymmetric if {{c1::\( a \ \rho \ b \wedge b \ \rho \ a \implies a = b\) is true, i.e. if \( \rho \cap \hat{\rho} \subseteq \text{id}\)}}
Back
A relation ρ on a set A is called antisymmetric if {{c1::\( a \ \rho \ b \wedge b \ \rho \ a \implies a = b\) is true, i.e. if \( \rho \cap \hat{\rho} \subseteq \text{id}\)}}
Example: \( \le, \ge\) and the division relation: \( a \mid b \wedge b \mid a \implies a = b\)
Example: \( \le, \ge\) and the division relation: \( a \mid b \wedge b \mid a \implies a = b\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A relation ρ on a set A is called {{c2::antisymmetric}} if {{c1::\( a \ \rho \ b \wedge b \ \rho \ a \implies a = b\) is true, i.e. if \( \rho \cap \hat{\rho} \subseteq \text{id}\)}} | |
| Extra | Example: \( \le, \ge\) and the division relation: \( a \mid b \wedge b \mid a \implies a = b\) |
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A relation is transitive if \((a \ \rho \ b \wedge b \ \rho \ c) \implies a \ \rho \ c \) is true.
Back
A relation is transitive if \((a \ \rho \ b \wedge b \ \rho \ c) \implies a \ \rho \ c \) is true.
Examples: \( \le, \ge, <, |, \equiv_m\)
Examples: \( \le, \ge, <, |, \equiv_m\)
Field-by-field Comparison
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|---|---|---|
| Text | A relation is {{c1::transitive}} if {{c2::\((a \ \rho \ b \wedge b \ \rho \ c) \implies a \ \rho \ c \) is true.}} | |
| Extra | Examples: \( \le, \ge, <, |, \equiv_m\) |
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Front
The inverse relation is {{c1::\( a \ \rho \ b \iff b \ \hat{\rho} \ a\)}}.
Back
The inverse relation is {{c1::\( a \ \rho \ b \iff b \ \hat{\rho} \ a\)}}.
Example: Inverse of parent relation is childhood relation. Also written as \( \rho^{-1}\)
Example: Inverse of parent relation is childhood relation. Also written as \( \rho^{-1}\)
Field-by-field Comparison
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|---|---|---|
| Text | The {{c2::inverse relation}} is {{c1::\( a \ \rho \ b \iff b \ \hat{\rho} \ a\)}}. | |
| Extra | Example: Inverse of parent relation is childhood relation. Also written as \( \rho^{-1}\) |
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A function is injective (or one-to-one) if for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"
Back
A function is injective (or one-to-one) if for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"
Example: \(f(x) = x\), counterexample: \(f(x) = x^2, x \in \mathbb{R}\)
Example: \(f(x) = x\), counterexample: \(f(x) = x^2, x \in \mathbb{R}\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A function is {{c1::injective (or one-to-one)}} if {{c2::for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"}} | |
| Extra | Example: \(f(x) = x\), counterexample: \(f(x) = x^2, x \in \mathbb{R}\) |
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A function \( f: A \rightarrow B\) is surjective (or onto) if \( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken
Back
A function \( f: A \rightarrow B\) is surjective (or onto) if \( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A function \( f: A \rightarrow B\) is {{c1::surjective (or onto)}} if {{c2::\( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken}} |
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A function is bijective (one-to-one correspondence) if it is both injective and surjective.
Back
A function is bijective (one-to-one correspondence) if it is both injective and surjective.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A function is {{c1::bijective (one-to-one correspondence)}} if it is {{c2::both injective and surjective.}} |
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The {{c2::power set of a set \(A\), denoted \(\mathcal{P}(A)\)}}, is the set of all subsets of \(A\).
Back
The {{c2::power set of a set \(A\), denoted \(\mathcal{P}(A)\)}}, is the set of all subsets of \(A\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | The {{c2::power set of a set \(A\), denoted \(\mathcal{P}(A)\)}}, is {{c1::the set of all subsets of \(A\)}}. |
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Front
Equivalence relation is a relation on a set \(A\) that is
* reflexive
* symmetric
* transitive
Back
Equivalence relation is a relation on a set \(A\) that is
Example: \(\equiv_m \)
* reflexive
* symmetric
* transitive
Example: \(\equiv_m \)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | {{c1::Equivalence relation}} is a relation on a set \(A\) that is<div>{{c2::<div>* reflexive</div><div>* symmetric</div><div>* transitive</div>}}<br></div> | |
| Extra | Example: \(\equiv_m \) |
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A partial order on a set \(A\) is a relation that is
* reflexive
* antisymmetric
* transitive
Back
A partial order on a set \(A\) is a relation that is
Examples: \(\leq, \geq\)
* reflexive
* antisymmetric
* transitive
Examples: \(\leq, \geq\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | {{c1::A partial order}} on a set \(A\) is a relation that is<div>{{c2::<div>* reflexive</div><div>* antisymmetric</div><div>* transitive</div>}}<br></div> | |
| Extra | Examples: \(\leq, \geq\) |
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A set together with a partial order \(\preceq\) is called a partially ordered set or simply poset.
Back
A set together with a partial order \(\preceq\) is called a partially ordered set or simply poset.
Denoted \((A; \preceq)\)
Denoted \((A; \preceq)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A set together with a partial order \(\preceq\) is called {{c1::a partially ordered set or simply poset.}} | |
| Extra | Denoted \((A; \preceq)\) |
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An algebra (also: algebraic structure, \( \Omega\)-algebra) is a pair \(\langle S, \Omega \rangle\) where S is a set and \(\Omega = (\omega_1, \ldots, \omega_n)\) is a list of operations on S.
Back
An algebra (also: algebraic structure, \( \Omega\)-algebra) is a pair \(\langle S, \Omega \rangle\) where S is a set and \(\Omega = (\omega_1, \ldots, \omega_n)\) is a list of operations on S.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | {{c1::An algebra (also: algebraic structure, \( \Omega\)-algebra)}} is a pair \(\langle S, \Omega \rangle\) {{c2::where S is a set and \(\Omega = (\omega_1, \ldots, \omega_n)\) is a list of operations on S.}} |
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In an algebra \(\langle S, \Omega \rangle\), how is S usually called?
Back
In an algebra \(\langle S, \Omega \rangle\), how is S usually called?
carrier (of the algebra)
carrier (of the algebra)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | In an algebra \(\langle S, \Omega \rangle\), how is S usually called? | |
| Back | carrier (of the algebra) |
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A monoid is an algebra \( \langle S; *, e \rangle\) where \(*\) is associative and \(e\) is the neutral element.
Back
A monoid is an algebra \( \langle S; *, e \rangle\) where \(*\) is associative and \(e\) is the neutral element.
Difference to group: Inverse missing
Difference to group: Inverse missing
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | {{c1::A <b>monoid</b>}}<b> </b>is an algebra {{c2::\( \langle S; *, e \rangle\) where \(*\) is associative and \(e\) is the neutral element.}} | |
| Extra | Difference to group: Inverse missing |
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In a poset \( ( A; \preceq )\), \(b\) covers \(a\) if \(a \prec b\) and there does not exist a \(c\) with \(a \prec c \land c \prec b \), so no elements are between \(a\) and \(b\).
Back
In a poset \( ( A; \preceq )\), \(b\) covers \(a\) if \(a \prec b\) and there does not exist a \(c\) with \(a \prec c \land c \prec b \), so no elements are between \(a\) and \(b\).
Example: direct superior in a company
Example: direct superior in a company
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In a poset \( ( A; \preceq )\), \(b\) <b>covers</b> \(a\) if {{c1::\(a \prec b\) and there does not exist a \(c\) with \(a \prec c \land c \prec b \), so no elements are between \(a\) and \(b\).}} | |
| Extra | Example: direct superior in a company |
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A poset \((A; \preceq)\) is called totally ordered (also: linearly ordered) by \(\preceq\) if any two elements of the poset are comparable.
Back
A poset \((A; \preceq)\) is called totally ordered (also: linearly ordered) by \(\preceq\) if any two elements of the poset are comparable.
Example: \((\mathbb{Z}; \ge)\)
Example: \((\mathbb{Z}; \ge)\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A poset \((A; \preceq)\) is called {{c2::<b>totally ordered</b> (also: linearly ordered) by \(\preceq\)}} if {{c1::any two elements of the poset are comparable.}} | |
| Extra | Example: \((\mathbb{Z}; \ge)\) |
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Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).
\(a \in A\) is a minimal (maximal) element of \(A\) if there exists no \(b \in A\) with \(b \prec a\) (\(b \succ a \) )
Back
Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).
\(a \in A\) is a minimal (maximal) element of \(A\) if there exists no \(b \in A\) with \(b \prec a\) (\(b \succ a \) )
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).<div>\(a \in A\) is a {{c1::<b>minimal (maximal) element</b> of \(A\)}} if {{c2::there exists no \(b \in A\) with \(b \prec a\) (\(b \succ a \) )}}<br></div> |
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Special elements in posets: \((A; \preceq)\) is a poset.
\(a \in A\) is the least (greatest) element of \(A\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
Back
Special elements in posets: \((A; \preceq)\) is a poset.
\(a \in A\) is the least (greatest) element of \(A\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
Field-by-field Comparison
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|---|---|---|
| Text | Special elements in posets: \((A; \preceq)\) is a poset.<div>\(a \in A\) is the {{c1::<b>least (greatest) element</b> of \(A\)}} if {{c2::\(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)}}</div> |
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Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).
\(a \in A\) is a lower (upper) bound of \(S\) if \(b \preceq a\) (\(b \succeq a) \) for all \(b \in S\)
Back
Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).
Note that a is not necessarily in the subset S (difference to the least and greatest elements).
\(a \in A\) is a lower (upper) bound of \(S\) if \(b \preceq a\) (\(b \succeq a) \) for all \(b \in S\)
Note that a is not necessarily in the subset S (difference to the least and greatest elements).
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|---|---|---|
| Text | Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).<div>\(a \in A\) is a {{c1::<b>lower (upper) bound</b> of \(S\)}} if {{c2::\(b \preceq a\) (\(b \succeq a) \) for all \(b \in S\)}}</div> | |
| Extra | Note that a is not necessarily in the subset S (difference to the least and greatest elements). |
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Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).
\(a \in A\) is the greatest lower (least upper) bound of \(S\) if \(a\) is the greatest (least) element of the set of all lower (upper) bounds of \(S\).
Back
Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).
\(a \in A\) is the greatest lower (least upper) bound of \(S\) if \(a\) is the greatest (least) element of the set of all lower (upper) bounds of \(S\).
Field-by-field Comparison
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| Text | Special elements in posets: \((A; \preceq)\) is a poset, \( S \subseteq A\).<div>\(a \in A\) is the {{c1::<b>greatest lower (least upper) bound</b> of \(S\)}} if {{c2::\(a\) is the greatest (least) element of the set of all lower (upper) bounds of \(S\). }}</div> |
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A poset \((A;\preceq)\) is well-ordered if it is totally ordered and every non-empty subset has a least element.
Back
A poset \((A;\preceq)\) is well-ordered if it is totally ordered and every non-empty subset has a least element.
Every totally ordered finite poset \(\rightarrow\) well-ordered
Every totally ordered finite poset \(\rightarrow\) well-ordered
Infinite example: \((\mathbb{N}; \le)\)
Infinite counterexample \((\mathbb{Z}; \le)\)
Infinite counterexample \((\mathbb{Z}; \le)\)
Field-by-field Comparison
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|---|---|---|
| Text | A poset \((A;\preceq)\) is <b>well-ordered </b>if {{c1::it is totally ordered and every non-empty subset has a least element.}} | |
| Extra | Every totally ordered finite poset \(\rightarrow\) well-ordered<div>Infinite example: \((\mathbb{N}; \le)\)<br>Infinite counterexample \((\mathbb{Z}; \le)\)</div> |
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\((A;\preceq)\) is a poset. If \(\{a,b\}\) have a greatest lower bound, then it is called the meet of \(a\) and \(b\) (also denoted \(a \land b\)).
Back
\((A;\preceq)\) is a poset. If \(\{a,b\}\) have a greatest lower bound, then it is called the meet of \(a\) and \(b\) (also denoted \(a \land b\)).
Field-by-field Comparison
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|---|---|---|
| Text | \((A;\preceq)\) is a poset. If \(\{a,b\}\) have a {{c2::greatest lower bound}}, then it is called the {{c1::<b>meet </b>of \(a\) and \(b\) (also denoted \(a \land b\)).}}<br> |
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\((A;\preceq)\) is a poset. If \(\{a,b\}\) have a least upper bound, then it is called the join of \(a\) and \(b\) (also denoted \(a \lor b\)).
Back
\((A;\preceq)\) is a poset. If \(\{a,b\}\) have a least upper bound, then it is called the join of \(a\) and \(b\) (also denoted \(a \lor b\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | \((A;\preceq)\) is a poset. If \(\{a,b\}\) have a {{c2::least upper bound}}, then it is called the {{c1::<b>join </b>of \(a\) and \(b\) (also denoted \(a \lor b\)).}} |
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A poset in which every pair of elements has a meet and a join is called a lattice.
Back
A poset in which every pair of elements has a meet and a join is called a lattice.
Examples: \((\mathbb{N}; \le), \ (\mathcal{P}(S); \subseteq)\)
Examples: \((\mathbb{N}; \le), \ (\mathcal{P}(S); \subseteq)\)
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|---|---|---|
| Text | A poset in which {{c2::every pair of elements has a meet and a join}} is called a {{c1::lattice}}. | |
| Extra | Examples: \((\mathbb{N}; \le), \ (\mathcal{P}(S); \subseteq)\) |
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The subset \(f(A)\) of \(B\) is called the image (also: range) of \(f\) and is also denoted \(Im(f)\).
Back
The subset \(f(A)\) of \(B\) is called the image (also: range) of \(f\) and is also denoted \(Im(f)\).
Example: \(f(x) = x^2\), the range of \(f\) is \(\mathbb{R}^{\ge 0}\)
Example: \(f(x) = x^2\), the range of \(f\) is \(\mathbb{R}^{\ge 0}\)
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|---|---|---|
| Text | The {{c2::subset \(f(A)\) of \(B\)}} is called the {{c1::<b>image</b> (also: range) of \(f\)}} and is also denoted \(Im(f)\). | |
| Extra | Example: \(f(x) = x^2\), the range of \(f\) is \(\mathbb{R}^{\ge 0}\) |
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For a subset \(T\) of \(B\), the {{c1::preimage (in Linalg: Urbild) of \(T\), denoted \(f^{-1}(T)\)}}, is the set of values in \(A\) that map into \(T\).
Back
For a subset \(T\) of \(B\), the {{c1::preimage (in Linalg: Urbild) of \(T\), denoted \(f^{-1}(T)\)}}, is the set of values in \(A\) that map into \(T\).
Example: \(f(x) = x^2\), the preimage of \([4,9]\) is \([-3,-2] \cup [2,3]\)
Example: \(f(x) = x^2\), the preimage of \([4,9]\) is \([-3,-2] \cup [2,3]\)
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|---|---|---|
| Text | For a subset \(T\) of \(B\), the {{c1::<b>preimage </b>(in Linalg: Urbild) of \(T\), denoted \(f^{-1}(T)\)}}, is {{c2::the set of values in \(A\) that map into \(T\).}} | |
| Extra | Example: \(f(x) = x^2\), the preimage of \([4,9]\) is \([-3,-2] \cup [2,3]\) |
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Two sets \(A, B\) are equinumerous (denoted \(A \sim B\)) if there exists a bijection \(A \rightarrow B\).
Back
Two sets \(A, B\) are equinumerous (denoted \(A \sim B\)) if there exists a bijection \(A \rightarrow B\).
Example: \(f: \mathbb{N} \rightarrow \mathbb{Z} = (-1)^n \lceil n/2 \rceil\)
Example: \(f: \mathbb{N} \rightarrow \mathbb{Z} = (-1)^n \lceil n/2 \rceil\)
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| Text | Two sets \(A, B\) are {{c1::<b>equinumerous </b>(denoted \(A \sim B\))}} if {{c2::there exists a bijection \(A \rightarrow B\).}} | |
| Extra | Example: \(f: \mathbb{N} \rightarrow \mathbb{Z} = (-1)^n \lceil n/2 \rceil\) |
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The set \(B\) dominates (denoted \(A \preceq B\)) if there exists an injective function \(A \rightarrow B\).
Back
The set \(B\) dominates (denoted \(A \preceq B\)) if there exists an injective function \(A \rightarrow B\).
Example: \(f(x): \mathbb{N} \rightarrow \mathbb{R} = x\)
Example: \(f(x): \mathbb{N} \rightarrow \mathbb{R} = x\)
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| Text | The set \(B\) {{c1::<b>dominates</b> (denoted \(A \preceq B\))}} if {{c2::there exists an injective function \(A \rightarrow B\).}} | |
| Extra | Example: \(f(x): \mathbb{N} \rightarrow \mathbb{R} = x\) |
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A set \(A\) is called countable if and only if {{c1::it is finite or\(A \sim \mathbb{N}\)}}, and uncountable otherwise.
Back
A set \(A\) is called countable if and only if {{c1::it is finite or\(A \sim \mathbb{N}\)}}, and uncountable otherwise.
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|---|---|---|
| Text | A set \(A\) is called <b>countable </b>if and only if {{c1::it is finite or\(A \sim \mathbb{N}\)}}, and <b>uncountable</b> otherwise. |
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Is the set \(\{0,1\}^*\) (finite binary sequences) countable?
Back
Is the set \(\{0,1\}^*\) (finite binary sequences) countable?
Yes. A possible injection to \(\mathbb{N}\) is to add a "1" at the beginning of each sequence and interpret it in binary.
Yes. A possible injection to \(\mathbb{N}\) is to add a "1" at the beginning of each sequence and interpret it in binary.
Field-by-field Comparison
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|---|---|---|
| Front | Is the set \(\{0,1\}^*\) (finite binary sequences) countable? | |
| Back | Yes. A possible injection to \(\mathbb{N}\) is to add a "1" at the beginning of each sequence and interpret it in binary. |
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Is the set \(\{0,1\}^\infty\) (semi-infinite binary sequences) countable?
Back
Is the set \(\{0,1\}^\infty\) (semi-infinite binary sequences) countable?
No. This can be proven by Cantor's diagonalization argument.
No. This can be proven by Cantor's diagonalization argument.
Field-by-field Comparison
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|---|---|---|
| Front | Is the set \(\{0,1\}^\infty\) (semi-infinite binary sequences) countable? | |
| Back | No. This can be proven by Cantor's diagonalization argument. |
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\(a \mid b\) means that {{c1::\(a\) divides \(b\), that is, there exists a \(c \in \mathbb{Z}\) such that \(b = ac\)}}
Back
\(a \mid b\) means that {{c1::\(a\) divides \(b\), that is, there exists a \(c \in \mathbb{Z}\) such that \(b = ac\)}}
\(\forall a \ne 0 \rightarrow a \mid 0\) and \(\forall a \quad 1 \mid a \land -1 \mid a\)
\(\forall a \ne 0 \rightarrow a \mid 0\) and \(\forall a \quad 1 \mid a \land -1 \mid a\)
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|---|---|---|
| Text | \(a \mid b\) means that {{c1::\(a\) divides \(b\), that is, there exists a \(c \in \mathbb{Z}\) such that \(b = ac\)}}<br> | |
| Extra | \(\forall a \ne 0 \rightarrow a \mid 0\) and \(\forall a \quad 1 \mid a \land -1 \mid a\)<br> |
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\(a \mod m\) is the same as \(R_m(a)\)
Back
\(a \mod m\) is the same as \(R_m(a)\)
Field-by-field Comparison
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|---|---|---|
| Text | {{c1::\(a \mod m\)}} is the same as {{c2::\(R_m(a)\)}}<br> |
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\(a \equiv_m b \stackrel{\text{def}}{\iff}\) \(m \mid (a-b)\)
Back
\(a \equiv_m b \stackrel{\text{def}}{\iff}\) \(m \mid (a-b)\)
Field-by-field Comparison
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| Text | \(a \equiv_m b \stackrel{\text{def}}{\iff}\) {{c1::\(m \mid (a-b)\)}}<br> |
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The equation \(ax \equiv_m 1\) has a unique solution \(x \in \mathbb{Z}_m\) if and only if \(\gcd(a,m) = 1\). This \(x\) is then called the multiplicative inverse of \(a \mod m\).
Back
The equation \(ax \equiv_m 1\) has a unique solution \(x \in \mathbb{Z}_m\) if and only if \(\gcd(a,m) = 1\). This \(x\) is then called the multiplicative inverse of \(a \mod m\).
Field-by-field Comparison
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|---|---|---|
| Text | The equation \(ax \equiv_m 1\) has a unique solution \(x \in \mathbb{Z}_m\) if and only if {{c1::\(\gcd(a,m) = 1\).}} This \(x\) is then called the {{c2::multiplicative inverse of \(a \mod m\)}}. |
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The direct product of \(n\) groups \(\langle G_1; *_1 \rangle, \ldots, \langle G_n; *_n \rangle\) is the algebra \(\langle G_1 \times \cdots \times G_n; \star\rangle\). The operation \(\star\) is component-wise.
Back
The direct product of \(n\) groups \(\langle G_1; *_1 \rangle, \ldots, \langle G_n; *_n \rangle\) is the algebra \(\langle G_1 \times \cdots \times G_n; \star\rangle\). The operation \(\star\) is component-wise.
Field-by-field Comparison
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|---|---|---|
| Text | The direct product of \(n\) groups \(\langle G_1; *_1 \rangle, \ldots, \langle G_n; *_n \rangle\) is {{c1::the algebra \(\langle G_1 \times \cdots \times G_n; \star\rangle\)}}. The operation \(\star\) is component-wise. |
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A subgroup \(H\) of a group \(G\) is a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, inverses exist).
Back
A subgroup \(H\) of a group \(G\) is a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, inverses exist).
Trivial subgroups: \(\{e\}, G\)
Trivial subgroups: \(\{e\}, G\)
Field-by-field Comparison
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|---|---|---|
| Text | A subgroup \(H\) of a group \(G\) is {{c1::a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, <b>inverses</b> exist).}} | |
| Extra | Trivial subgroups: \(\{e\}, G\) |
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The order of an element \(a\) in a group (denoted \(\text{ord}(a)\)) is {{c1::the smallest \(m \ge 1\) such that \(a^m = e\). If such an \(m\) does not exist, \(\text{ord}(a) = \infty\)}}
Back
The order of an element \(a\) in a group (denoted \(\text{ord}(a)\)) is {{c1::the smallest \(m \ge 1\) such that \(a^m = e\). If such an \(m\) does not exist, \(\text{ord}(a) = \infty\)}}
\(\text{ord}(e) = 1\) in any group
\(\text{ord}(e) = 1\) in any group
Field-by-field Comparison
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|---|---|---|
| Text | The order of an element \(a\) in a group (denoted \(\text{ord}(a)\)) is {{c1::the smallest \(m \ge 1\) such that \(a^m = e\). If such an \(m\) does not exist, \(\text{ord}(a) = \infty\)}} | |
| Extra | \(\text{ord}(e) = 1\) in any group<br> |
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For a finite group \(G\), we call \(|G|\) the order of \(G\).
Back
For a finite group \(G\), we call \(|G|\) the order of \(G\).
Field-by-field Comparison
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| Text | For a finite group \(G\), we call \(|G|\) the {{c1::order of \(G\)}}. |
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A group \(G = \langle g \rangle\) generated by an element \(g \in G\) is called cyclic, and \(g\) is called a generator of \(G\).
Back
A group \(G = \langle g \rangle\) generated by an element \(g \in G\) is called cyclic, and \(g\) is called a generator of \(G\).
Examples:
\(\langle \mathbb{Z}_n;\oplus\rangle\) (cyclic for every \(n\), 1 is a generator)
\(\langle\mathbb{Z}_n; +,-,0\rangle\)(infinite cyclic group with generators 1 and -1)
Examples:
\(\langle \mathbb{Z}_n;\oplus\rangle\) (cyclic for every \(n\), 1 is a generator)
\(\langle\mathbb{Z}_n; +,-,0\rangle\)(infinite cyclic group with generators 1 and -1)
Field-by-field Comparison
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|---|---|---|
| Text | A group \(G = \langle g \rangle\) generated by an element \(g \in G\) is called {{c1::cyclic}}, and \(g\) is called {{c1::a <b>generator</b> of \(G\)}}. | |
| Extra | Examples:<br>\(\langle \mathbb{Z}_n;\oplus\rangle\) (cyclic for every \(n\), 1 is a generator)<br>\(\langle\mathbb{Z}_n; +,-,0\rangle\)(infinite cyclic group with generators 1 and -1) |
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Lagrange's theorem: If \(G\) is a finite group and \(H\) is a subgroup, then the order of \(H\) divides the order of \(G\), i.e. \(|H|\) divides \(|G|\).
Back
Lagrange's theorem: If \(G\) is a finite group and \(H\) is a subgroup, then the order of \(H\) divides the order of \(G\), i.e. \(|H|\) divides \(|G|\).
Field-by-field Comparison
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|---|---|---|
| Text | Lagrange's theorem: If \(G\) is a finite group and \(H\) is a subgroup, then {{c1::the order of \(H\) divides the order of \(G\), i.e. \(|H|\) divides \(|G|\).}} |
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The group \(\mathbb{Z}^*_m\) contains all numbers \(a \in \mathbb{Z}_m\) that are coprime to \(m\), that is, \(\gcd(a,m) = 1\).
Back
The group \(\mathbb{Z}^*_m\) contains all numbers \(a \in \mathbb{Z}_m\) that are coprime to \(m\), that is, \(\gcd(a,m) = 1\).
As they are coprime, they are invertible. Thus its the set of units.
As they are coprime, they are invertible. Thus its the set of units.
Field-by-field Comparison
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|---|---|---|
| Text | The group \(\mathbb{Z}^*_m\) contains all numbers \(a \in \mathbb{Z}_m\) that are {{c1::coprime to \(m\), that is, \(\gcd(a,m) = 1\).}} | |
| Extra | As they are coprime, they are invertible. Thus its the set of units. |
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The Euler function \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) (also called Euler's totient function) is defined as {{c1::the cardinality of \(\mathbb{Z}^*_m\).}}
Back
The Euler function \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) (also called Euler's totient function) is defined as {{c1::the cardinality of \(\mathbb{Z}^*_m\).}}
Example: \(\mathbb{Z}_{18}^* = \{1,5,7,11,13,17\}\), so \(\varphi(18) = 6\)
Example: \(\mathbb{Z}_{18}^* = \{1,5,7,11,13,17\}\), so \(\varphi(18) = 6\)
Field-by-field Comparison
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|---|---|---|
| Text | The Euler function \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) (also called Euler's totient function) is defined as {{c1::the cardinality of \(\mathbb{Z}^*_m\).}} | |
| Extra | Example: \(\mathbb{Z}_{18}^* = \{1,5,7,11,13,17\}\), so \(\varphi(18) = 6\) |
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A ring \(\langle R, +, -, 0, \cdot, 1 \rangle\) is an algebra with the properties that
- \(\langle R, +, -, 0 \rangle\) is a commutative group
- \(\langle R, \cdot, 1 \rangle\) is a monoid
- \( a(b+c) = (ab) + (ac), (b+c)a = (ba) + (ca)\) (left and right distributive laws)
Back
A ring \(\langle R, +, -, 0, \cdot, 1 \rangle\) is an algebra with the properties that
Examples: \(\mathbb{Z}, \mathbb{R}\)
- \(\langle R, +, -, 0 \rangle\) is a commutative group
- \(\langle R, \cdot, 1 \rangle\) is a monoid
- \( a(b+c) = (ab) + (ac), (b+c)a = (ba) + (ca)\) (left and right distributive laws)
Examples: \(\mathbb{Z}, \mathbb{R}\)
Field-by-field Comparison
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|---|---|---|
| Text | {{c1::A ring \(\langle R, +, -, 0, \cdot, 1 \rangle\)}} is an algebra with the properties that<br><ul><li>{{c2::\(\langle R, +, -, 0 \rangle\) is a commutative group}}<br></li><li>{{c3::\(\langle R, \cdot, 1 \rangle\) is a monoid}}</li><li>{{c4::\( a(b+c) = (ab) + (ac), (b+c)a = (ba) + (ca)\) (left and right distributive laws)}}</li></ul> | |
| Extra | Examples: \(\mathbb{Z}, \mathbb{R}\) |
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A ring is called commutative if \(ab = ba\).
Back
A ring is called commutative if \(ab = ba\).
Field-by-field Comparison
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|---|---|---|
| Text | A ring is called {{c1::commutative}} if {{c2::\(ab = ba\).}} |
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An element \(u\) of a ring is called a unit if \(u\) {{c2::is invertible, so \(uu^{-1} = u^{-1}u = 1\).}}
Back
An element \(u\) of a ring is called a unit if \(u\) {{c2::is invertible, so \(uu^{-1} = u^{-1}u = 1\).}}
Example: Units of \(\mathbb{Z}\) are \(-1, 1\)
Example: Units of \(\mathbb{Z}\) are \(-1, 1\)
Field-by-field Comparison
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|---|---|---|
| Text | An element \(u\) of a ring is called a {{c1::unit}} if \(u\) {{c2::is invertible, so \(uu^{-1} = u^{-1}u = 1\).}} | |
| Extra | Example: Units of \(\mathbb{Z}\) are \(-1, 1\) |
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The set of units of \(R\) is denoted by \(R^*\)
Back
The set of units of \(R\) is denoted by \(R^*\)
Field-by-field Comparison
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| Text | The set of units of \(R\) is denoted by {{c1::\(R^*\)}} |
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An integral domain is a commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).
Back
An integral domain is a commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).
Examples: \(\mathbb{Z}, \mathbb{R}\)
Examples: \(\mathbb{Z}, \mathbb{R}\)
Counterexample: \(\mathbb{Z}_m, m\) not prime
Field-by-field Comparison
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|---|---|---|
| Text | An {{c1::integral domain}} is a {{c2::commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).}} | |
| Extra | Examples: \(\mathbb{Z}, \mathbb{R}\)<div>Counterexample: \(\mathbb{Z}_m, m\) not prime</div> |
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A field (Körper) is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
Back
A field (Körper) is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)
Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)
Field-by-field Comparison
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|---|---|---|
| Text | A {{c1::field (<i>Körper</i>)}} is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}} | |
| Extra | Example: \(\mathbb{R}\), but not \(\mathbb{Z}\) |
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\(\mathbb{Z}_p\) is a field if and only if \(p\) is prime.
Back
\(\mathbb{Z}_p\) is a field if and only if \(p\) is prime.
Field-by-field Comparison
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|---|---|---|
| Text | \(\mathbb{Z}_p\) is a field if and only if {{c1::\(p\) is prime.}}<br> |
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A root (also: zero) of \(a(x) \in \mathbb{R}[x]\) is {{c2::an element \(y \in \mathbb{R}\) for which \(a(y) = 0\).}}
Back
A root (also: zero) of \(a(x) \in \mathbb{R}[x]\) is {{c2::an element \(y \in \mathbb{R}\) for which \(a(y) = 0\).}}
Example: \(x^4 + x^3 + x + 1\) in GF(2)\([x]\) has root 1.
Example: \(x^4 + x^3 + x + 1\) in GF(2)\([x]\) has root 1.
Field-by-field Comparison
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|---|---|---|
| Text | A {{c1::root (also: zero)}} of \(a(x) \in \mathbb{R}[x]\) is {{c2::an element \(y \in \mathbb{R}\) for which \(a(y) = 0\).}} | |
| Extra | Example: \(x^4 + x^3 + x + 1\) in GF(2)\([x]\) has root 1. |
Note 269: ETH::DiskMat
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In a field F, \(y \in F\) is a root of \(a(x)\) if and only if \(x - y\) divides \(a(x)\) or \(a(y) = 0\)
Back
In a field F, \(y \in F\) is a root of \(a(x)\) if and only if \(x - y\) divides \(a(x)\) or \(a(y) = 0\)
Field-by-field Comparison
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|---|---|---|
| Text | In a field F, \(y \in F\) is a root of \(a(x)\) if and only if {{c1::\(x - y\) divides \(a(x)\) or \(a(y) = 0\)}} |
Note 270: ETH::DiskMat
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The idea of Universal Instantiation is that if a statement is true for all elements, it is also true for a particular element, so \(\forall x F \models F[x/t]\).
Back
The idea of Universal Instantiation is that if a statement is true for all elements, it is also true for a particular element, so \(\forall x F \models F[x/t]\).
Example: All elements in \(\mathbb{R}\) are invertible. Thus, 2 is also invertible.
Example: All elements in \(\mathbb{R}\) are invertible. Thus, 2 is also invertible.
Field-by-field Comparison
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|---|---|---|
| Text | The idea of {{c2::Universal Instantiation}} is that {{c1::if a statement is true for all elements, it is also true for a particular element, so \(\forall x F \models F[x/t]\).}} | |
| Extra | Example: All elements in \(\mathbb{R}\) are invertible. Thus, 2 is also invertible. |
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Predicate logic: A formula in prenex form has all quantifiers in front and none afterwards.
Back
Predicate logic: A formula in prenex form has all quantifiers in front and none afterwards.
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|---|---|---|
| Text | Predicate logic: A formula in {{c2::prenex form}} has {{c1::all quantifiers in front and none afterwards. }} |
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A mathematical statement (also proposition) is a statement that is true or false in a mathematical sense.
Back
A mathematical statement (also proposition) is a statement that is true or false in a mathematical sense.
Example: 5 is a prime number.
Example: 5 is a prime number.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A <i>mathematical statement</i> (also <i>proposition</i>) is {{c1::a statement that is true or false in a mathematical sense}}. | |
| Extra | Example: 5 is a prime number. |
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A mathematical statement not known, but believed, to be true or false is called conjecture.
Back
A mathematical statement not known, but believed, to be true or false is called conjecture.
Example: Collatz conjecture.
Example: Collatz conjecture.
Field-by-field Comparison
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|---|---|---|
| Text | A mathematical statement not known, but believed, to be true or false is called {{c1::<i>conjecture</i>}}. | |
| Extra | Example: Collatz conjecture. |
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An axiom or postulate is a statement that is taken to be true.
Back
An axiom or postulate is a statement that is taken to be true.
Example: All right angles are equal to each other.
Example: All right angles are equal to each other.
Field-by-field Comparison
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|---|---|---|
| Text | An <i>axiom</i> or <i>postulate</i> is {{c1::a statement that is taken to be true}}. | |
| Extra | Example: All right angles are equal to each other. |
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An expression using the propositional symbols \(A, B, C, \dots\) and logical operators \(\land, \lor, \lnot, \ldots\) is called a formula (of propositional logic).
Back
An expression using the propositional symbols \(A, B, C, \dots\) and logical operators \(\land, \lor, \lnot, \ldots\) is called a formula (of propositional logic).
Field-by-field Comparison
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|---|---|---|
| Text | An {{c2::expression using the propositional symbols \(A, B, C, \dots\) and logical operators \(\land, \lor, \lnot, \ldots\)}} is called a {{c1::<i>formula</i> (of propositional logic)}}. |
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In propositional logic, a formula \(G\) is a logical consequence of a formula \(F\) if for all truth assignments to the propositional symbols appearing in \(F\) or \(G\), the truth value of \(G\) is \(1\) if the truth value of \(F\) is \(1\). This is denoted with \(F \models G\).
Back
In propositional logic, a formula \(G\) is a logical consequence of a formula \(F\) if for all truth assignments to the propositional symbols appearing in \(F\) or \(G\), the truth value of \(G\) is \(1\) if the truth value of \(F\) is \(1\). This is denoted with \(F \models G\).
Example: \(A \land B \models A \lor B\)
Example: \(A \land B \models A \lor B\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In <b>propositional logic</b>, a formula \(G\) is a <i>logical consequence</i> of a formula \(F\) {{c1::if for all truth assignments to the propositional symbols appearing in \(F\) or \(G\), the truth value of \(G\) is \(1\) if the truth value of \(F\) is \(1\)}}. This is denoted with \(F \models G\). | |
| Extra | Example: \(A \land B \models A \lor B\) |
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\(\models F\) means that \(F\) is a tautology.
Back
\(\models F\) means that \(F\) is a tautology.
Field-by-field Comparison
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|---|---|---|
| Text | \(\models F\) means that \(F\) is a {{c1::tautology}}. |
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A \(k\)-ary predicate \(P\) on \(U\) is a {{c1::function \(U^k \to \{0, 1\}\)}}.
Back
A \(k\)-ary predicate \(P\) on \(U\) is a {{c1::function \(U^k \to \{0, 1\}\)}}.
Example: \(\text{prime}(x)=\begin{cases}1 & \text{if } x \text{ is prime}\\0 & \text{else}\end{cases}\)
Example: \(\text{prime}(x)=\begin{cases}1 & \text{if } x \text{ is prime}\\0 & \text{else}\end{cases}\)
Field-by-field Comparison
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|---|---|---|
| Text | A \(k\)-ary <i>predicate</i> \(P\) on \(U\) is a {{c1::function \(U^k \to \{0, 1\}\)}}. | |
| Extra | Example: \(\text{prime}(x)=\begin{cases}1 & \text{if } x \text{ is prime}\\0 & \text{else}\end{cases}\) |
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A logical formula is generally not a mathematical statement, because the truth value depends on the interpretation of the symbols.
Back
A logical formula is generally not a mathematical statement, because the truth value depends on the interpretation of the symbols.
(so we can't prove/disprove it)
(so we can't prove/disprove it)
Field-by-field Comparison
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|---|---|---|
| Text | A logical formula is generally <i>not</i> a mathematical statement, because {{c1::the truth value depends on the interpretation of the symbols}}. | |
| Extra | (so we can't prove/disprove it) |
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A proof of \(S\) by case distinction has three steps:
- Find a finite list \(R_1,\ldots,R_k\) of mathematical statements, the cases.
- Prove that at least one of the \(R_i\) is true (at least one case occurs).
- Prove \(R_i \implies S\) for \(i = 1,\ldots,k\).
Back
A proof of \(S\) by case distinction has three steps:
- Find a finite list \(R_1,\ldots,R_k\) of mathematical statements, the cases.
- Prove that at least one of the \(R_i\) is true (at least one case occurs).
- Prove \(R_i \implies S\) for \(i = 1,\ldots,k\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A proof of \(S\) by <i>case distinction</i> has three steps:<br><ol><li>{{c1::Find a finite list \(R_1,\ldots,R_k\) of mathematical statements, the cases.}}<br></li><li>{{c2::Prove that at least one of the \(R_i\) is true (at least one case occurs).}}<br></li><li>{{c3::Prove \(R_i \implies S\) for \(i = 1,\ldots,k\).}}<br></li></ol> |
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Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(\{ A \mid A \notin A\}\)}}.
Back
Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(\{ A \mid A \notin A\}\)}}.
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| Text | Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(\{ A \mid A \notin A\}\)}}. |
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\(2^A\) is an alternatively used notation that denotes {{c1::the power set of \(A\), so \(\mathcal{P}(A))\)}}.
Back
\(2^A\) is an alternatively used notation that denotes {{c1::the power set of \(A\), so \(\mathcal{P}(A))\)}}.
Field-by-field Comparison
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|---|---|---|
| Text | \(2^A\) is an alternatively used notation that denotes {{c1::the power set of \(A\), so \(\mathcal{P}(A))\)}}. |
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The Cartesian product \(A \times B\) of sets \(A, B\) is {{c1::the set of all ordered pairs with the first component from \(A\) and the second component from \(B\): \(A\times B = \{(a,b)\mid a\in A \land b \in B\}\)}}.
Back
The Cartesian product \(A \times B\) of sets \(A, B\) is {{c1::the set of all ordered pairs with the first component from \(A\) and the second component from \(B\): \(A\times B = \{(a,b)\mid a\in A \land b \in B\}\)}}.
Field-by-field Comparison
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|---|---|---|
| Text | The <b>Cartesian product </b>\(A \times B\) of sets \(A, B\) is {{c1::the set of all ordered pairs with the first component from \(A\) and the second component from \(B\): \(A\times B = \{(a,b)\mid a\in A \land b \in B\}\)}}. |
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A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of \(A\times B\). If \(A = B\), then \(\rho\) is called a relation on \(A\).
Back
A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of \(A\times B\). If \(A = B\), then \(\rho\) is called a relation on \(A\).
Field-by-field Comparison
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|---|---|---|
| Text | A <b>relation </b>\(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is {{c1::a subset of \(A\times B\).}} If \(A = B\), then \(\rho\) is called {{c1::a <i>relation on</i> \(A\).}} |
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Name four examples for (binary) relations as defined in Discrete Mathematics.
Back
Name four examples for (binary) relations as defined in Discrete Mathematics.
\(=, \ne, \le, \ge, <, >, \mid, \dots\)
\(=, \ne, \le, \ge, <, >, \mid, \dots\)
Field-by-field Comparison
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|---|---|---|
| Front | Name four examples for (binary) relations as defined in Discrete Mathematics. | |
| Back | \(=, \ne, \le, \ge, <, >, \mid, \dots\) |
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The transitive closure of a relation \(\rho\) on a set \(A\), denoted \(\rho^*\), is defined as {{c1::\(\rho^* = \bigcup_{n\in\mathbb{N}\setminus \{0\} } \rho^n\)}}.
Back
The transitive closure of a relation \(\rho\) on a set \(A\), denoted \(\rho^*\), is defined as {{c1::\(\rho^* = \bigcup_{n\in\mathbb{N}\setminus \{0\} } \rho^n\)}}.
Field-by-field Comparison
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|---|---|---|
| Text | The <b>transitive closure </b>of a relation \(\rho\) on a set \(A\), denoted \(\rho^*\), is defined as {{c1::\(\rho^* = \bigcup_{n\in\mathbb{N}\setminus \{0\} } \rho^n\)}}. |
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For a poset \((A;\preceq)\), two elements \(a,b\) are comparable if \(a \preceq b\) or \(b \preceq a\), otherwise they are incomparable.
Back
For a poset \((A;\preceq)\), two elements \(a,b\) are comparable if \(a \preceq b\) or \(b \preceq a\), otherwise they are incomparable.
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|---|---|---|
| Text | For a poset \((A;\preceq)\), two elements \(a,b\) are <b>comparable</b> if {{c1::\(a \preceq b\) or \(b \preceq a\),}} otherwise they are <b>incomparable</b>. |
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The Hasse diagram of a poset \((A; \preceq)\) is the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).
Back
The Hasse diagram of a poset \((A; \preceq)\) is the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).
Field-by-field Comparison
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|---|---|---|
| Text | The <i>Hasse diagram</i> of a poset \((A; \preceq)\) is {{c1::the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).}} |
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A function \(f: A\to B\) from a domain \(A\) to a codomain \(B\) is a relation from \(A\) to \(B\) with the special properties:
{{c1::1. (totally defined) \(\forall a\in A \; \exists b \in B \quad a \mathop{f} b\)
2. (well-defined) \(\forall a\in A \; \forall b, b' \in B \quad (a \mathop{f} b \land a\mathop{f}b' \to b = b')\)}}
{{c1::1. (totally defined) \(\forall a\in A \; \exists b \in B \quad a \mathop{f} b\)
2. (well-defined) \(\forall a\in A \; \forall b, b' \in B \quad (a \mathop{f} b \land a\mathop{f}b' \to b = b')\)}}
Back
A function \(f: A\to B\) from a domain \(A\) to a codomain \(B\) is a relation from \(A\) to \(B\) with the special properties:
{{c1::1. (totally defined) \(\forall a\in A \; \exists b \in B \quad a \mathop{f} b\)
2. (well-defined) \(\forall a\in A \; \forall b, b' \in B \quad (a \mathop{f} b \land a\mathop{f}b' \to b = b')\)}}
{{c1::1. (totally defined) \(\forall a\in A \; \exists b \in B \quad a \mathop{f} b\)
2. (well-defined) \(\forall a\in A \; \forall b, b' \in B \quad (a \mathop{f} b \land a\mathop{f}b' \to b = b')\)}}
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A <b>function</b> \(f: A\to B\) from a <i>domain</i> \(A\) to a <i>codomain</i> \(B\) is {{c1::a relation from \(A\) to \(B\)}} with the special properties:<br>{{c1::1. (totally defined) \(\forall a\in A \; \exists b \in B \quad a \mathop{f} b\)<br>2. (well-defined) \(\forall a\in A \; \forall b, b' \in B \quad (a \mathop{f} b \land a\mathop{f}b' \to b = b')\)}} |
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The set of all functions \(A\to B\) is denoted as \(B^A\).
Back
The set of all functions \(A\to B\) is denoted as \(B^A\).
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|---|---|---|
| Text | The set of all functions \(A\to B\) is denoted as {{c1::\(B^A\).}} |
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A partial function \(A \to B\) is a relation from \(A\) to \(B\) such that{{c1::\(\forall a \in A \; \forall b,b' \in B \; (a \mathop{f} b \land a\mathop{f} b' \to b = b')\) (well-defined).}}
Back
A partial function \(A \to B\) is a relation from \(A\) to \(B\) such that{{c1::\(\forall a \in A \; \forall b,b' \in B \; (a \mathop{f} b \land a\mathop{f} b' \to b = b')\) (well-defined).}}
Field-by-field Comparison
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|---|---|---|
| Text | A <b>partial function </b>\(A \to B\) is a relation from \(A\) to \(B\) such that{{c1::\(\forall a \in A \; \forall b,b' \in B \; (a \mathop{f} b \land a\mathop{f} b' \to b = b')\) (well-defined).}} |
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A function \(f:\mathbb{N}\to\{0,1\}\) is called computable if {{c1::there is a computer program that, for every \(n\in\mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).}}
Back
A function \(f:\mathbb{N}\to\{0,1\}\) is called computable if {{c1::there is a computer program that, for every \(n\in\mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).}}
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A function \(f:\mathbb{N}\to\{0,1\}\) is called <b>computable</b> if {{c1::there is a computer program that, for every \(n\in\mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).}} |
Note 293: ETH::DiskMat
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There are uncomputable functions \(\mathbb{N} \to \{0, 1\}\) because {{c1::the set of functions \(\mathbb{N} \to \{0, 1\}\) is uncountable (Cantor's diagonalization argument), but the set of programs \(\{0, 1\}^*\) computing them is countable.}}
Back
There are uncomputable functions \(\mathbb{N} \to \{0, 1\}\) because {{c1::the set of functions \(\mathbb{N} \to \{0, 1\}\) is uncountable (Cantor's diagonalization argument), but the set of programs \(\{0, 1\}^*\) computing them is countable.}}
Field-by-field Comparison
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|---|---|---|
| Text | There are <i>uncomputable functions</i> \(\mathbb{N} \to \{0, 1\}\) because {{c1::the set of functions \(\mathbb{N} \to \{0, 1\}\) is uncountable (<i>Cantor's diagonalization argument</i>), but the set of programs \(\{0, 1\}^*\) computing them is countable.}} |
Note 294: ETH::DiskMat
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A prominent example for an uncomputable function is the Halting problem.
Back
A prominent example for an uncomputable function is the Halting problem.
Field-by-field Comparison
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|---|---|---|
| Text | A prominent example for an uncomputable function is {{c1::the <i>Halting problem</i>}}<i>.</i> |
Note 295: ETH::DiskMat
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Sketch step-by-step how Cantor's diagonalization argument can be used to prove that the set \(\{0,1\}^\infty\) is uncountable.
Back
Sketch step-by-step how Cantor's diagonalization argument can be used to prove that the set \(\{0,1\}^\infty\) is uncountable.
- Proof by contradiction: Assume a bijection to \(\mathbb{N}\) exists.
- That means there exists for each \(n\in \mathbb{N}\) a corresponding sequence of 0 and 1s, and vice-versa.
- We now construct a new sequence \(\alpha\) of 0s and 1s, by always taking the \(i\)-th bit from the \(i\)-th sequence, and inverting it.
- This new sequence does not agree with every existing sequence in at least one place.
- However, there is no \(n\in\mathbb{N}\) such that \(\alpha = f(n)\) since \(\alpha\) disagrees with every \(f(n)\) in at least one place.
- Thus, no bijection to \(\mathbb{N}\) exists, which means \(\{0,1\}^\infty\) is uncountable.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Sketch step-by-step how <b>Cantor's diagonalization argument</b> can be used to prove that the set \(\{0,1\}^\infty\) is uncountable. | |
| Back | <ul><li>Proof by contradiction: Assume a bijection to \(\mathbb{N}\) exists.</li><li>That means there exists for each \(n\in \mathbb{N}\) a corresponding sequence of 0 and 1s, and vice-versa.</li><li>We now construct a new sequence \(\alpha\) of 0s and 1s, by always taking the \(i\)-th bit from the \(i\)-th sequence, and inverting it.</li><li>This new sequence does not agree with every existing sequence in at least one place.</li><li>However, there is no \(n\in\mathbb{N}\) such that \(\alpha = f(n)\) since \(\alpha\) disagrees with every \(f(n)\) in at least one place.</li><li>Thus, no bijection to \(\mathbb{N}\) exists, which means \(\{0,1\}^\infty\) is uncountable.</li></ul> |
Note 296: ETH::DiskMat
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An integer greater than \(1\) that is not a prime is called composite.
Back
An integer greater than \(1\) that is not a prime is called composite.
Field-by-field Comparison
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|---|---|---|
| Text | An integer greater than \(1\) that is not a prime is called {{c1::composite}}. |
Note 297: ETH::DiskMat
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What is the fundamental theorem of arithmetic?
Back
What is the fundamental theorem of arithmetic?
Every positive integer can be written uniquely as the product of primes.
Every positive integer can be written uniquely as the product of primes.
Field-by-field Comparison
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|---|---|---|
| Front | What is the <i>fundamental theorem of arithmetic</i>? | |
| Back | Every positive integer can be written uniquely as the product of primes. |
Note 298: ETH::DiskMat
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It follow from the respective definitions that \(\gcd(a,b) \times \text{lcm}(a,b) =\) \(ab\).
Back
It follow from the respective definitions that \(\gcd(a,b) \times \text{lcm}(a,b) =\) \(ab\).
Field-by-field Comparison
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|---|---|---|
| Text | It follow from the respective definitions that \(\gcd(a,b) \times \text{lcm}(a,b) =\) {{c1:: \(ab\)}}. |
Note 299: ETH::DiskMat
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For \(a,b,m\in\mathbb{Z}\) with \(m\ge1\), we say that \(a\) is congruent to \(b\) modulo \(m\) if \(m\) divides \(a-b\). Written as an expression: \(a\equiv_mb \iff m \mid (a-b)\).
Back
For \(a,b,m\in\mathbb{Z}\) with \(m\ge1\), we say that \(a\) is congruent to \(b\) modulo \(m\) if \(m\) divides \(a-b\). Written as an expression: \(a\equiv_mb \iff m \mid (a-b)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | For \(a,b,m\in\mathbb{Z}\) with \(m\ge1\), we say that \(a\) is <i>congruent to </i>\(b\) <i>modulo </i>\(m\) if {{c1:: \(m\) divides \(a-b\)}}. Written as an expression:{{c1:: \(a\equiv_mb \iff m \mid (a-b)\).}} |
Note 300: ETH::DiskMat
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A group or monoid \(\langle G;* \rangle\) is called commutative or abelian if \(a * b = b * a\) for all \(a,b \in G\).
Back
A group or monoid \(\langle G;* \rangle\) is called commutative or abelian if \(a * b = b * a\) for all \(a,b \in G\).
Field-by-field Comparison
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|---|---|---|
| Text | A group or monoid \(\langle G;* \rangle\) is called <i>commutative</i> or <i>abelian</i> if {{c1::\(a * b = b * a\) for all \(a,b \in G\)}}. |
Note 301: ETH::DiskMat
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For two groups \(\langle G;*;\widehat{};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a group homomorphism if for all \(a\) and \(b\):
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
Back
For two groups \(\langle G;*;\widehat{};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a group homomorphism if for all \(a\) and \(b\):
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | For two groups \(\langle G;*;\widehat{};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a <i>group homomorphism</i> if for all \(a\) and \(b\):<br>{{c1::\(\psi(a*b) = \psi(a)\star\psi(b)\)}}.<br>If \(\psi\) is {{c2::a bijection from \(G\) to \(H\)}}, then it is called an <i>isomorphism</i>. |
Note 302: ETH::DiskMat
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If for two groups \(G\) and \(H\) there is a function \(\psi: G\to H\) which is an isomorphism, then we say that \(G\) and \(H\) are isomorphic and we write this as \(G \simeq H\).
Back
If for two groups \(G\) and \(H\) there is a function \(\psi: G\to H\) which is an isomorphism, then we say that \(G\) and \(H\) are isomorphic and we write this as \(G \simeq H\).
Field-by-field Comparison
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|---|---|---|
| Text | If for two groups \(G\) and \(H\) there is a function \(\psi: G\to H\) which is an isomorphism, then we say that {{c1::\(G\) and \(H\) are <i>isomorphic</i>}} and we write this as {{c1::\(G \simeq H\)}}. |
Note 303: ETH::DiskMat
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Diffie-Hellman is used to securely create a shared secret between two parties over a public channel.
Back
Diffie-Hellman is used to securely create a shared secret between two parties over a public channel.
Field-by-field Comparison
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|---|---|---|
| Text | Diffie-Hellman is used to {{c1::securely create a shared secret between two parties over a public channel::do what?}}. |
Note 304: ETH::DiskMat
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Both RSA and Diffie-Hellman use modular exponentiation for their main operation.
Back
Both RSA and Diffie-Hellman use modular exponentiation for their main operation.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Both RSA and Diffie-Hellman use {{c1::modular exponentiation}} for their main operation. |
Note 305: ETH::DiskMat
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Explain the mechanical analog of the Diffie-Hellman protocol.
Back
Explain the mechanical analog of the Diffie-Hellman protocol.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Explain the mechanical analog of the Diffie-Hellman protocol. | |
| Back | <img src="paste-39931b24c512906843c903f461b7c1cc9f5a6685.jpg"> |
Note 306: ETH::DiskMat
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An operation on a set \(S\) is a function \(S^n \to S\), where \(n \ge 0\) is called the arity of the operation.
Back
An operation on a set \(S\) is a function \(S^n \to S\), where \(n \ge 0\) is called the arity of the operation.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | An <i>operation</i> on a set \(S\) is {{c1::a function \(S^n \to S\), where \(n \ge 0\) is called the <i>arity</i> of the operation::what (include arity)?}}. |
Note 307: ETH::DiskMat
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A cyclic group of order \(n\) {{c1::is isomorphic to \(\langle \mathbb{Z}_n,\oplus)\), and hence commutative::has which useful property?}}.
Back
A cyclic group of order \(n\) {{c1::is isomorphic to \(\langle \mathbb{Z}_n,\oplus)\), and hence commutative::has which useful property?}}.
Field-by-field Comparison
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|---|---|---|
| Text | A cyclic group of order \(n\) {{c1::is isomorphic to \(\langle \mathbb{Z}_n,\oplus)\), and hence commutative::has which useful property?}}. |
Note 308: ETH::DiskMat
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\(\mathbb{Z}_m^* \stackrel{\text{def}}{=}\) {{c1::\(\{a\in \mathbb{Z}_m \mid \gcd(a,m) = 1\}\)}}
Back
\(\mathbb{Z}_m^* \stackrel{\text{def}}{=}\) {{c1::\(\{a\in \mathbb{Z}_m \mid \gcd(a,m) = 1\}\)}}
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | \(\mathbb{Z}_m^* \stackrel{\text{def}}{=}\) {{c1::\(\{a\in \mathbb{Z}_m \mid \gcd(a,m) = 1\}\)}} |
Note 309: ETH::DiskMat
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\(\mathbb{Z}_m^*\) is useful compared to \(\mathbb{Z}_m\)because {{c1::\(\mathbb{Z}_m\)is not a group with respect to multiplication modulo \(m\), but we would like to have this for building RSA}}.
Back
\(\mathbb{Z}_m^*\) is useful compared to \(\mathbb{Z}_m\)because {{c1::\(\mathbb{Z}_m\)is not a group with respect to multiplication modulo \(m\), but we would like to have this for building RSA}}.
Not all element in Zm have an inverse, something which Zm* guarantees by bezout.
Not all element in Zm have an inverse, something which Zm* guarantees by bezout.
Field-by-field Comparison
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|---|---|---|
| Text | \(\mathbb{Z}_m^*\) is useful compared to \(\mathbb{Z}_m\)because {{c1::\(\mathbb{Z}_m\)is not a group with respect to multiplication modulo \(m\), but we would like to have this for building RSA}}. | |
| Extra | Not all element in Zm have an inverse, something which Zm* guarantees by bezout. |
Note 310: ETH::DiskMat
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The Fermat-Euler theorem states that for all \(m\ge 2\) and all \(a\) with \(\gcd(a,m) = 1\),{{c1:: \[a^{\varphi(m)} \equiv_m 1\]and so in particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \(a^{p-1} \equiv_p 1\).}}
Back
The Fermat-Euler theorem states that for all \(m\ge 2\) and all \(a\) with \(\gcd(a,m) = 1\),{{c1:: \[a^{\varphi(m)} \equiv_m 1\]and so in particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \(a^{p-1} \equiv_p 1\).}}
This theorem is used for RSA.
This theorem is used for RSA.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | The Fermat-Euler theorem states that for all \(m\ge 2\) and all \(a\) with \(\gcd(a,m) = 1\),{{c1:: \[a^{\varphi(m)} \equiv_m 1\]and so in particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \(a^{p-1} \equiv_p 1\).}} | |
| Extra | This theorem is used for RSA. |
Note 311: ETH::DiskMat
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For any prime \(p\), the Euler totient function \(\varphi(p)\) is equal to \(p-1\).
Back
For any prime \(p\), the Euler totient function \(\varphi(p)\) is equal to \(p-1\).
Field-by-field Comparison
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|---|---|---|
| Text | For any prime \(p\), the Euler totient function \(\varphi(p)\) is equal to {{c1::\(p-1\)}}. |
Note 312: ETH::DiskMat
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The characteristic of a ring is the order of \(1\) in the additive group if it is finite, and 0 if it is infinite.
Back
The characteristic of a ring is the order of \(1\) in the additive group if it is finite, and 0 if it is infinite.
Example: the characteristic of \(\langle \mathbb{Z}_m;\oplus,\ominus,0,\odot,1\rangle\)is \(m\).
Example: the characteristic of \(\langle \mathbb{Z}_m;\oplus,\ominus,0,\odot,1\rangle\)is \(m\).
Field-by-field Comparison
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|---|---|---|
| Text | The <i>characteristic</i> of a ring is {{c1::the order of \(1\) in the additive group if it is finite, and 0 if it is infinite.}} | |
| Extra | Example: the characteristic of \(\langle \mathbb{Z}_m;\oplus,\ominus,0,\odot,1\rangle\)is \(m\). |
Note 313: ETH::DiskMat
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An element \(a\ne0\) of a commutative ring \(R\) is called a zerodivisor if \(ab=0\) for some \(b\ne0\) in \(R\).
Back
An element \(a\ne0\) of a commutative ring \(R\) is called a zerodivisor if \(ab=0\) for some \(b\ne0\) in \(R\).
Field-by-field Comparison
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|---|---|---|
| Text | An element \(a\ne0\) of a commutative ring \(R\) is called a <i>zerodivisor</i> if {{c1:: \(ab=0\) for some \(b\ne0\) in \(R\)}}. |
Note 314: ETH::DiskMat
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The degree of the polynomial \(0\) is defined as \(-\infty\).
Back
The degree of the polynomial \(0\) is defined as \(-\infty\).
Field-by-field Comparison
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|---|---|---|
| Text | The degree of the polynomial \(0\) is defined as {{c1::\(-\infty\)}}. |
Note 315: ETH::DiskMat
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Front
How do I show the injectivity of a function?
Back
How do I show the injectivity of a function?
Show that if \(a \not= b\) then under that assumption, if \(f(a) = f(b)\) we get a contradiction as this implies \(a = b\).
Example: \(f(x) = 2x\), then if \(a \not = b\) then if \(f(a) = f(b) \ \implies \ 2a = 2b\). This however \( \ \implies a = b\).
Show that if \(a \not= b\) then under that assumption, if \(f(a) = f(b)\) we get a contradiction as this implies \(a = b\).
Example: \(f(x) = 2x\), then if \(a \not = b\) then if \(f(a) = f(b) \ \implies \ 2a = 2b\). This however \( \ \implies a = b\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How do I show the injectivity of a function? | |
| Back | Show that if \(a \not= b\) then under that assumption, if \(f(a) = f(b)\) we get a contradiction as this implies \(a = b\).<br><br><b>Example: </b>\(f(x) = 2x\), then if \(a \not = b\) then if \(f(a) = f(b) \ \implies \ 2a = 2b\). This however \( \ \implies a = b\). |
Note 316: ETH::DiskMat
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Front
Reduce \(R_{11}(9^{2024})\)
Back
Reduce \(R_{11}(9^{2024})\)
As $9^{10} \equiv_{11} 1$ (see Fermat little theorem and 11 prime), we can reduce the exponent modulo $10$ (see Lagrange's theorem in chapter 5). Thus $R_{11}(9^{2024}) = R_{11}(9^{4}) = R_{11}(-2^{4}) = 5$.
For this to work however, we need the *number and the order of the group* (modulo remainder) to be *coprime*, i.e. $\gcd(9, 11) = 1$.
As $9^{10} \equiv_{11} 1$ (see Fermat little theorem and 11 prime), we can reduce the exponent modulo $10$ (see Lagrange's theorem in chapter 5). Thus $R_{11}(9^{2024}) = R_{11}(9^{4}) = R_{11}(-2^{4}) = 5$.
For this to work however, we need the *number and the order of the group* (modulo remainder) to be *coprime*, i.e. $\gcd(9, 11) = 1$.
If the modulus itself is prime then it always works and the order of the element can be used to reduce the exponent as 9^{11-1} = 1 by FLT.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Reduce \(R_{11}(9^{2024})\) | |
| Back | As $9^{10} \equiv_{11} 1$ (see Fermat little theorem and 11 prime), we can reduce the exponent modulo $10$ (see Lagrange's theorem in chapter 5). Thus $R_{11}(9^{2024}) = R_{11}(9^{4}) = R_{11}(-2^{4}) = 5$.<br><br>For this to work however, we need the *number and the order of the group* (modulo remainder) to be *coprime*, i.e. $\gcd(9, 11) = 1$.<div>If the modulus itself is prime then it always works and the order of the element can be used to reduce the exponent as 9^{11-1} = 1 by FLT.</div> |
Note 317: ETH::DiskMat
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We can reduce the exponent \(a^m\) modulo \(n\) by {{c1::the \(\text{ord}(a)\)}} iff. \(\gcd(a, n) = 1\), i.e. \(a\) and \(n\) are coprime.
Back
We can reduce the exponent \(a^m\) modulo \(n\) by {{c1::the \(\text{ord}(a)\)}} iff. \(\gcd(a, n) = 1\), i.e. \(a\) and \(n\) are coprime.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | We can reduce the exponent \(a^m\) modulo \(n\) by {{c1::the \(\text{ord}(a)\)}} iff. {{c2::\(\gcd(a, n) = 1\), i.e. \(a\) and \(n\) are coprime}}. |
Note 318: ETH::DiskMat
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Front
For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
Back
For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
The group ℤ*_m is cyclic if and only if:
• \(m = 2\)
• \(m = 4\)
• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))
• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\)) Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).
2 is a generator.
Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1
Other generators: 3, 10, 13, 14, 15
The group ℤ*_m is cyclic if and only if:
• \(m = 2\)
• \(m = 4\)
• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))
• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\)) Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).
2 is a generator.
Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1
Other generators: 3, 10, 13, 14, 15
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15) | |
| Back | The group ℤ*_m is cyclic if and only if:<br>• \(m = 2\)<br>• \(m = 4\)<br>• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))<br>• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\)) Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).<br><br>2 is a generator.<br><br>Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1<br><br>Other generators: 3, 10, 13, 14, 15 |
Note 319: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
sXwCtB@o/s
Previous
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Front
What \(a \in \mathbb{Z}_n\) generate \(\mathbb{Z}_n\)?
Back
What \(a \in \mathbb{Z}_n\) generate \(\mathbb{Z}_n\)?
All \(a \in \mathbb{Z}_n\) such that \(\gcd(a, n) = 1\).
All \(a \in \mathbb{Z}_n\) such that \(\gcd(a, n) = 1\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What \(a \in \mathbb{Z}_n\) generate \(\mathbb{Z}_n\)? | |
| Back | All \(a \in \mathbb{Z}_n\) such that \(\gcd(a, n) = 1\). |
Note 320: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
q+Di!TuDdT
Previous
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Front
How many divisors does \(n\) expressed as a factor of prime numbers \(n = \prod_{i = 1}^m p_i^{e_i}\) have?
Back
How many divisors does \(n\) expressed as a factor of prime numbers \(n = \prod_{i = 1}^m p_i^{e_i}\) have?
\(n\) has \(\# _ {\text{div}(n)} = \prod_{i = 1}^m (e_i + 1)\) divisors.
\(n\) has \(\# _ {\text{div}(n)} = \prod_{i = 1}^m (e_i + 1)\) divisors.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | How many divisors does \(n\) expressed as a factor of prime numbers \(n = \prod_{i = 1}^m p_i^{e_i}\) have? | |
| Back | \(n\) has \(\# _ {\text{div}(n)} = \prod_{i = 1}^m (e_i + 1)\) divisors. |
Note 321: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
d_Wm7Nf:G2
Previous
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Front
What do we need to state before using the decomposition of an \(n \in \mathbb{Z}\) into prime factors?
Back
What do we need to state before using the decomposition of an \(n \in \mathbb{Z}\) into prime factors?
We need to state that this is allowed by the fundamental theorem of arithmetic.
We need to state that this is allowed by the fundamental theorem of arithmetic.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What do we need to state before using the decomposition of an \(n \in \mathbb{Z}\) into prime factors? | |
| Back | We need to state that this is allowed by the fundamental theorem of arithmetic. |
Note 322: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
H*J4QbU35P
Previous
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Front
What exponentiation operation is valid in modular arithmetic?
Back
What exponentiation operation is valid in modular arithmetic?
I can do:
I can do:
- \(a \equiv_n b\) and then \(a^x \equiv_n b^x\)
What is illegal is:
- \(a \equiv_n b\) and \(c \equiv_n d\) and then doing \(a^c \equiv_n b^d\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What exponentiation operation is valid in modular arithmetic? | |
| Back | I can do:<br><ul><li>\(a \equiv_n b\) and then \(a^x \equiv_n b^x\)<br></li></ul><div>What is illegal is:</div><div><ul><li>\(a \equiv_n b\) and \(c \equiv_n d\) and then doing \(a^c \equiv_n b^d\)</li></ul></div> |
Note 323: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
zkj+2s}Km%
Previous
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Front
The gcd does not change if we subract a multiple of the first number from the second.
Back
The gcd does not change if we subract a multiple of the first number from the second.
\(\text{gcd}(m, n - qm) = \text{gcd}(m,n) \), which is why reduction modulo \(m\) preserves the gcd, which is what makes Euclid's algorithm work.
\(\text{gcd}(m, n - qm) = \text{gcd}(m,n) \), which is why reduction modulo \(m\) preserves the gcd, which is what makes Euclid's algorithm work.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | The gcd does <b>not</b> change if we {{c1:: subract a multiple of the first number from the second}}. | |
| Back Extra | \(\text{gcd}(m, n - qm) = \text{gcd}(m,n) \), which is why reduction modulo \(m\) preserves the gcd, which is what makes Euclid's algorithm work. |
Note 324: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
vi7xPhAi#`
Previous
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Front
A left neutral element (or identity element) of an algebra \(\langle S; * \rangle\) is an element \(e\) such that \(e * a = a\) for all \(a \in S\).
A right neutral element satisfies \(a * e = a\) for all \(a \in S\).
If \(e * a = a * e = a\) for all \(a \in S\), then \(e\) is simply called a neutral element.
Back
A left neutral element (or identity element) of an algebra \(\langle S; * \rangle\) is an element \(e\) such that \(e * a = a\) for all \(a \in S\).
A right neutral element satisfies \(a * e = a\) for all \(a \in S\).
If \(e * a = a * e = a\) for all \(a \in S\), then \(e\) is simply called a neutral element.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A {{c1::left neutral element}} (or {{c1::identity element}}) of an algebra \(\langle S; * \rangle\) is an element \({{c2::e}}\) such that {{c3::\(e * a = a\)}} for all \({{c4::a}} \in S\).</p> <p>A {{c1::right neutral element}} satisfies {{c2::\(a * e = a\)}} for all \({{c3::a}} \in S\).</p> <p>If {{c2::\(e * a = a * e = a\)}} for all \({{c3::a}} \in S\), then \({{c4::e}}\) is simply called a {{c1::neutral element}}.</p> |
Note 325: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
c.BJE1FC)A
Previous
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Front
A binary operation \(*\) on a set \(S\) is associative if \(a * (b * c) = (a * b) * c\) for all \(a, b, c\) in \(S\).
Back
A binary operation \(*\) on a set \(S\) is associative if \(a * (b * c) = (a * b) * c\) for all \(a, b, c\) in \(S\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A binary operation \(*\) on a set \(S\) is {{c1::associative}} if {{c2::\(a * (b * c) = (a * b) * c\)}} for all \({{c3::a, b, c}}\) in \(S\).</p> |
Note 326: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
qi1M.
Previous
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Front
When an operation is associative, \(a_1 * a_2 * \dots * a_n\) is independent of the order of execution.
Back
When an operation is associative, \(a_1 * a_2 * \dots * a_n\) is independent of the order of execution.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>When an operation is associative, \(a_1 * a_2 * \dots * a_n\) is {{c1::independent of the order of execution}}.</p> |
Note 327: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
D0k7OAp
Previous
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Front
What happens if there is a left and right neutral element in a group?
Back
What happens if there is a left and right neutral element in a group?
Lemma 5.1: If \(\langle S; * \rangle\) has both a left and a right neutral element, then they are equal.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What happens if there is a left and right neutral element in a group? | |
| Back | <p><strong>Lemma 5.1</strong>: If \(\langle S; * \rangle\) has both a left and a right neutral element, then they are <strong>equal</strong>.</p> |
Note 328: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
uFN2l+&cfr
Previous
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Front
Can there be more than one neutral element?
Back
Can there be more than one neutral element?
\(\langle S; * \rangle\) can have at most one neutral element.
There can be a distinct left and right neutral though.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Can there be more than one neutral element?</p> | |
| Back | <p>\(\langle S; * \rangle\) can have <strong>at most one neutral element</strong>.</p><p><br></p><p>There can be a distinct left and right neutral though.</p> |
Note 329: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
u${[$*iYrd
Previous
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Front
Does the uniqueness of the neutral element imply that a group is abelian (commutative)?
I.e. does a*e = e*a mean G is abelian?
Back
Does the uniqueness of the neutral element imply that a group is abelian (commutative)?
I.e. does a*e = e*a mean G is abelian?
No! The uniqueness of the neutral element does not imply commutativity.
Counterexample: The identity matrix \(I_3\) is the unique neutral element for the group of \(3 \times 3\) real matrices under multiplication. We have \(A \cdot I_3 = I_3 \cdot A\) for all matrices \(A\), even though matrix multiplication is not commutative in general.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Does the uniqueness of the neutral element imply that a group is abelian (commutative)?</p><br><br>I.e. does a*e = e*a mean G is abelian? | |
| Back | <p><strong>No!</strong> The uniqueness of the neutral element does <strong>not</strong> imply commutativity.</p><br><p><strong>Counterexample</strong>: The identity matrix \(I_3\) is the unique neutral element for the group of \(3 \times 3\) real matrices under multiplication. We have \(A \cdot I_3 = I_3 \cdot A\) for all matrices \(A\), even though matrix multiplication is <strong>not commutative</strong> in general.</p> |
Note 330: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
lzYGHiH>1u
Previous
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Front
Give an example of a binary operation that is not associative and demonstrate why.
Back
Give an example of a binary operation that is not associative and demonstrate why.
Exponentiation on the integers is not associative.
Example:
- \((2^3)^2 = 8^2 = 64\)
- \(2^{(3^2)} = 2^9 = 512\)
Since \((2^3)^2 \neq 2^{(3^2)}\), exponentiation is not associative.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Give an example of a binary operation that is <strong>not</strong> associative and demonstrate why.</p> | |
| Back | <p><strong>Exponentiation</strong> on the integers is not associative.</p> <p><strong>Example</strong>:<br> - \((2^3)^2 = 8^2 = 64\)<br> - \(2^{(3^2)} = 2^9 = 512\)</p> <p>Since \((2^3)^2 \neq 2^{(3^2)}\), exponentiation is not associative.</p> |
Note 331: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
u`Y+W
Previous
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Front
A left inverse element of \(a\) in \(\langle S; *, e \rangle\) is an element \(b\) such that \(b * a = e\).
Back
A left inverse element of \(a\) in \(\langle S; *, e \rangle\) is an element \(b\) such that \(b * a = e\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A {{c1::left inverse element}} of \(a\) in \(\langle S; *, e \rangle\) is an element \({{c2::b}}\) such that {{c3::\(b * a = e\)}}.</p> |
Note 332: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
iltVkN7$2X
Previous
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Front
A right inverse element of \(a\) in \(\langle S; *, e \rangle\) is an element \(b\) such that \(a * b = e\).
Back
A right inverse element of \(a\) in \(\langle S; *, e \rangle\) is an element \(b\) such that \(a * b = e\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A {{c1::right inverse element}} of \(a\) in \(\langle S; *, e \rangle\) is an element \({{c2::b}}\) such that {{c3::\(a * b = e\)}}.</p> |
Note 333: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
s`dg
Previous
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New Note
Front
If both \(b * a = e\) and \(a * b = e\), then \(b\) is simply called an inverse of \(a\).
Back
If both \(b * a = e\) and \(a * b = e\), then \(b\) is simply called an inverse of \(a\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>If both {{c1::\(b * a = e\)}} and {{c2::\(a * b = e\)}}, then \({{c3::b}}\) is simply called an {{c4::inverse}} of \(a\).</p> |
Note 334: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
C}sy0oyJ5m
Previous
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Front
Lemma about uniqueness of the inverse:
Back
Lemma about uniqueness of the inverse:
Lemma 5.2: In a monoid \(\langle M; *, e \rangle\), if \(a \in M\) has a left inverse and a right inverse, then they are equal. In particular, \(a\) has at most one inverse.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Lemma about uniqueness of the inverse:</p> | |
| Back | <p><strong>Lemma 5.2</strong>: In a monoid \(\langle M; *, e \rangle\), if \(a \in M\) has a left inverse and a right inverse, then they are <strong>equal</strong>. In particular, \(a\) has <strong>at most one inverse</strong>.</p> |
Note 335: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
Ma#P3o/Xx{
Previous
Note did not exist
New Note
Front
A group is an algebra \(\langle G; *, \widehat{\ \ }, e \rangle\) satisfying three axioms: G1 (associativity), G2 (neutral element), and G3 (inverse elements).
Back
A group is an algebra \(\langle G; *, \widehat{\ \ }, e \rangle\) satisfying three axioms: G1 (associativity), G2 (neutral element), and G3 (inverse elements).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A {{c1::group}} is an algebra \(\langle G; *, \widehat{\ \ }, e \rangle\) satisfying {{c2::three}} axioms: {{c3::G1 (associativity)}}, {{c4::G2 (neutral element)}}, and {{c5::G3 (inverse elements)}}.</p> |
Note 336: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
GE_=q.pKz`
Previous
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New Note
Front
Group axiom G1 states that the operation \(*\) is associative: \(a * (b * c) = (a * b) * c\) for all \(a, b, c\) in \(G\).
Back
Group axiom G1 states that the operation \(*\) is associative: \(a * (b * c) = (a * b) * c\) for all \(a, b, c\) in \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>Group axiom <strong>G1</strong> states that the operation \(*\) is {{c1::associative}}: {{c2::\(a * (b * c) = (a * b) * c\)}} for all \(a, b, c\) in \(G\).</p> |
Note 337: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
FZV7*~vSAW
Previous
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Front
Group axiom G2 states that \(e\) is a neutral element: \(a * e = e * a = a\) for all \(a\) in \(G\).
Back
Group axiom G2 states that \(e\) is a neutral element: \(a * e = e * a = a\) for all \(a\) in \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>Group axiom <strong>G2</strong> states that {{c1::\(e\) is a neutral element: \(a * e = e * a = a\)}} for all \(a\) in \(G\).</p> |
Note 338: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
C&Xw,j%hGf
Previous
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Front
Group axiom G3 states that {{c1::every element \(a\) in \(G\) has an inverse element \(\widehat{a}\) such that \(a * \widehat{a} = \widehat{a} * a = e\)}}.
Back
Group axiom G3 states that {{c1::every element \(a\) in \(G\) has an inverse element \(\widehat{a}\) such that \(a * \widehat{a} = \widehat{a} * a = e\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>Group axiom {{c2::<strong>G3</strong>}} states that {{c1::every element \(a\) in \(G\) has an inverse element \(\widehat{a}\) such that \(a * \widehat{a} = \widehat{a} * a = e\)}}.</p> |
Note 339: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
oFF!4
Previous
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Front
A group \(\langle G; * \rangle\) (or monoid) is called commutative or abelian if \(a * b = b * a\) for all \(a, b \in G\).
Back
A group \(\langle G; * \rangle\) (or monoid) is called commutative or abelian if \(a * b = b * a\) for all \(a, b \in G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A group \(\langle G; * \rangle\) (or monoid) is called {{c1::commutative}} or {{c1::abelian}} if {{c2::\(a * b = b * a\)}} for all \({{c3::a, b}} \in G\).</p> |
Note 340: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
xH`d$W-97_
Previous
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Front
In a Group:
\(\widehat{(\widehat{a})} =\) \(a\) (inverse of inverse is the original element). This is a property from a Lemma.
Back
In a Group:
\(\widehat{(\widehat{a})} =\) \(a\) (inverse of inverse is the original element). This is a property from a Lemma.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a Group:<br> \(\widehat{(\widehat{a})} =\){{c1:: \(a\) }} {{c1:: (inverse of inverse is the original element)}}. This is a property from a Lemma.</p> |
Note 341: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
C18gm]huq&
Previous
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New Note
Front
In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.
Back
In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a Group:<br> {{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.</p> |
Note 342: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
G|6fl[78G`
Previous
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Front
To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:
- G1 (associativity)
- G2 (neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).
- G3 (inverse) G3' -> you only need to prove the existence of a right inverse
Back
To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:
- G1 (associativity)
- G2 (neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).
- G3 (inverse) G3' -> you only need to prove the existence of a right inverse
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:<br> - {{c2::G1 (associativity)}}<br> - {{c3::G2 (neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).}}<br> - {{c4::G3 (inverse) G3' -> you only need to prove the existence of a right inverse}}</p> |
Note 343: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
s*8T*K?3f=
Previous
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Front
A function \(f: A \rightarrow A\) has a left inverse if and only if \(f\) is injective.
Back
A function \(f: A \rightarrow A\) has a left inverse if and only if \(f\) is injective.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A function \(f: A \rightarrow A\) has a {{c1::left inverse}} if and only if \(f\) is {{c2::injective}}.</p> |
Note 344: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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added
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GUID:
Mqpm#lUsGl
Previous
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Front
A function \(f: A \rightarrow A\) has a right inverse if and only if \(f\) is surjective.
Back
A function \(f: A \rightarrow A\) has a right inverse if and only if \(f\) is surjective.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A function \(f: A \rightarrow A\) has a {{c1::right inverse}} if and only if \(f\) is {{c2::surjective}}.</p> |
Note 345: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
HM@g5s7n?R
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Front
A function \(f: A \rightarrow A\) has an inverse \(f^{-1}\) if and only if \(f\) is bijective.
Back
A function \(f: A \rightarrow A\) has an inverse \(f^{-1}\) if and only if \(f\) is bijective.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A function \(f: A \rightarrow A\) has an {{c1::inverse}} \(f^{-1}\) if and only if \(f\) is {{c2::bijective}}.</p> |
Note 346: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
HbBihH#d!&
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Front
The direct product of \(n\) groups \(\langle G_1; *_1 \rangle, \dots, \langle G_n; *_n \rangle\) is the algebra \(\langle G_1 \times \dots \times G_n; \star \rangle\) where the operation \(\star\) is component-wise.
Back
The direct product of \(n\) groups \(\langle G_1; *_1 \rangle, \dots, \langle G_n; *_n \rangle\) is the algebra \(\langle G_1 \times \dots \times G_n; \star \rangle\) where the operation \(\star\) is component-wise.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c1::direct product}} of \(n\) groups \(\langle G_1; *_1 \rangle, \dots, \langle G_n; *_n \rangle\) is the algebra {{c2::\(\langle G_1 \times \dots \times G_n; \star \rangle\)}} where the operation \(\star\) is {{c3::component-wise}}.</p> |
Note 347: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
F5|a$AQwO,
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Front
Give an example of a direct product of groups and explain its structure.
Back
Give an example of a direct product of groups and explain its structure.
The group \(\langle \mathbb{Z}_5; \oplus \rangle \times \langle \mathbb{Z}_7; \oplus \rangle\):
- Carrier: \(\mathbb{Z}_5 \times \mathbb{Z}_7\)
- Neutral element: \((0, 0)\)
- Operation is component-wise: \((a, b) \star (c, d) = (a \oplus_5 c, b \oplus_7 d)\)
By the Chinese Remainder Theorem, this group is isomorphic to \(\langle \mathbb{Z}_{35}; \oplus \rangle\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Give an example of a direct product of groups and explain its structure.</p> | |
| Back | <p>The group \(\langle \mathbb{Z}_5; \oplus \rangle \times \langle \mathbb{Z}_7; \oplus \rangle\):<br> - Carrier: \(\mathbb{Z}_5 \times \mathbb{Z}_7\)<br> - Neutral element: \((0, 0)\)<br> - Operation is component-wise: \((a, b) \star (c, d) = (a \oplus_5 c, b \oplus_7 d)\)</p> <p>By the <strong>Chinese Remainder Theorem</strong>, this group is isomorphic to \(\langle \mathbb{Z}_{35}; \oplus \rangle\).</p> |
Note 348: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
6o/GG^(~_
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Front
In a group, the left cancellation law states: \(a = b\) \(\Leftrightarrow\) \(ca = cb\).
Back
In a group, the left cancellation law states: \(a = b\) \(\Leftrightarrow\) \(ca = cb\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a group, the {{c1::left cancellation}} law states: \(a = b\) {{c2::\(\Leftrightarrow\)}} {{c3::\(ca = cb\)}}.</p> |
Note 349: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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DPl{@]cnAw
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Front
In a group, the right cancellation law states: \(a = b\) \(\Leftrightarrow\) \(ac = bc\).
Back
In a group, the right cancellation law states: \(a = b\) \(\Leftrightarrow\) \(ac = bc\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a group, the {{c1::right cancellation}} law states: \(a = b\) {{c2::\(\Leftrightarrow\)}} {{c3::\(ac = bc\)}}.</p> |
Note 350: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
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Front
In a group's operation table, every row and every column must contain every element exactly once.
Back
In a group's operation table, every row and every column must contain every element exactly once.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a group's operation table, every {{c1::row}} and every {{c1::column}} must contain {{c2::every element exactly once}}.</p> |
Note 351: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
qo:})x5a6`
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Front
In a group, the equations \(a * x = b\) and \(x * a = b\) have unique solutions for all \(a, b\) (property G3(v)).
Back
In a group, the equations \(a * x = b\) and \(x * a = b\) have unique solutions for all \(a, b\) (property G3(v)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a group, the equations \({{c1::a * x = b}}\) and \({{c2::x * a = b}}\) have {{c3::unique solutions}} for all \(a, b\) (property G3(v)).</p> |
Note 352: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
lx/&=nJI{d
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Front
The inverse in a group with addition is denoted \(-a\), and the inverse in a group with multiplication is denoted {{c2::\(a^{-1}\)}}.
If the operation is addition, the neutral element is usually denoted \(0\). If the operation is multiplication, the neutral element is denoted \(1\).
Back
The inverse in a group with addition is denoted \(-a\), and the inverse in a group with multiplication is denoted {{c2::\(a^{-1}\)}}.
If the operation is addition, the neutral element is usually denoted \(0\). If the operation is multiplication, the neutral element is denoted \(1\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The inverse in a group with addition is denoted {{c1::\(-a\)}}, and the inverse in a group with multiplication is denoted {{c2::\(a^{-1}\)}}.</p> <p>If the operation is addition, the neutral element is usually denoted {{c3::\(0\)}}. If the operation is multiplication, the neutral element is denoted {{c4::\(1\)}}.</p> |
Note 353: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
oIQZcTr*H#
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Front
For two groups \(\langle G; *, \widehat{\ \ }, e \rangle\) and \(\langle H; \star, \tilde{\ \ }, e' \rangle\), a function \(\psi: G \rightarrow H\) is called a group homomorphism if for all \(a\) and \(b\): \[.
This means the operation can be applied before or after the function with the same result.
Back
For two groups \(\langle G; *, \widehat{\ \ }, e \rangle\) and \(\langle H; \star, \tilde{\ \ }, e' \rangle\), a function \(\psi: G \rightarrow H\) is called a group homomorphism if for all \(a\) and \(b\): \[.
This means the operation can be applied before or after the function with the same result.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>For two groups \(\langle G; *, \widehat{\ \ }, e \rangle\) and \(\langle H; \star, \tilde{\ \ }, e' \rangle\), a function \(\psi: G \rightarrow H\) is called a {{c1::group homomorphism}} if {{c2:: for all \(a\) and \(b\): \[{{c2::\psi(a * b) = \psi(a) \star \psi(b)}}\]}}.</p> <p>This means the operation can be applied {{c3::before or after}} the function with the same result.</p> |
Note 354: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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Note Type: Cloze (with MathJax)
GUID:
oh?4Rvv7tZ
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Front
If \(\psi: G \rightarrow H\) is a bijection and a homomorphism, then it is called an isomorphism, and we say that \(G\) and \(H\) are isomorphic and write \(G \simeq H\).
Back
If \(\psi: G \rightarrow H\) is a bijection and a homomorphism, then it is called an isomorphism, and we say that \(G\) and \(H\) are isomorphic and write \(G \simeq H\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>If \(\psi: G \rightarrow H\) is a {{c1::bijection}} and a homomorphism, then it is called an {{c2::isomorphism}}, and we say that \(G\) and \(H\) are {{c2::isomorphic}} and write {{c2::\(G \simeq H\)}}.</p> |
Note 355: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
mH2hUq)MAg
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Front
Lemma 5.5(i): A group homomorphism \(\psi: G \rightarrow H\) maps the neutral element to the neutral element: \(\psi(e) = e'\).
Back
Lemma 5.5(i): A group homomorphism \(\psi: G \rightarrow H\) maps the neutral element to the neutral element: \(\psi(e) = e'\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p><strong>Lemma 5.5(i)</strong>: A group homomorphism \(\psi: G \rightarrow H\) maps the neutral element to {{c1::the neutral element: \(\psi(e) = e'\)}}.</p> |
Note 356: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
MUE8!)s%
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Front
Lemma 5.5(ii): A group homomorphism \(\psi: G \rightarrow H\) maps inverses to {{c1::inverses: \(\psi(\widehat{a}) = \widetilde{\psi(a)}\)}} for all \(a\).
Back
Lemma 5.5(ii): A group homomorphism \(\psi: G \rightarrow H\) maps inverses to {{c1::inverses: \(\psi(\widehat{a}) = \widetilde{\psi(a)}\)}} for all \(a\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p><strong>Lemma 5.5(ii)</strong>: A group homomorphism \(\psi: G \rightarrow H\) maps inverses to {{c1::inverses: \(\psi(\widehat{a}) = \widetilde{\psi(a)}\)}} for all \(a\).</p> |
Note 357: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
C8cvZ0}Mn#
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Front
Give an example of a group homomorphism involving the logarithm function.
Back
Give an example of a group homomorphism involving the logarithm function.
The logarithm function is a group homomorphism from \(\langle \mathbb{R}^{>0}; \cdot \rangle\) to \(\langle \mathbb{R}; + \rangle\) because: \[\log(a \cdot b) = \log a + \log b\]
It's also an isomorphism because the logarithm is bijective on positive reals.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Give an example of a group homomorphism involving the logarithm function.</p> | |
| Back | <p>The logarithm function is a group homomorphism from \(\langle \mathbb{R}^{>0}; \cdot \rangle\) to \(\langle \mathbb{R}; + \rangle\) because: \[\log(a \cdot b) = \log a + \log b\]</p> <p>It's also an <strong>isomorphism</strong> because the logarithm is bijective on positive reals.</p> |
Note 358: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
FMdIN{F>5c
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Front
Is the projection of points from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) a homomorphism? Is it an isomorphism?
Back
Is the projection of points from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) a homomorphism? Is it an isomorphism?
The projection is a homomorphism (it preserves the group operation of vector addition).
However, it is not an isomorphism because it's not a bijection (not injective - many 3D points project to the same 2D point).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Is the projection of points from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) a homomorphism? Is it an isomorphism?</p> | |
| Back | <p>The projection is a <strong>homomorphism</strong> (it preserves the group operation of vector addition).</p> <p>However, it is <strong>not an isomorphism</strong> because it's not a bijection (not injective - many 3D points project to the same 2D point).</p> |
Note 359: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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GUID:
zKcmqH!B|+
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Front
To prove \(\phi: G \rightarrow H\) is an isomorphism, you must verify two main properties:
- \(\phi\) is a homomorphism
- \(\phi\) is a bijection.
Back
To prove \(\phi: G \rightarrow H\) is an isomorphism, you must verify two main properties:
- \(\phi\) is a homomorphism
- \(\phi\) is a bijection.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>To prove \(\phi: G \rightarrow H\) is an {{c2:: isomorphism}}, you must verify two main properties:<br /> - \(\phi\) is a {{c1::homomorphism}}<br /> - \(\phi\) is a {{c1::bijection}}.</p> |
Note 360: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
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Front
To verify the homomorphism property, check that: \(\phi(g_1 \cdot g_2) = \phi(g_1) + \phi(g_2)\) for all \(g_1, g_2\) in \(G\).
Back
To verify the homomorphism property, check that: \(\phi(g_1 \cdot g_2) = \phi(g_1) + \phi(g_2)\) for all \(g_1, g_2\) in \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>To verify the {{c1::homomorphism property}}, check that: {{c2::\(\phi(g_1 \cdot g_2) = \phi(g_1) + \phi(g_2)\)}} for all \(g_1, g_2\) in \(G\).</p> |
Note 361: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
bzpi*}NRv1
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Front
Does every homomorphism have to be injective? Give an example.
Back
Does every homomorphism have to be injective? Give an example.
No, homomorphisms do not need to be injective.
Example: We could map all elements of \(G\) to the neutral element \(e'\) in \(H\). This satisfies the homomorphism property: \[\psi(a * b) = e' = e' \star e' = \psi(a) \star \psi(b)\] but is clearly not injective.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Does every homomorphism have to be injective? Give an example.</p> | |
| Back | <p><strong>No</strong>, homomorphisms do not need to be injective.</p> <p><strong>Example</strong>: We could map all elements of \(G\) to the neutral element \(e'\) in \(H\). This satisfies the homomorphism property: \[\psi(a * b) = e' = e' \star e' = \psi(a) \star \psi(b)\] but is clearly not injective.</p> |
Note 362: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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GUID:
pL8pp7+)x{
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Front
For a homomorphism \(h: G \rightarrow H\), the kernel \(\ker h\) is the set of all elements mapped to the neutral element (essentially the nullspace).
Back
For a homomorphism \(h: G \rightarrow H\), the kernel \(\ker h\) is the set of all elements mapped to the neutral element (essentially the nullspace).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>For a homomorphism \(h: G \rightarrow H\), the {{c1::kernel \(\ker h\)}} is the set of all elements mapped to the {{c2::neutral element}} (essentially the {{c2::nullspace}}).</p> |
Note 363: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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GUID:
BA!Uj{h&4e
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Front
For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{im} h\)}} is the set of all elements in \(H\) that are mapped to by some element in \(G\).
Back
For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{im} h\)}} is the set of all elements in \(H\) that are mapped to by some element in \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{im} h\)}} is the {{c2::set of all elements in \(H\) that are mapped to by some element in \(G\)}}.</p> |
Note 364: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
B7f%iVC#KH
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Front
What does it mean intuitively for two groups to be isomorphic?
Back
What does it mean intuitively for two groups to be isomorphic?
Two groups are isomorphic if they have the same structure - they "behave identically" even if they look different.
Analogy: Like two jigsaw puzzles that look completely different, but use the same cutout pattern. The same piece goes into the same place on both puzzles.
The bijection between them preserves all group operations and relationships.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What does it mean intuitively for two groups to be isomorphic?</p> | |
| Back | <p>Two groups are isomorphic if they have the <strong>same structure</strong> - they "behave identically" even if they look different.</p> <p><strong>Analogy</strong>: Like two jigsaw puzzles that look completely different, but use the same cutout pattern. The same piece goes into the same place on both puzzles.</p> <p>The bijection between them preserves all group operations and relationships.</p> |
Note 365: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
GUID:
p$?:uS#|X
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Front
In a homomorphism \(\psi: G \rightarrow H\), does it matter whether you apply the operation before or after applying the function?
Back
In a homomorphism \(\psi: G \rightarrow H\), does it matter whether you apply the operation before or after applying the function?
No, it doesn't matter! That's exactly what defines a homomorphism:
\[\psi(a *_G b) = \psi(a) *_H \psi(b)\]
You get the same result whether you:
- First operate in \(G\), then map to \(H\), OR
- First map both elements to \(H\), then operate in \(H\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>In a homomorphism \(\psi: G \rightarrow H\), does it matter whether you apply the operation before or after applying the function?</p> | |
| Back | <p><strong>No</strong>, it doesn't matter! That's exactly what defines a homomorphism:</p> <p>\[\psi(a *_G b) = \psi(a) *_H \psi(b)\]</p> <p>You get the same result whether you:<br> - First operate in \(G\), then map to \(H\), OR<br> - First map both elements to \(H\), then operate in \(H\)</p> |
Note 366: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
.d,WRJq.}
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Front
A subset \(H \subseteq G\) of a group is called a subgroup if \(H\) is: closed with respect to all operations (operation, neutral, inverse).
Back
A subset \(H \subseteq G\) of a group is called a subgroup if \(H\) is: closed with respect to all operations (operation, neutral, inverse).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A subset \(H \subseteq G\) of a group is called a {{c1::subgroup}} if \(H\) is: {{c2::closed with respect to all operations (operation, neutral, inverse)}}.</p> |
Note 367: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
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Front
For \(H\) to be a subgroup, the neutral element must be in \(H\): \(e \in H\).
Back
For \(H\) to be a subgroup, the neutral element must be in \(H\): \(e \in H\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>For \(H\) to be a subgroup, the {{c1::neutral element}} must be in \(H\): {{c1::\(e \in H\)}}.</p> |
Note 368: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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Note Type: Cloze (with MathJax)
GUID:
N9}Teh]+={
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Front
For \(H\) to be a subgroup, it must have closure under inverses: {{c2:: \(\widehat{a} \in H\) for all \(a \in H\)}}.
Back
For \(H\) to be a subgroup, it must have closure under inverses: {{c2:: \(\widehat{a} \in H\) for all \(a \in H\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>For \(H\) to be a subgroup, it must have {{c1::closure under inverses}}: {{c2:: \(\widehat{a} \in H\) for all \({{c3::a \in H}}\)}}.</p> |
Note 369: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
CL8*F@7NV5
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Front
For any group \(G\), there exist two trivial subgroups:
- {{c2::The set \(\{e\}\) (containing only the neutral element)}}
- \(G\) itself
Back
For any group \(G\), there exist two trivial subgroups:
- {{c2::The set \(\{e\}\) (containing only the neutral element)}}
- \(G\) itself
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <!-- Card 38: Trivial Subgroups (Cloze) --> <p>For any group \(G\), there exist two trivial subgroups:<br /> - {{c2::The set \(\{e\}\) (containing only the neutral element)}}<br /> - {{c3::\(G\) itself}}</p> |
Note 370: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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GUID:
xp_U|[rM*4
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Front
List all subgroups of \(\mathbb{Z}_{12}\).
Back
List all subgroups of \(\mathbb{Z}_{12}\).
The subgroups of \(\mathbb{Z}_{12}\) are:
- \(\{0\}\) (trivial subgroup)
- \(\{0, 6\}\)
- \(\{0, 4, 8\}\)
- \(\{0, 3, 6, 9\}\)
- \(\{0, 2, 4, 6, 8, 10\}\)
- \(\mathbb{Z}_{12}\) (trivial subgroup - the whole group)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>List all subgroups of \(\mathbb{Z}_{12}\).</p> | |
| Back | <p>The subgroups of \(\mathbb{Z}_{12}\) are:<br> - \(\{0\}\) (trivial subgroup)<br> - \(\{0, 6\}\)<br> - \(\{0, 4, 8\}\)<br> - \(\{0, 3, 6, 9\}\)<br> - \(\{0, 2, 4, 6, 8, 10\}\)<br> - \(\mathbb{Z}_{12}\) (trivial subgroup - the whole group)</p> |
Note 371: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
GUID:
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Previous
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Front
Why is closure important when verifying that \(H\) is a subgroup of \(G\)?
Back
Why is closure important when verifying that \(H\) is a subgroup of \(G\)?
Closure ensures that when you apply operations within \(H\), you stay within \(H\).
Without closure:
- \(a * b\) might not be in \(H\) (operation closure)
- \(\widehat{a}\) might not be in \(H\) (inverse closure)
- The neutral element \(e\) might not be in \(H\)
If \(H\) lacks closure, it cannot form a group on its own.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Why is closure important when verifying that \(H\) is a subgroup of \(G\)?</p> | |
| Back | <p>Closure ensures that when you apply operations within \(H\), you <strong>stay within</strong> \(H\).</p> <p>Without closure:<br> - \(a * b\) might not be in \(H\) (operation closure)<br> - \(\widehat{a}\) might not be in \(H\) (inverse closure)<br> - The neutral element \(e\) might not be in \(H\)</p> <p>If \(H\) lacks closure, it cannot form a group on its own.</p> |
Note 372: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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Note Type: Cloze (with MathJax)
GUID:
PI2pek)9t%
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Front
In a group, \(a^0\) is defined as the identity element \(e\).
Back
In a group, \(a^0\) is defined as the identity element \(e\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a group, \({{c1::a^0}}\) is defined as the {{c2::identity element}} \({{c3::e}}\).</p> |
Note 373: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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G#]T4?!iZs
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Front
In a group, for \(n \geq 1\), the positive power is defined recursively: {{c1::\(a^n = \cdot a^{n-1}\)}}.
Back
In a group, for \(n \geq 1\), the positive power is defined recursively: {{c1::\(a^n = \cdot a^{n-1}\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a group, for \(n \geq 1\), the positive power is defined recursively: {{c1::\(a^n = \cdot a^{n-1}\)}}.</p> |
Note 374: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
jbk$(]c7J_
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Front
What is the order of \(\text{ord}(a)\) for \(a \in G\) in a group?
Back
What is the order of \(\text{ord}(a)\) for \(a \in G\) in a group?
Let \(G\) be a group and let \(a\) be an element of \(G\). The order of \(a\), denoted \(\text{ord}(a)\), is the least \(m \geq 1\) such that \(a^m = e\), if such an \(m\) exists: \[\text{ord}(a) = \min \{n \geq 1 \ | \ a^n = e\} \cup \{\infty\}\]
If no such \(m\) exists, \(G\)0 is said to be infinite, written \(G\)1.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the order of \(\text{ord}(a)\) for \(a \in G\) in a group?</p> | |
| Back | <p>Let \(G\) be a group and let \(a\) be an element of \(G\). The order of \(a\), denoted \(\text{ord}(a)\), is the least \(m \geq 1\) such that \(a^m = e\), if such an \(m\) exists: \[\text{ord}(a) = \min \{n \geq 1 \ | \ a^n = e\} \cup \{\infty\}\]</p> <p>If no such \(m\) exists, \(G\)0 is said to be infinite, written \(G\)1.</p> |
Note 375: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
uaVso1SVrk
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Front
If no \(m\) exists, such that \(a^m = e\) in a group, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.
Back
If no \(m\) exists, such that \(a^m = e\) in a group, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>If {{c2:: no \(m\) exists, such that \(a^m = e\) in a group}}, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.</p> |
Note 376: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
HI{+|nsE+a
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Front
The order \(\text{ord}(e)\) of \(e \in G\) is 1 by definition.
Back
The order \(\text{ord}(e)\) of \(e \in G\) is 1 by definition.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The order \(\text{ord}(e)\) of \(e \in G\) is {{c1:: 1 by definition}}.</p> |
Note 377: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
hJb:YVj|nK
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Front
If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is 2}}, a is it's own self-inverse.
Back
If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is 2}}, a is it's own self-inverse.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is 2}}, {{c1:: a is it's own self-inverse}}.</p> |
Note 378: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
m=x|N12mNI
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Front
If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is \(|G|\)}}, it has "volle Ordung".
Back
If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is \(|G|\)}}, it has "volle Ordung".
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is \(|G|\)}}, {{c1:: it has "volle Ordung"}}.</p> |
Note 379: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
PU}Z|agcHs
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Front
For a finite group \(G\), \(|G|\) is called the order of \(G\).
Back
For a finite group \(G\), \(|G|\) is called the order of \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>For a finite group \(G\), {{c1::\(|G|\)}} is called the {{c2::order of \(G\)}}.</p> |
Note 380: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
hAzQO,E_+E
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Front
What is the order of elements in finite groups.
Back
What is the order of elements in finite groups.
Lemma 5.6: In a finite group \(G\), every element has a finite order.
(This doesn't hold for infinite groups - elements can have infinite order.)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the order of elements in finite groups.</p> | |
| Back | <p><strong>Lemma 5.6</strong>: In a <strong>finite group</strong> \(G\), every element has a <strong>finite order</strong>.</p> <p>(This doesn't hold for infinite groups - elements can have infinite order.)</p> |
Note 381: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
cV1b,==V*(
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Front
Give an example of an element with infinite order.
Back
Give an example of an element with infinite order.
In the group \(\langle \mathbb{Z}; + \rangle\), any integer \(a \neq 0\) has infinite order.
Explanation: The carrier \(\mathbb{Z}\) is infinite, and we never "loop around" to reach \(0\) by repeatedly adding a non-zero integer. For any \(n \geq 1\), we have \(na \neq 0\) if \(a \neq 0\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Give an example of an element with infinite order.</p> | |
| Back | <p>In the group \(\langle \mathbb{Z}; + \rangle\), any integer \(a \neq 0\) has <strong>infinite order</strong>.</p> <p><strong>Explanation</strong>: The carrier \(\mathbb{Z}\) is infinite, and we never "loop around" to reach \(0\) by repeatedly adding a non-zero integer. For any \(n \geq 1\), we have \(na \neq 0\) if \(a \neq 0\).</p> |
Note 382: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
hx=y:u%$sF
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Front
Front
By what can we reduce the exponent of an element in a finite order Group?
Back
In a group \(G\) of finite order, for \(a \in G\): \[a^m = a^{{{c1::R_{\text{ord}(a)}(m)}}}\]
This holds because {{c2::\(a^m = a^{m + \text{ord}(a)} = a^m \cdot a^{\text{ord}(a)} = a^m \cdot e = a^m\)}}.
Back
Front
By what can we reduce the exponent of an element in a finite order Group?
Back
In a group \(G\) of finite order, for \(a \in G\): \[a^m = a^{{{c1::R_{\text{ord}(a)}(m)}}}\]
This holds because {{c2::\(a^m = a^{m + \text{ord}(a)} = a^m \cdot a^{\text{ord}(a)} = a^m \cdot e = a^m\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>By what can we reduce the exponent of an element in a <strong>finite order</strong> Group?</p> <h1>Back</h1> <p>In a group \(G\) of finite order, for \(a \in G\): \[a^m = a^{{{c1::R_{\text{ord}(a)}(m)}}}\]</p> <p>This holds because {{c2::\(a^m = a^{m + \text{ord}(a)} = a^m \cdot a^{\text{ord}(a)} = a^m \cdot e = a^m\)}}.</p> |
Note 383: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
N,%,xCUm($
Previous
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Front
What is the group generated by a, denoted \(\langle a \rangle\) defined as?
Back
What is the group generated by a, denoted \(\langle a \rangle\) defined as?
For a group \(G\) and \(a \in G\), the group generated by \(a\), denoted \(\langle a \rangle\), is defined as: \[\langle a \rangle \ \overset{\text{def}}{=} \ \{a^n \ | \ n \in \mathbb{Z}\}\]
This is a group, the smallest subgroup of \(G\) containing the element \(a\).
For finite groups: \(\langle a \rangle = \{e, a, a^2, \dots, a^{\text{ord}(a)-1}\}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the group <em>generated by a</em>, denoted \(\langle a \rangle\) defined as?</p> | |
| Back | <p>For a group \(G\) and \(a \in G\), the group generated by \(a\), denoted \(\langle a \rangle\), is defined as: \[\langle a \rangle \ \overset{\text{def}}{=} \ \{a^n \ | \ n \in \mathbb{Z}\}\]</p> <p>This is a group, the smallest subgroup of \(G\) containing the element \(a\).</p> <p>For finite groups: \(\langle a \rangle = \{e, a, a^2, \dots, a^{\text{ord}(a)-1}\}\).</p> |
Note 384: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
vRto[%;el{
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Front
The smallest subgroup of a group \(G\) containing \(a \in G\) is the group generated by \(a\), \(\langle a \rangle\).
Back
The smallest subgroup of a group \(G\) containing \(a \in G\) is the group generated by \(a\), \(\langle a \rangle\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c2:: smallest}} subgroup of a group \(G\) containing \(a \in G\) is {{c1:: the group <em>generated by \(a\)</em>, \(\langle a \rangle\)}}.</p> |
Note 385: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
H*rUl{%/Iu
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Front
We denote the group generated by \(a\) as \(\langle a \rangle\).
Back
We denote the group generated by \(a\) as \(\langle a \rangle\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>We denote the {{c2:: group generated}} by \(a\) as {{c1:: \(\langle a \rangle\)}}.</p> |
Note 386: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
G3,dI)){d{
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Front
Which elements generate \(\mathbb{Z}_n\)?
Back
Which elements generate \(\mathbb{Z}_n\)?
\(\mathbb{Z}_n\) is generated by all \(a \in \mathbb{Z}_n\) for which \(\gcd(n, a) = 1\) (all elements coprime to \(n\)).
Proof idea:
- If \(\mathbb{Z}_n = \langle a \rangle\), then \(1 \in \langle a \rangle\)
- This means \(au \equiv_n 1\) for some \(u\)
- By Bézout's identity, this implies \(\gcd(a, n) = 1\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Which elements generate \(\mathbb{Z}_n\)?</p> | |
| Back | <p>\(\mathbb{Z}_n\) is generated by <strong>all \(a \in \mathbb{Z}_n\) for which \(\gcd(n, a) = 1\)</strong> (all elements coprime to \(n\)).</p> <p><strong>Proof idea</strong>:<br> - If \(\mathbb{Z}_n = \langle a \rangle\), then \(1 \in \langle a \rangle\)<br> - This means \(au \equiv_n 1\) for some \(u\)<br> - By Bézout's identity, this implies \(\gcd(a, n) = 1\)</p> |
Note 387: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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op}IVwoXF>
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Front
A group \(G = \) \(\langle g \rangle\) generated by an element \(g\) is called cyclic.
Back
A group \(G = \) \(\langle g \rangle\) generated by an element \(g\) is called cyclic.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A group \(G = \) {{c2:: \(\langle g \rangle\) generated by an element}} \(g\) is called {{c1::cyclic}}.</p> |
Note 388: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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rMrK0z!nVO
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Front
In a cyclic group \(\langle g \rangle\), associativity is inherited from the parent group \(G\).
Back
In a cyclic group \(\langle g \rangle\), associativity is inherited from the parent group \(G\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a cyclic group \(\langle g \rangle\), {{c1::associativity}} is {{c2::inherited}} from the parent group \({{c3::G}}\).</p> |
Note 389: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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KE]BoQ-4oe
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Front
The neutral element is always in \(\langle g \rangle\) because \(g^0 = e\).
Back
The neutral element is always in \(\langle g \rangle\) because \(g^0 = e\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c1::neutral element}} is always in \(\langle g \rangle\) because {{c2::\(g^0 = e\)}}.</p> |
Note 390: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
Q~CcPI;l0U
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Front
In a cyclic group, the inverse of \(a^n\) is {{c2::\(a^{-n}\)}}.
Back
In a cyclic group, the inverse of \(a^n\) is {{c2::\(a^{-n}\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>In a cyclic group, the {{c1::inverse}} of \(a^n\) is {{c2::\(a^{-n}\)}}.</p> |
Note 391: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
Gdk~1PL`6=
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Front
A cyclic group can have more than one generator.
Back
A cyclic group can have more than one generator.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A cyclic group can have {{c1::more than one}} {{c2::generator}}.</p> |
Note 392: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
JOM4&m6Z&$
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Front
What is a cyclic group of order \(n\) isomorphic to?
Back
What is a cyclic group of order \(n\) isomorphic to?
Theorem 5.7: A cyclic group of order \(n\) is isomorphic to \(\langle \mathbb{Z}_n; \oplus \rangle\) (and hence abelian).
This means all cyclic groups of the same order have the same structure.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is a cyclic group of order \(n\) isomorphic to?</p> | |
| Back | <p><strong>Theorem 5.7</strong>: A cyclic group of order \(n\) is <strong>isomorphic</strong> to \(\langle \mathbb{Z}_n; \oplus \rangle\) (and hence <strong>abelian</strong>).</p> <p>This means all cyclic groups of the same order have the same structure.</p> |
Note 393: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
eS{U|$mPp_
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Front
What is a special property of the group \(\langle \mathbb{Z}_n; \oplus \rangle\) for all \(n\)?
Back
What is a special property of the group \(\langle \mathbb{Z}_n; \oplus \rangle\) for all \(n\)?
It is abelian!
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is a special property of the group \(\langle \mathbb{Z}_n; \oplus \rangle\) for all \(n\)?</p> | |
| Back | <p>It is <strong>abelian</strong>!</p> |
Note 394: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
EDPL:,`:xZ
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Front
The group {{c2:: \(\langle \mathbb{Z}_n; \oplus \rangle\)}} is generated by \(1\).
Back
The group {{c2:: \(\langle \mathbb{Z}_n; \oplus \rangle\)}} is generated by \(1\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The group {{c2:: \(\langle \mathbb{Z}_n; \oplus \rangle\)}} is generated by {{c1:: \(1\)}}.</p> |
Note 395: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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GUID:
zK!1=,UI{i
Previous
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Front
All generators of {{c2:: \(\langle \mathbb{Z}_n; \oplus \rangle\)}} are coprime to \(n\).
Back
All generators of {{c2:: \(\langle \mathbb{Z}_n; \oplus \rangle\)}} are coprime to \(n\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>All generators of {{c2:: \(\langle \mathbb{Z}_n; \oplus \rangle\)}} are {{c1:: <strong>coprime</strong> to \(n\)}}.</p> |
Note 396: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
m+a(a1n4{R
Previous
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Front
State Lagrange's Theorem (Theorem 5.8).
Back
State Lagrange's Theorem (Theorem 5.8).
Theorem 5.8 (Lagrange's Theorem): Let \(G\) be a finite group and let \(H\) be a subgroup of \(G\). Then the order of \(H\) divides the order of \(G\), i.e., \(|H|\) divides \(|G|\).
Written: \(|H| \ | \ |G|\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Lagrange's Theorem (Theorem 5.8).</p> | |
| Back | <p><strong>Theorem 5.8 (Lagrange's Theorem)</strong>: Let \(G\) be a finite group and let \(H\) be a subgroup of \(G\). Then the order of \(H\) <strong>divides</strong> the order of \(G\), i.e., \(|H|\) divides \(|G|\).</p> <p>Written: \(|H| \ | \ |G|\)</p> |
Note 397: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
D:fQHHFS8g
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Front
State Corollary 5.9 about the relationship between element order and group order for a finite group \(G\).
Back
State Corollary 5.9 about the relationship between element order and group order for a finite group \(G\).
Corollary 5.9: For a finite group \(G\), the order of every element divides the group order, i.e., \(\text{ord}(a)\) divides \(|G|\) for every \(a \in G\).
Proof: \(\langle a \rangle\) is a subgroup of \(G\) of order \(\text{ord}(a)\), which by Lagrange's Theorem must divide \(|G|\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Corollary 5.9 about the relationship between element order and group order for a finite group \(G\).</p> | |
| Back | <p><strong>Corollary 5.9</strong>: For a finite group \(G\), the order of every element <strong>divides</strong> the group order, i.e., \(\text{ord}(a)\) divides \(|G|\) for every \(a \in G\).</p> <p><strong>Proof</strong>: \(\langle a \rangle\) is a subgroup of \(G\) of order \(\text{ord}(a)\), which by Lagrange's Theorem must divide \(|G|\).</p> |
Note 398: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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eQKQ_hr,6l
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Front
We have the order {{c1::\(\text{ord}(a)\)}} = \(|\langle a \rangle|\).
Back
We have the order {{c1::\(\text{ord}(a)\)}} = \(|\langle a \rangle|\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>We have the order {{c1::\(\text{ord}(a)\)}} = {{c2::\(|\langle a \rangle|\)}}.</p> |
Note 399: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
f_%oe]V2X6
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Front
State Corollary 5.10 about raising elements to the power of the group order.
Back
State Corollary 5.10 about raising elements to the power of the group order.
Corollary 5.10: Let \(G\) be a finite group. Then \(a^{|G|} = e\) for every \(a \in G\).
Proof: By Corollary 5.9, \(|G| = k \cdot \text{ord}(a)\) for some \(k\). Thus: \[a^{|G|} = a^{k \cdot \text{ord}(a)} = (a^{\text{ord}(a)})^k = e^k = e\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Corollary 5.10 about raising elements to the power of the group order.</p> | |
| Back | <p><strong>Corollary 5.10</strong>: Let \(G\) be a finite group. Then \(a^{|G|} = e\) for every \(a \in G\).</p> <p><strong>Proof</strong>: By Corollary 5.9, \(|G| = k \cdot \text{ord}(a)\) for some \(k\). Thus: \[a^{|G|} = a^{k \cdot \text{ord}(a)} = (a^{\text{ord}(a)})^k = e^k = e\]</p> |
Note 400: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
jngIBgkHz<
Previous
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Front
State Corollary 5.11 about groups of prime order (what property, waht does each element satisfy).
Back
State Corollary 5.11 about groups of prime order (what property, waht does each element satisfy).
Corollary 5.11: Every group of prime order is cyclic, and in such a group every element except the neutral element is a generator.
Proof: Only \(1 | p\) and \(p | p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Corollary 5.11 about groups of prime order (what property, waht does each element satisfy).</p> | |
| Back | <p><strong>Corollary 5.11</strong>: Every group of <strong>prime order</strong> is cyclic, and in such a group <strong>every element except the neutral element is a generator</strong>.</p> <p><strong>Proof</strong>: Only \(1 | p\) and \(p | p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group).</p> |
Note 401: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
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Front
For what order is every group cyclic?
Back
For what order is every group cyclic?
If the order of the group is prime, it is cyclic!
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>For what order is every group cyclic?</p> | |
| Back | <p>If the <strong>order of the group</strong> is <strong>prime</strong>, it is cyclic!</p> |
Note 402: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
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S^AYekncO
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Front
We use {{c1::\(\langle \mathbb{Z}_n; \oplus \rangle\)}} as our standard notation for cyclic groups of order \(n\).
Back
We use {{c1::\(\langle \mathbb{Z}_n; \oplus \rangle\)}} as our standard notation for cyclic groups of order \(n\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>We use {{c1::\(\langle \mathbb{Z}_n; \oplus \rangle\)}} as our standard notation for cyclic groups of order \(n\).</p> |
Note 403: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Front
The group denoted by \(\mathbb{Z}_n\) is always meant in conjunction with the operation \(\oplus\) modulo \(n\).
Back
The group denoted by \(\mathbb{Z}_n\) is always meant in conjunction with the operation \(\oplus\) modulo \(n\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The group denoted by \(\mathbb{Z}_n\) is always meant in conjunction with the operation {{c2::\(\oplus\) modulo \(n\)}}.</p> |
Note 404: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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zqVqxXe~xC
Previous
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Front
The group \(\mathbb{Z}_n\) also only contains the positive numbers up to \(n\) \(\{0, 1, 2, \dots, n-1\}\), as the negatives \(\{-n+1, \dots, -2, -1\}\) are equal to a positive number \(\equiv_n\).
Back
The group \(\mathbb{Z}_n\) also only contains the positive numbers up to \(n\) \(\{0, 1, 2, \dots, n-1\}\), as the negatives \(\{-n+1, \dots, -2, -1\}\) are equal to a positive number \(\equiv_n\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The group \(\mathbb{Z}_n\) also {{c3::only contains the positive numbers up to \(n\)}} \(\{0, 1, 2, \dots, n-1\}\), as the negatives \(\{-n+1, \dots, -2, -1\}\) are equal to a positive number \(\equiv_n\).</p> |
Note 405: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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GUID:
k3`On,s-[c
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Front
The generators of \(\langle \mathbb{Z}_n; \oplus \rangle\) are all \(g \in \mathbb{Z}_n\) for which \(\gcd(g, n) = 1\) (i.e., \(g\) is coprime to \(n\)).
The group \(\langle \mathbb{Z}_n; \oplus \rangle\) is cyclic for every \(n\), where \(1\) is always a generator.
Back
The generators of \(\langle \mathbb{Z}_n; \oplus \rangle\) are all \(g \in \mathbb{Z}_n\) for which \(\gcd(g, n) = 1\) (i.e., \(g\) is coprime to \(n\)).
The group \(\langle \mathbb{Z}_n; \oplus \rangle\) is cyclic for every \(n\), where \(1\) is always a generator.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The generators of \(\langle \mathbb{Z}_n; \oplus \rangle\) are all \(g \in \mathbb{Z}_n\) for which {{c1::\(\gcd(g, n) = 1\)}} (i.e., \(g\) is {{c2::coprime}} to \(n\)).</p> <p>The group \(\langle \mathbb{Z}_n; \oplus \rangle\) is cyclic for every \(n\), where {{c3::\(1\)}} is always a generator.</p> |
Note 406: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
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Note Type: Basic (with MathJax)
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omy86HKbVn
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Front
Is \(\langle \mathbb{Z}_n; \oplus \rangle\) abelian (commutative)?
Back
Is \(\langle \mathbb{Z}_n; \oplus \rangle\) abelian (commutative)?
Yes, \(\langle \mathbb{Z}_n; \oplus \rangle\) is abelian because addition modulo \(n\) is commutative: \[a \oplus b = (a + b) \bmod n = (b + a) \bmod n = b \oplus a\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Is \(\langle \mathbb{Z}_n; \oplus \rangle\) abelian (commutative)?</p> | |
| Back | <p><strong>Yes</strong>, \(\langle \mathbb{Z}_n; \oplus \rangle\) is <strong>abelian</strong> because addition modulo \(n\) is commutative: \[a \oplus b = (a + b) \bmod n = (b + a) \bmod n = b \oplus a\]</p> |
Note 407: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
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sxW-Trt$`+
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Front
Front
\(\mathbb{Z}_m^*\) is defined as?
Back
\[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ | \ \gcd(a, m) = 1\} \]
This is the set of all elements in \(\mathbb{Z}_m\) that are coprime to \(m\).
Back
Front
\(\mathbb{Z}_m^*\) is defined as?
Back
\[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ | \ \gcd(a, m) = 1\} \]
This is the set of all elements in \(\mathbb{Z}_m\) that are coprime to \(m\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>\(\mathbb{Z}_m^*\) is defined as?</p> <h1>Back</h1> <p>\[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ | \ \gcd(a, m) = 1\} \]</p> <p>This is the set of all elements in \(\mathbb{Z}_m\) that are {{c2::coprime}} to \(m\).</p> |
Note 408: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
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q|}rXYFly~
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Front
The Euler function \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) is defined as {{c2::the cardinality of \(\mathbb{Z}_m^*\): \[\varphi(m) = |\mathbb{Z}_m^*|\]}}.
Back
The Euler function \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) is defined as {{c2::the cardinality of \(\mathbb{Z}_m^*\): \[\varphi(m) = |\mathbb{Z}_m^*|\]}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c1::Euler function}} \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) is defined as {{c2::the cardinality of \(\mathbb{Z}_m^*\): \[\varphi(m) = |\mathbb{Z}_m^*|\]}}.</p> |
Note 409: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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G&Y|dtr7^k
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Front
Front
If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?
Back
\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]
For a prime \(p\) and \(e \geq 1\): \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]
This comes from the fact that for prime \(p\) and \(e \geq 1\) we have \[ \varphi(p^e) = p^{e-1}(p - 1) \]since exactly every \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]0th integer in \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]1 contains a factor \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]2 and thus \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]3 elements don't contain a factor \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]4, i.e. are in \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]5.
Since the \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]6's are pairwise relatively prime (obviously) by the CRT we have that \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]7. This holds as the CRT allows us to establish a bijection between each \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]8 and a unique tuple \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]9 where \[ \varphi(p^e) = p^{e-1}(p - 1) \]0.
Back
Front
If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?
Back
\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]
For a prime \(p\) and \(e \geq 1\): \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]
This comes from the fact that for prime \(p\) and \(e \geq 1\) we have \[ \varphi(p^e) = p^{e-1}(p - 1) \]since exactly every \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]0th integer in \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]1 contains a factor \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]2 and thus \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]3 elements don't contain a factor \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]4, i.e. are in \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]5.
Since the \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]6's are pairwise relatively prime (obviously) by the CRT we have that \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]7. This holds as the CRT allows us to establish a bijection between each \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]8 and a unique tuple \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]9 where \[ \varphi(p^e) = p^{e-1}(p - 1) \]0.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?</p> <h1>Back</h1> <p>\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]</p> <p>For a prime \(p\) and \(e \geq 1\): \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]</p> <p>This comes from the fact that for prime \(p\) and \(e \geq 1\) we have \[ \varphi(p^e) = p^{e-1}(p - 1) \]since exactly every \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]0th integer in \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]1 contains a factor \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]2 and thus \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]3 elements don't contain a factor \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]4, i.e. are in \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]5.<br> Since the \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]6's are pairwise relatively prime (obviously) by the CRT we have that \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]7. This holds as the CRT allows us to establish a bijection between each \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]8 and a unique tuple \[\varphi(p^e) = {{c2::p^{e-1}(p-1)}}\]9 where \[ \varphi(p^e) = p^{e-1}(p - 1) \]0.</p> |
Note 410: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
eJwT]j&5OY
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Front
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
Back
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
\(\mathbb{Z}_m\) (with \(\oplus\)) is not a group with respect to multiplication modulo \(m\) because elements that are not coprime to \(m\) don't have a multiplicative inverse.
For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).
Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(\oplus\)0).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?</p> | |
| Back | <p>\(\mathbb{Z}_m\) (with \(\oplus\)) is <strong>not a group</strong> with respect to multiplication modulo \(m\) because elements that are <strong>not coprime</strong> to \(m\) don't have a <strong>multiplicative inverse</strong>.</p> <p>For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).</p> <p>Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(\oplus\)0).</p> |
Note 411: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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ivOfI913lL
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Front
Is \(\mathbb{Z}_m^*\) a group?.
Back
Is \(\mathbb{Z}_m^*\) a group?.
Theorem 5.13: \(\langle \mathbb{Z}_m^*; \odot, \text{ }^{-1}, 1 \rangle\) is a group.
Proof idea: For \(a, b \in \mathbb{Z}_m^*\), if \(\gcd(a, m) = 1\) and \(\gcd(b, m) = 1\), then \(\gcd(ab, m) = 1\). Thus the group is closed under multiplication.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Is \(\mathbb{Z}_m^*\) a group?.</p> | |
| Back | <p><strong>Theorem 5.13</strong>: \(\langle \mathbb{Z}_m^*; \odot, \text{ }^{-1}, 1 \rangle\) is a <strong>group</strong>.</p> <p><strong>Proof idea</strong>: For \(a, b \in \mathbb{Z}_m^*\), if \(\gcd(a, m) = 1\) and \(\gcd(b, m) = 1\), then \(\gcd(ab, m) = 1\). Thus the group is closed under multiplication.</p> |
Note 412: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
FSUY[I=V>]
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Front
State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
Back
State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
Corollary 5.14 (Fermat's Little Theorem): For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]
In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]
Proof: This follows from Corollary 5.10 (\(a^{|G|} = e\)) since the order of \(\mathbb{Z}_m^*\) is \[a^{p-1} \equiv_p 1\]0 (and \[a^{p-1} \equiv_p 1\]1 for prime \[a^{p-1} \equiv_p 1\]2).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):</p> | |
| Back | <p><strong>Corollary 5.14 (Fermat's Little Theorem)</strong>: For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]</p> <p>In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]</p> <p><strong>Proof</strong>: This follows from Corollary 5.10 (\(a^{|G|} = e\)) since the order of \(\mathbb{Z}_m^*\) is \[a^{p-1} \equiv_p 1\]0 (and \[a^{p-1} \equiv_p 1\]1 for prime \[a^{p-1} \equiv_p 1\]2).</p> |
Note 413: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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m5H0_vW**A
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Front
Compute \(\varphi(60)\) using the prime factorization method.
Back
Compute \(\varphi(60)\) using the prime factorization method.
First, find the prime factorization: \(60 = 2^2 \cdot 3 \cdot 5\)
\[\varphi(60) = \varphi(2^2) \cdot \varphi(3) \cdot \varphi(5)\]
\[= 2^{2-1}(2-1) \cdot 3^{1-1}(3-1) \cdot 5^{1-1}(5-1)\]
\[= 2 \cdot 1 \cdot 1 \cdot 2 \cdot 1 \cdot 4 = 2 \cdot 2 \cdot 4 = 16\]
So \(\varphi(60) = 16\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Compute \(\varphi(60)\) using the prime factorization method.</p> | |
| Back | <p>First, find the prime factorization: \(60 = 2^2 \cdot 3 \cdot 5\)</p> <p>\[\varphi(60) = \varphi(2^2) \cdot \varphi(3) \cdot \varphi(5)\]</p> <p>\[= 2^{2-1}(2-1) \cdot 3^{1-1}(3-1) \cdot 5^{1-1}(5-1)\]</p> <p>\[= 2 \cdot 1 \cdot 1 \cdot 2 \cdot 1 \cdot 4 = 2 \cdot 2 \cdot 4 = 16\]</p> <p>So \(\varphi(60) = 16\).</p> |
Note 414: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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PjfIvXynOi
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Front
Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?
Back
Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?
| \(\mathbb{Z}_m\) | \(\mathbb{Z}_m^*\) | |
|---|---|---|
| \(\oplus\) | Yes (forms a group) | No |
| \(\odot\) | No | Yes (forms a group) |
Key point: \(\mathbb{Z}_m\) with addition \(\oplus\) is a group. \(\mathbb{Z}_m^*\) (coprime elements) with multiplication \(\odot\) is a group.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?</p> | |
| Back | <table> <thead> <tr> <th></th> <th>\(\mathbb{Z}_m\)</th> <th>\(\mathbb{Z}_m^*\)</th> </tr> </thead> <tbody> <tr> <td>\(\oplus\)</td> <td><strong>Yes</strong> (forms a group)</td> <td>No</td> </tr> <tr> <td>\(\odot\)</td> <td>No</td> <td><strong>Yes</strong> (forms a group)</td> </tr> </tbody> </table> <p><strong>Key point</strong>: \(\mathbb{Z}_m\) with addition \(\oplus\) is a group. \(\mathbb{Z}_m^*\) (coprime elements) with multiplication \(\odot\) is a group.</p> |
Note 415: ETH::DiskMat
Deck: ETH::DiskMat
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IL*Um}/|aF
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Front
A ring is called commutative if multiplication is commutative: \(ab = ba\).
Back
A ring is called commutative if multiplication is commutative: \(ab = ba\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A ring is called {{c1::commutative}} if {{c2::multiplication is commutative}}: {{c2::\(ab = ba\)}}.</p> |
Note 416: ETH::DiskMat
Deck: ETH::DiskMat
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Lemma 5.17(4): If a ring \(R\) is non-trivial (has more than one element), then \(1 \neq 0\).
Back
Lemma 5.17(4): If a ring \(R\) is non-trivial (has more than one element), then \(1 \neq 0\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p><strong>Lemma 5.17(4)</strong>: If a ring \(R\) is {{c1::non-trivial (has more than one element)}}, then {{c2::\(1 \neq 0\)}}.</p> |
Note 417: ETH::DiskMat
Deck: ETH::DiskMat
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An element \(u\) of a ring \(R\) is called a unit if \(u\) is invertible.
Back
An element \(u\) of a ring \(R\) is called a unit if \(u\) is invertible.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>An element \(u\) of a ring \(R\) is called a {{c1::unit}} if \(u\) is {{c2::invertible}}.</p> |
Note 418: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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nf).~SMKa%
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Front
If \(uv = vu = 1\) for some \(v \in R\) (we write \(v = u^{-1}\)), then \(u\) is a?
Back
If \(uv = vu = 1\) for some \(v \in R\) (we write \(v = u^{-1}\)), then \(u\) is a?
Unit.
Example The units of \(\mathbb{Z}\) are \(-1\) and \(1\). Therefore \(\mathbb{Z}^* = \{-1, 1\}\). In contrast, \(\mathbb{R}^* = \mathbb{R} \backslash \{0\}\), as we can divide any two numbers.
The set of units of \(R\) is denoted by \(R^*\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>If \(uv = vu = 1\) for some \(v \in R\) (we write \(v = u^{-1}\)), then \(u\) is a?</p> | |
| Back | <p>Unit.</p> <p><strong>Example</strong> The units of \(\mathbb{Z}\) are \(-1\) and \(1\). Therefore \(\mathbb{Z}^* = \{-1, 1\}\). In contrast, \(\mathbb{R}^* = \mathbb{R} \backslash \{0\}\), as we can divide any two numbers.</p> <p>The set of units of \(R\) is denoted by \(R^*\).</p> |
Note 419: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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cAat^jY(>E
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Front
The set of units of \(R\) is denoted by \(R^*\) and \(R^*\) is a group. This holds as we can easily see that every element of \(R^*\) has an inverse by definition. Thus the axiom \(G3\) holds.
Back
The set of units of \(R\) is denoted by \(R^*\) and \(R^*\) is a group. This holds as we can easily see that every element of \(R^*\) has an inverse by definition. Thus the axiom \(G3\) holds.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c1::set of units}} of \(R\) is denoted by {{c2::\(R^*\)}} and {{c3::\(R^*\) is a group. This holds as we can easily see that every element of \(R^*\) has an inverse by definition. Thus the axiom \(G3\) holds}}.</p> |
Note 420: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Front
State Lemma 5.18 about the units of a ring.
Back
State Lemma 5.18 about the units of a ring.
Lemma 5.18: For a ring \(R\), \(R^*\) is a group (the multiplicative group of units of \(R\)).
Proof idea: Every element of \(R^*\) has an inverse by definition, so axiom G3 holds. The other group axioms (associativity, neutral element) are inherited from the ring.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Lemma 5.18 about the units of a ring.</p> | |
| Back | <p><strong>Lemma 5.18</strong>: For a ring \(R\), \(R^*\) is a <strong>group</strong> (the multiplicative group of units of \(R\)).</p> <p><strong>Proof idea</strong>: Every element of \(R^*\) has an inverse by definition, so axiom <strong>G3</strong> holds. The other group axioms (associativity, neutral element) are inherited from the ring.</p> |
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For \(a, b\) in a commutative ring \(R\), we say that \(a\) divides \(b\), denoted \(a \ | \ b\), if there exists a \(c \in R\) such that \(b = ac\).
Back
For \(a, b\) in a commutative ring \(R\), we say that \(a\) divides \(b\), denoted \(a \ | \ b\), if there exists a \(c \in R\) such that \(b = ac\).
Field-by-field Comparison
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|---|---|---|
| Text | <p>For \(a, b\) in a <strong>commutative</strong> ring \(R\), we say that {{c1::\(a\) divides \(b\), denoted \(a \ | \ b\)}}, if {{c2:: there exists a \(c \in R\) such that \(b = ac\)}}.</p> |
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In a ring, \(d\) is a gcd of \(a\) and \(b\) if:
Back
In a ring, \(d\) is a gcd of \(a\) and \(b\) if:
For any ring elements \(a\) and \(b\) in \(R\) (not both \(0\)), a ring element \(d\) is called a greatest common divisor of \(a\) and \(b\) if:
- \(d\) divides both \(a\) and \(a\)0
- Every common divisor of \(a\)1 and \(a\)2 divides \(a\)3
Formally: \[d \ | \ a \ \land \ d \ | \ b \ \land \ \forall c ((c \ | \ a \ \land \ c \ | \ b) \rightarrow c \ | \ d)\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>In a ring, \(d\) is a gcd of \(a\) and \(b\) if:</p> | |
| Back | <p>For any ring elements \(a\) and \(b\) in \(R\) (not both \(0\)), a ring element \(d\) is called a greatest common divisor of \(a\) and \(b\) if:<br> - \(d\) divides both \(a\) and \(a\)0<br> - Every common divisor of \(a\)1 and \(a\)2 divides \(a\)3</p> <p>Formally: \[d \ | \ a \ \land \ d \ | \ b \ \land \ \forall c ((c \ | \ a \ \land \ c \ | \ b) \rightarrow c \ | \ d)\]</p> |
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What are the units of \(\mathbb{Z}\) and \(\mathbb{R}\)?
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What are the units of \(\mathbb{Z}\) and \(\mathbb{R}\)?
- Units of \(\mathbb{Z}\): \(\mathbb{Z}^* = \{-1, 1\}\) (only elements with multiplicative inverse in \(\mathbb{Z}\))
- Units of \(\mathbb{R}\): \(\mathbb{R}^* = \mathbb{R} \setminus \{0\}\) (all non-zero reals have multiplicative inverse)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What are the units of \(\mathbb{Z}\) and \(\mathbb{R}\)?</p> | |
| Back | <ul> <li><strong>Units of \(\mathbb{Z}\)</strong>: \(\mathbb{Z}^* = \{-1, 1\}\) (only elements with multiplicative inverse in \(\mathbb{Z}\))</li> <li><strong>Units of \(\mathbb{R}\)</strong>: \(\mathbb{R}^* = \mathbb{R} \setminus \{0\}\) (all non-zero reals have multiplicative inverse)</li> </ul> |
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What is the characteristic of \(\mathbb{Z}_m\)?
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What is the characteristic of \(\mathbb{Z}_m\)?
The characteristic of \(\mathbb{Z}_m\) is \(m\).
Explanation: The characteristic is the order of \(1\) in the additive group. In \(\mathbb{Z}_m\), adding \(1\) to itself \(m\) times gives: \[\underbrace{1 + 1 + \cdots + 1}_{m \text{ times}} = m \equiv_m 0\]
So \(\text{ord}(1) = m\).
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| Field | Before | After |
|---|---|---|
| Front | <p>What is the characteristic of \(\mathbb{Z}_m\)?</p> | |
| Back | <p>The characteristic of \(\mathbb{Z}_m\) is <strong>\(m\)</strong>.</p> <p><strong>Explanation</strong>: The characteristic is the order of \(1\) in the additive group. In \(\mathbb{Z}_m\), adding \(1\) to itself \(m\) times gives: \[\underbrace{1 + 1 + \cdots + 1}_{m \text{ times}} = m \equiv_m 0\]</p> <p>So \(\text{ord}(1) = m\).</p> |
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An integral domain \(D\) is a (nontrivial, \(0 \neq 1\)) commutative ring without c3:
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An integral domain \(D\) is a (nontrivial, \(0 \neq 1\)) commutative ring without c3:
Field-by-field Comparison
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|---|---|---|
| Text | <p>An {{c1::integral domain \(D\)}} is a {{c2::(nontrivial, \(0 \neq 1\)) commutative ring without c3::zerodivisors (\(ab = 0 \implies a = 0 \lor b = 0\))}}:</p> |
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What is a zerodivisor?
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What is a zerodivisor?
A zerodivisor \(a \in R, a \neq 0\) in a commutative ring such that \(ab = ba = 0\) for \(b \neq 0\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is a zerodivisor?</p> | |
| Back | <p>A zerodivisor \(a \in R, a \neq 0\) in a commutative ring such that \(ab = ba = 0\) for \(b \neq 0\).</p> |
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Name a zerodivisor in a Ring.
Back
Name a zerodivisor in a Ring.
\(2\) is a zerodivisor of \(\mathbb_{Z}_4\), as \(2*2 = 0\).
Field-by-field Comparison
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|---|---|---|
| Front | <p>Name a zerodivisor in a Ring.</p> | |
| Back | <p>\(2\) is a zerodivisor of \(\mathbb_{Z}_4\), as \(2*2 = 0\).</p> |
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Which of the following are integral domains: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_6, \mathbb{Z}_7\)?
Back
Which of the following are integral domains: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_6, \mathbb{Z}_7\)?
Integral domains: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_7\) (where \(7\) is prime)
Not integral domains: \(\mathbb{Z}_6\) (since \(6\) is not prime)
Explanation: \(\mathbb{Z}_m\) is an integral domain if and only if \(m\) is prime. For \(\mathbb{Z}_6\), we have \(2 \cdot 3 \equiv_6 0\), so \(2\) and \(3\) are zerodivisors.
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|---|---|---|
| Front | <p>Which of the following are integral domains: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_6, \mathbb{Z}_7\)?</p> | |
| Back | <p><strong>Integral domains</strong>: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_7\) (where \(7\) is prime)</p> <p><strong>Not integral domains</strong>: \(\mathbb{Z}_6\) (since \(6\) is not prime)</p> <p><strong>Explanation</strong>: \(\mathbb{Z}_m\) is an integral domain if and only if \(m\) is prime. For \(\mathbb{Z}_6\), we have \(2 \cdot 3 \equiv_6 0\), so \(2\) and \(3\) are zerodivisors.</p> |
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State Lemma 5.20 about division in integral domains: (The quotient has what property?)
Back
State Lemma 5.20 about division in integral domains: (The quotient has what property?)
Lemma 5.20: In an integral domain, if \(a | b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is unique and is denoted by \(c = b/a\) (the quotient).
Explanation: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\), so \(b = ac\)0.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Lemma 5.20 about division in integral domains: (The quotient has what property?)</p> | |
| Back | <p><strong>Lemma 5.20</strong>: In an integral domain, if \(a | b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is <strong>unique</strong> and is denoted by \(c = b/a\) (the quotient).</p> <p><strong>Explanation</strong>: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\), so \(b = ac\)0.</p> |
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Front
What is a polynomial over a commutative ring?
Back
A polynomial \(a(x)\) over a commutative ring \(R\) in the indeterminate \(x\) is a formal expression of the form: \[ a(x) = {{c2::a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1 x + a_0}} = \sum_{i=0}^d a_i x^i \] for some non-negative integer \(d\), with \(a_i \in R\).
The set of polynomials in \(x\) over \(R\) is denoted \(R[x]\).
Back
Front
What is a polynomial over a commutative ring?
Back
A polynomial \(a(x)\) over a commutative ring \(R\) in the indeterminate \(x\) is a formal expression of the form: \[ a(x) = {{c2::a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1 x + a_0}} = \sum_{i=0}^d a_i x^i \] for some non-negative integer \(d\), with \(a_i \in R\).
The set of polynomials in \(x\) over \(R\) is denoted \(R[x]\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>What is a polynomial over a commutative ring?</p> <h1>Back</h1> <p>A polynomial \(a(x)\) over a commutative ring \(R\) in the indeterminate \(x\) is a formal expression of the form: \[ a(x) = {{c2::a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1 x + a_0}} = \sum_{i=0}^d a_i x^i \] for some non-negative integer \(d\), with \(a_i \in R\).</p> <p>The set of polynomials in \(x\) over \(R\) is denoted {{c4::\(R[x]\)}}.</p> |
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The degree of \(a(x)\), denoted \(\deg(a(x))\), is the greatest \(i\) for which \(a_i \neq 0\).
Back
The degree of \(a(x)\), denoted \(\deg(a(x))\), is the greatest \(i\) for which \(a_i \neq 0\).
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|---|---|---|
| Text | <p>The {{c1::degree of \(a(x)\), denoted \(\deg(a(x))\)}}, is the {{c3::greatest \(i\) for which \(a_i \neq 0\)}}.</p> |
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The polynomial \(0\) (all \(a_i\) are \(0\)) is defined to have degree \(-\infty\).
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The polynomial \(0\) (all \(a_i\) are \(0\)) is defined to have degree \(-\infty\).
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|---|---|---|
| Text | <p>The polynomial {{c1::\(0\) (all \(a_i\) are \(0\))}} is defined to have degree {{c2::\(-\infty\)}}.</p> |
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The degree of the sum of two polynomials is at most the maximum (can be smaller if the biggest coefficients cancel) of their degrees.
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The degree of the sum of two polynomials is at most the maximum (can be smaller if the biggest coefficients cancel) of their degrees.
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| Text | <p>The degree of the sum of two polynomials is {{c2::at most the maximum (can be smaller if the biggest coefficients cancel)}} of their degrees.</p> |
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The degree of the product of two polynomials is at most the sum of their degrees.
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The degree of the product of two polynomials is at most the sum of their degrees.
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| Text | <p>The degree of the {{c1::product}} of two polynomials is {{c2::at most the sum}} of their degrees.</p> |
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The degree of the product of two polynomials is equal to the sum of their degrees if \(R\) is an integral domain.
Back
The degree of the product of two polynomials is equal to the sum of their degrees if \(R\) is an integral domain.
Field-by-field Comparison
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| Text | <p>The degree of the product of two polynomials is {{c1::equal}} to the sum of their degrees if \(R\) is an {{c2::integral domain}}.</p> |
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For any commutative ring \(R\), \(R[x]\) is a?
Back
For any commutative ring \(R\), \(R[x]\) is a?
Theorem 5.21: For any commutative ring \(R\), \(R[x]\) is a commutative ring.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>For any <em>commutative ring</em> \(R\), \(R[x]\) is a?</p> | |
| Back | <p><strong>Theorem 5.21</strong>: For any <strong>commutative</strong> ring \(R\), \(R[x]\) is a <strong>commutative ring</strong>.</p> |
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Lemma 5.22(1): If \(D\) is an integral domain, then \(D[x]\) is also an integral domain.
Back
Lemma 5.22(1): If \(D\) is an integral domain, then \(D[x]\) is also an integral domain.
Field-by-field Comparison
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| Text | <p><strong>Lemma 5.22(1)</strong>: If \(D\) is an {{c1::integral domain}}, then {{c2::\(D[x]\) is also an integral domain}}.</p> |
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Lemma 5.22(2): In \(D[x]\) where \(D\) is an integral domain, the degree of the product of two polynomials is?
Back
Lemma 5.22(2): In \(D[x]\) where \(D\) is an integral domain, the degree of the product of two polynomials is?
The degree of their product is exactly the sum (not just at most) of their degrees.
This holds because \(ab \neq 0\) for all \(a,b \neq 0\) in an integral domain (no zerodivisors).
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| Front | <p><strong>Lemma 5.22(2)</strong>: In \(D[x]\) where \(D\) is an integral domain, the degree of the product of two polynomials is?</p> | |
| Back | <p>The degree of their product is exactly the sum (not just at most) of their degrees.</p> <p>This holds because \(ab \neq 0\) for all \(a,b \neq 0\) in an integral domain (no zerodivisors).</p> |
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Lemma 5.22(3): The units of \(D[x]\) are the constant polynomials that are units of \(D\): \(D[x]^* = D^*\).
Back
Lemma 5.22(3): The units of \(D[x]\) are the constant polynomials that are units of \(D\): \(D[x]^* = D^*\).
Field-by-field Comparison
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|---|---|---|
| Text | <p><strong>Lemma 5.22(3)</strong>: The {{c1::units of \(D[x]\)}} are the {{c2::constant polynomials that are units of \(D\): \(D[x]^* = D^*\)}}.</p> |
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A polynomial \(a(x)\) is called monic if the leading coefficient is \(1\).
Back
A polynomial \(a(x)\) is called monic if the leading coefficient is \(1\).
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| Text | <p>A polynomial \(a(x)\) is called {{c1::monic}} if the {{c2::leading coefficient is \(1\)}}.</p> |
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A polynomial \(a(x) \in F[x]\) with degree at least \(1\) is called irreducible if it is divisible only by:
Back
A polynomial \(a(x) \in F[x]\) with degree at least \(1\) is called irreducible if it is divisible only by:
- Constant polynomials (\(\deg = 0\))
- Constant multiples \(a(x)\) (itself)
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|---|---|---|
| Front | <p>A polynomial \(a(x) \in F[x]\) with degree at least \(1\) is called irreducible if it is divisible only by:</p> | |
| Back | <ul> <li>Constant polynomials (\(\deg = 0\))</li> <li>Constant multiples \(a(x)\) (itself)</li> </ul> |
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How can you check if a polynomial of degree \(d\) is irreducible?
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How can you check if a polynomial of degree \(d\) is irreducible?
To check if a polynomial of degree \(d\) is irreducible, check all monic irreducible polynomials of degree \(\leq d/2\) as possible divisors.
Why \(d/2\)? If \(a(x) = b(x) \cdot c(x)\) where \(b\) and \(c\) are non-constant, then \(\deg(b) + \deg(c) = \deg(a) = d\). So at least one of \(b\) or \(c\) has degree \(\leq d/2\).
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|---|---|---|
| Front | <p>How can you check if a polynomial of degree \(d\) is irreducible?</p> | |
| Back | <p>To check if a polynomial of degree \(d\) is irreducible, check all <strong>monic irreducible</strong> polynomials of degree \(\leq d/2\) as possible divisors.</p> <p><strong>Why \(d/2\)?</strong> If \(a(x) = b(x) \cdot c(x)\) where \(b\) and \(c\) are non-constant, then \(\deg(b) + \deg(c) = \deg(a) = d\). So at least one of \(b\) or \(c\) has degree \(\leq d/2\).</p> |
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What is the GCD in a polynomial Field
Back
What is the GCD in a polynomial Field
The monic polynomial \(g(x)\) of largest degree such that \(g(x) \ | \ a(x)\) and \(g(x) \ | \ b(x)\) is called the greatest common divisor of \(a(x)\) and \(b(x)\), denoted \(\gcd(a(x), b(x))\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the GCD in a polynomial Field</p> | |
| Back | <p>The <em>monic</em> polynomial \(g(x)\) of <em>largest degree</em> such that \(g(x) \ | \ a(x)\) and \(g(x) \ | \ b(x)\) is called the <em>greatest common divisor</em> of \(a(x)\) and \(b(x)\), denoted \(\gcd(a(x), b(x))\).</p> |
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If \(b(x)\) divides \(a(x)\), then so does:
Back
If \(b(x)\) divides \(a(x)\), then so does:
\(v \cdot b(x)\) for any nonzero \(v \in F\).
This holds because if \(a(x) = b(x) \cdot c(x)\), then \(a(x) = vb(x) \cdot (v^{-1} c(x))\).
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| Field | Before | After |
|---|---|---|
| Front | <p>If \(b(x)\) divides \(a(x)\), then so does:</p> | |
| Back | <p>\(v \cdot b(x)\) for any nonzero \(v \in F\).</p> <p>This holds because if \(a(x) = b(x) \cdot c(x)\), then \(a(x) = vb(x) \cdot (v^{-1} c(x))\).</p> |
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Euclidian Division of polynomials in a Field:
Back
Euclidian Division of polynomials in a Field:
Theorem 5.25: Let \(F\) be a field. For any \(a(x)\) and \(b(x) \neq 0\) in \(F[x]\), there exists a unique \(q(x)\) (quotient) and unique \(r(x)\) (remainder) such that: \[ a(x) = b(x) \cdot q(x) + r(x) \quad \text{and} \quad \deg(r(x)) < \deg(b(x)) \]
This is analogous to integer division with remainder.
Field-by-field Comparison
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|---|---|---|
| Front | <p>Euclidian Division of polynomials in a Field:</p> | |
| Back | <p><strong>Theorem 5.25</strong>: Let \(F\) be a field. For any \(a(x)\) and \(b(x) \neq 0\) in \(F[x]\), there exists a <strong>unique</strong> \(q(x)\) (quotient) and <strong>unique</strong> \(r(x)\) (remainder) such that: \[ a(x) = b(x) \cdot q(x) + r(x) \quad \text{and} \quad \deg(r(x)) < \deg(b(x)) \]</p> <p>This is analogous to integer division with remainder.</p> |
Note 446: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
u1$>B^csAD
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Front
How do you perform polynomial division when the divisor is not monic (e.g., in \(\text{GF}(7)[x]\))?
Back
How do you perform polynomial division when the divisor is not monic (e.g., in \(\text{GF}(7)[x]\))?
If dividing by a non-monic polynomial like \(4x + 2\) in \(\text{GF}(7)[x]\):
- Find the multiplicative inverse of the leading coefficient in the field
- For \(4\) in \(\text{GF}(7)\): \(4 \cdot 2 \equiv_7 1\), so \(4^{-1} = 2\)
- Multiply the polynomial by this inverse to make it monic
- \(2 \cdot (4x + 2) = 8x + 4 \equiv_7 x + 4\)
- Now divide by the monic polynomial
Example: \(3x^2 + 6x + 5\) divided by \(4x + 2\) becomes \(3x^2 + 6x + 5\) divided by \(\text{GF}(7)[x]\)0 (after multiplying by \(\text{GF}(7)[x]\)1).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>How do you perform polynomial division when the divisor is not monic (e.g., in \(\text{GF}(7)[x]\))?</p> | |
| Back | <p>If dividing by a non-monic polynomial like \(4x + 2\) in \(\text{GF}(7)[x]\):</p> <ol> <li>Find the multiplicative inverse of the leading coefficient in the field</li> <li>For \(4\) in \(\text{GF}(7)\): \(4 \cdot 2 \equiv_7 1\), so \(4^{-1} = 2\)</li> <li>Multiply the polynomial by this inverse to make it monic</li> <li>\(2 \cdot (4x + 2) = 8x + 4 \equiv_7 x + 4\)</li> <li>Now divide by the monic polynomial</li> </ol> <p><strong>Example</strong>: \(3x^2 + 6x + 5\) divided by \(4x + 2\) becomes \(3x^2 + 6x + 5\) divided by \(\text{GF}(7)[x]\)0 (after multiplying by \(\text{GF}(7)[x]\)1).</p> |
Note 447: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
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xkr{;Hl0wh
Previous
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Front
How do you find the GCD of two polynomials?
Back
How do you find the GCD of two polynomials?
To find \(\gcd(a(x), b(x))\):
- Find a common factor \((x - \alpha)\) using the roots method:
- Try all possible elements of the field to find roots
- If \(\alpha\) is a root of both, then \((x - \alpha)\) is a common factor
- Use division with remainder to reduce to smaller polynomials
- Repeat the process on the smaller polynomials
- Important: Don't just find all roots and multiply! A root might be repeated (e.g., \((x+1)^2\)), and you'd miss the higher multiplicity
Example: For \(a(x) = (x+1)(x+1)(x+2)\), the GCD with another polynomial might be \((x+1)^2\), not just \((x+1)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>How do you find the GCD of two polynomials?</p> | |
| Back | <p>To find \(\gcd(a(x), b(x))\):</p> <ol> <li>Find a common factor \((x - \alpha)\) using the <strong>roots method</strong>:</li> <li>Try all possible elements of the field to find roots</li> <li>If \(\alpha\) is a root of both, then \((x - \alpha)\) is a common factor</li> <li>Use <strong>division with remainder</strong> to reduce to smaller polynomials</li> <li>Repeat the process on the smaller polynomials</li> <li><strong>Important</strong>: Don't just find all roots and multiply! A root might be repeated (e.g., \((x+1)^2\)), and you'd miss the higher multiplicity</li> </ol> <p><strong>Example</strong>: For \(a(x) = (x+1)(x+1)(x+2)\), the GCD with another polynomial might be \((x+1)^2\), not just \((x+1)\).</p> |
Note 448: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
dy9U=xZ%`c
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Front
What does polynomial evaluation preserve?
Back
What does polynomial evaluation preserve?
Lemma 5.28: Polynomial evaluation is compatible with the ring operations:
- If \(c(x) = a(x) + b(x)\) then \(c(\alpha) = a(\alpha) + b(\alpha)\) for any \(\alpha\)
- If \(c(x) = a(x) \cdot b(x)\) then \(c(\alpha) = a(\alpha) \cdot b(\alpha)\) for any \(\alpha\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What does polynomial evaluation preserve?</p> | |
| Back | <p><strong>Lemma 5.28</strong>: Polynomial evaluation is compatible with the ring operations:<br> - If \(c(x) = a(x) + b(x)\) then \(c(\alpha) = a(\alpha) + b(\alpha)\) for any \(\alpha\)<br> - If \(c(x) = a(x) \cdot b(x)\) then \(c(\alpha) = a(\alpha) \cdot b(\alpha)\) for any \(\alpha\)</p> |
Note 449: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
EXrM_MIDyC
Previous
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Front
Let \(a(x) \in R[x]\). An element \(\alpha \in R\) for which \(a(\alpha) = 0\) is called a root of \(a(x)\).
Back
Let \(a(x) \in R[x]\). An element \(\alpha \in R\) for which \(a(\alpha) = 0\) is called a root of \(a(x)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>Let \(a(x) \in R[x]\). An element \(\alpha \in R\) for which {{c1::\(a(\alpha) = 0\) is called a root of \(a(x)\)}}.</p> |
Note 450: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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Front
\(\alpha \in F\) is a root of \(a(x)\) if and only if:
Back
\(\alpha \in F\) is a root of \(a(x)\) if and only if:
\((x - \alpha)\) divides \(a(x)\).
Corollary: An irreducible polynomial of degree \(\geq 2\) has no roots.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>\(\alpha \in F\) is a root of \(a(x)\) <em>if and only if</em>:</p> | |
| Back | <p>\((x - \alpha)\) divides \(a(x)\).</p> <p><strong>Corollary</strong>: An <strong>irreducible</strong> polynomial of degree \(\geq 2\) has <strong>no roots</strong>.</p> |
Note 451: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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s2,pAWT0;U
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Front
An irreducible polynomial of degree \(\geq 2\) has no roots.
Back
An irreducible polynomial of degree \(\geq 2\) has no roots.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>An <strong>irreducible</strong> polynomial of degree \(\geq 2\) has {{c1:: <strong>no roots</strong>}}.</p> |
Note 452: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
D=/jeb&[,)
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Front
When is a polynomial of degree \(2\) or \(3\) irreducible?
Back
When is a polynomial of degree \(2\) or \(3\) irreducible?
Corollary 5.30: A polynomial \(a(x)\) of degree \(2\) or \(3\) over a field \(F\) is irreducible if and only if it has no root.
Important: This doesn't work for polynomials of higher degrees! A degree \(4\) polynomial might be the product of two irreducible degree \(2\) polynomials, each with no roots.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>When is a polynomial of degree \(2\) or \(3\) irreducible?</p> | |
| Back | <p><strong>Corollary 5.30</strong>: A polynomial \(a(x)\) of degree \(2\) or \(3\) over a field \(F\) is irreducible <strong>if and only if</strong> it has <strong>no root</strong>.</p> <p><strong>Important</strong>: This doesn't work for polynomials of higher degrees! A degree \(4\) polynomial might be the product of two irreducible degree \(2\) polynomials, each with no roots.</p> |
Note 453: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
GO/(AI35~Q
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Front
If we want to use roots to check that a polynomial is irreducible, it has to have?
Back
If we want to use roots to check that a polynomial is irreducible, it has to have?
Degree \(2\) or \(3\).
Important: This doesn't work for polynomials of higher degrees! A degree \(4\) polynomial might be the product of two irreducible degree \(2\) polynomials, each with no roots.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>If we want to use roots to check that a polynomial is irreducible, it has to have?</p> | |
| Back | <p>Degree \(2\) or \(3\).</p> <p><strong>Important</strong>: This doesn't work for polynomials of higher degrees! A degree \(4\) polynomial might be the product of two irreducible degree \(2\) polynomials, each with no roots.</p> |
Note 454: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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added
Note Type: Basic (with MathJax)
GUID:
J*162N]zbU
Previous
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Front
State Theorem 5.31 about the number of roots a polynomial can have.
Back
State Theorem 5.31 about the number of roots a polynomial can have.
Theorem 5.31: For a field \(F\), a nonzero polynomial \(a(x) \in F[x]\) of degree \(d\) has at most \(d\) roots.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Theorem 5.31 about the number of roots a polynomial can have.</p> | |
| Back | <p><strong>Theorem 5.31</strong>: For a field \(F\), a nonzero polynomial \(a(x) \in F[x]\) of degree \(d\) has <strong>at most \(d\) roots</strong>.</p> |
Note 455: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
C$zk7TjRBE
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Front
A polynomial \(a(x) \in F[x]\) of degree at most \(d\) is uniquely determined by:
Back
A polynomial \(a(x) \in F[x]\) of degree at most \(d\) is uniquely determined by:
By any \(d + 1\) values of \(a(x)\), i.e., by \(a(\alpha_1), \dots, a(\alpha_{d+1})\) for any distinct \(\alpha_1, \dots, \alpha_{d+1} \in F\).
This is the basis for polynomial interpolation.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>A polynomial \(a(x) \in F[x]\) of degree <strong>at most \(d\)</strong> is <strong>uniquely determined</strong> by:</p> | |
| Back | <p>By any \(d + 1\) values of \(a(x)\), i.e., by \(a(\alpha_1), \dots, a(\alpha_{d+1})\) for any <strong>distinct</strong> \(\alpha_1, \dots, \alpha_{d+1} \in F\).</p> <p>This is the basis for polynomial interpolation.</p> |
Note 456: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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lq:b}[Y<9t
Previous
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Front
Front
Lagrange Interpolation for polynomials in a Field
Back
Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\).
Then \(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{{{c2::(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}}}{{{c3::(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}}}\]
Note that for \(u_i(x)\) to be well-defined, all constant terms \(\alpha_i - \alpha_j\) in the denominator must be invertible. This is guaranteed in a field since \(a_i - a_j \neq 0\) for \(i \neq j\) (as they are all distinct).
Back
Front
Lagrange Interpolation for polynomials in a Field
Back
Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\).
Then \(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{{{c2::(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}}}{{{c3::(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}}}\]
Note that for \(u_i(x)\) to be well-defined, all constant terms \(\alpha_i - \alpha_j\) in the denominator must be invertible. This is guaranteed in a field since \(a_i - a_j \neq 0\) for \(i \neq j\) (as they are all distinct).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>Lagrange Interpolation for polynomials in a Field</p> <h1>Back</h1> <p>Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\).</p> <p>Then \(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{{{c2::(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}}}{{{c3::(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}}}\]</p> <p>Note that for \(u_i(x)\) to be well-defined, all constant terms \(\alpha_i - \alpha_j\) in the denominator must be invertible. This is guaranteed in a field since \(a_i - a_j \neq 0\) for \(i \neq j\) (as they are all distinct).</p> |
Note 457: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
B|?G*=z[c4
Previous
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Front
Why is a polynomial of degree \(d\) uniquely determined by \(d + 1\) values of \(a(x)\)?
Back
Why is a polynomial of degree \(d\) uniquely determined by \(d + 1\) values of \(a(x)\)?
This \(a(x)\) is unique since if there was another \(a'(x)\) then \(a(x) - a'(x)\) would have at most degree \(d\) and thus at most \(d\) roots. But since \(a(x) - a'(x)\) has the same \(d + 1\) roots, it's \(0 \implies a(x) = a'(x)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Why is a polynomial of degree \(d\) <strong>uniquely</strong> determined by \(d + 1\) values of \(a(x)\)?</p> | |
| Back | <p>This \(a(x)\) is unique since if there was another \(a'(x)\) then \(a(x) - a'(x)\) would have at most degree \(d\) and thus at most \(d\) roots. But since \(a(x) - a'(x)\) has the same \(d + 1\) roots, it's \(0 \implies a(x) = a'(x)\).</p> |
Note 458: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
oPaK;$.R2B
Previous
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Front
An irreducible polynomial of degree \(\geq 2\) has no roots in the field.
Proof: If it had a root \(\alpha\), then \((x - \alpha)\) would divide it by Lemma 5.29, contradicting irreducibility.
Back
An irreducible polynomial of degree \(\geq 2\) has no roots in the field.
Proof: If it had a root \(\alpha\), then \((x - \alpha)\) would divide it by Lemma 5.29, contradicting irreducibility.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>An {{c1::irreducible}} polynomial of degree {{c2::\(\geq 2\)}} has {{c3::no roots}} in the field.</p> <p><strong>Proof</strong>: If it had a root \(\alpha\), then \((x - \alpha)\) would divide it by Lemma 5.29, contradicting irreducibility.</p> |
Note 459: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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Note Type: Cloze (with MathJax)
GUID:
y&2ryUB}aI
Previous
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Front
A ring \(R\) is a field if and only if {{c1:: \(\langle R \setminus \{0\}; \cdot, \text{ }^{-1}, 1 \rangle\) is an abelian group}}.
Back
A ring \(R\) is a field if and only if {{c1:: \(\langle R \setminus \{0\}; \cdot, \text{ }^{-1}, 1 \rangle\) is an abelian group}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A ring \(R\) is a field if and only if {{c1:: \(\langle R \setminus \{0\}; \cdot, \text{ }^{-1}, 1 \rangle\) is an abelian group}}.</p> |
Note 460: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
GUID:
q]nbKvbP{^
Previous
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Front
In a field, you can:
Back
In a field, you can:
- add
- subtract
- multiply
- divide by any nonzero element.
You can divide as in a field, the multiplicative monoid is also a group (without \(0\), thus \(0\) cannot be divided by - no inverse).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>In a field, you can:</p> | |
| Back | <ul> <li>add</li> <li>subtract</li> <li>multiply</li> <li><em>divide</em> by any nonzero element.</li> </ul> <p>You can divide as in a field, the multiplicative monoid is also a <em>group</em> (without \(0\), thus \(0\) cannot be divided by - no inverse).</p> |
Note 461: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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Gt)<8bFII>
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Front
Which of the following are fields: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5, \mathbb{Z}_6, R[x]\)?
Back
Which of the following are fields: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5, \mathbb{Z}_6, R[x]\)?
Fields: \(\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5\) (where \(5\) is prime)
Not fields:
- \(\mathbb{Z}\) (not all nonzero elements have multiplicative inverse, e.g., \(2\))
- \(\mathbb{Z}_6\) (since \(6\) is not prime, e.g., \(2\) has no inverse)
- \(R[x]\) for any ring \(R\) (polynomials don't have multiplicative inverses)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Which of the following are fields: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5, \mathbb{Z}_6, R[x]\)?</p> | |
| Back | <p><strong>Fields</strong>: \(\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5\) (where \(5\) is prime)</p> <p><strong>Not fields</strong>:<br> - \(\mathbb{Z}\) (not all nonzero elements have multiplicative inverse, e.g., \(2\))<br> - \(\mathbb{Z}_6\) (since \(6\) is not prime, e.g., \(2\) has no inverse)<br> - \(R[x]\) for any ring \(R\) (polynomials don't have multiplicative inverses)</p> |
Note 462: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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QUR}wUg]J-
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Front
State Theorem 5.23 about when \(\mathbb{Z}_p\) is a field.
Back
State Theorem 5.23 about when \(\mathbb{Z}_p\) is a field.
Theorem 5.23: \(\mathbb{Z}_p\) is a field if and only if \(p\) is prime.
Explanation: When \(p\) is prime, every non-zero element is coprime to \(p\) and thus has a multiplicative inverse. When \(p\) is composite, there exist elements without inverses (the factors of \(p\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Theorem 5.23 about when \(\mathbb{Z}_p\) is a field.</p> | |
| Back | <p><strong>Theorem 5.23</strong>: \(\mathbb{Z}_p\) is a field <strong>if and only if</strong> \(p\) is prime.</p> <p><strong>Explanation</strong>: When \(p\) is prime, every non-zero element is coprime to \(p\) and thus has a multiplicative inverse. When \(p\) is composite, there exist elements without inverses (the factors of \(p\)).</p> |
Note 463: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
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t$?o*APa?u
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Front
We denote the field with \(p\) elements (where \(p\) is prime) by {{c2::\(\text{GF}(p)\) rather than \(\mathbb{Z}_p\)}}.
Back
We denote the field with \(p\) elements (where \(p\) is prime) by {{c2::\(\text{GF}(p)\) rather than \(\mathbb{Z}_p\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>We denote the {{c1:: field with \(p\) elements (where \(p\) is prime)}} by {{c2::\(\text{GF}(p)\) rather than \(\mathbb{Z}_p\)}}.</p> |
Note 464: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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zic@0yO~I[
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Front
When is a field an integral domain?
Back
When is a field an integral domain?
Theorem 5.24: A field is always an integral domain.
Proof idea: If \(ab = 0\) in a field and \(a \neq 0\), then \(a\) has an inverse \(a^{-1}\). Multiplying both sides by \(a^{-1}\) gives \(b = a^{-1} \cdot 0 = 0\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>When is a field an integral domain?</p> | |
| Back | <p><strong>Theorem 5.24</strong>: A field is <strong>always</strong> an <strong>integral domain</strong>.</p> <p><strong>Proof idea</strong>: If \(ab = 0\) in a field and \(a \neq 0\), then \(a\) has an inverse \(a^{-1}\). Multiplying both sides by \(a^{-1}\) gives \(b = a^{-1} \cdot 0 = 0\).</p> |
Note 465: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
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Note Type: Basic (with MathJax)
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q?95w$fJDO
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Front
What is \(F[x]_{m(x)}\)?
Back
What is \(F[x]_{m(x)}\)?
Let \(m(x)\) be a polynomial of degree \(d\) over \(F\). Then: \[F[x]_{m(x)} \ \overset{\text{def}}{=} \ \{a(x) \in F[x] \ | \ \deg(a(x)) < d\}\]
This is the set of all polynomials over \(F\) with degree strictly less than \(d\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is \(F[x]_{m(x)}\)?</p> | |
| Back | <p>Let \(m(x)\) be a polynomial of degree \(d\) over \(F\). Then: \[F[x]_{m(x)} \ \overset{\text{def}}{=} \ \{a(x) \in F[x] \ | \ \deg(a(x)) < d\}\]</p> <p>This is the set of all polynomials over \(F\) with <strong>degree strictly less than \(d\)</strong>.</p> |
Note 466: ETH::DiskMat
Deck: ETH::DiskMat
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FjP~)Df]`o
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Front
All polynomials in \(F[x]_{m(x)}\)? have {{c1:: degree \(< \text{deg}(m(x))\)}}.
Back
All polynomials in \(F[x]_{m(x)}\)? have {{c1:: degree \(< \text{deg}(m(x))\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>All polynomials in \(F[x]_{m(x)}\)? have {{c1:: degree \(< \text{deg}(m(x))\)}}.</p> |
Note 467: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
q[i&,qVWc9
Previous
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Front
What is modular congruence in a field?
Back
What is modular congruence in a field?
\[a(x) \equiv_{m(x)} b(x) \quad \overset{\text{def}}{\Leftrightarrow} \quad m(x) \ | \ (a(x) - b(x))\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is modular congruence in a field?</p> | |
| Back | <p>\[a(x) \equiv_{m(x)} b(x) \quad \overset{\text{def}}{\Leftrightarrow} \quad m(x) \ | \ (a(x) - b(x))\]</p> |
Note 468: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
OkpH]*vEZ%
Previous
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Front
What are the equivalence classes modulo \(m(x)\) in a polynomial field.
Back
What are the equivalence classes modulo \(m(x)\) in a polynomial field.
Lemma 5.33: Congruence modulo \(m(x)\) is an equivalence relation on \(F[x]\), and each equivalence class has a unique representation of degree less than \(\deg(m(x))\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What are the equivalence classes modulo \(m(x)\) in a polynomial field.</p> | |
| Back | <p><strong>Lemma 5.33</strong>: Congruence modulo \(m(x)\) is an <strong>equivalence relation</strong> on \(F[x]\), and each equivalence class has a <strong>unique representation</strong> of degree less than \(\deg(m(x))\).</p> |
Note 469: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
r3)589~fN6
Previous
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New Note
Front
What is the cardinality of \(F[x]_{m(x)}\)?
Back
What is the cardinality of \(F[x]_{m(x)}\)?
Lemma 5.34: Let \(F\) be a finite field with \(q\) elements and let \(m(x)\) be a polynomial of degree \(d\) over \(F\). Then: \[|F[x]_{m(x)}| = q^d\]
Explanation: Each polynomial has \(d\) coefficients, and each coefficient can be any of \(q\) elements from \(F\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the cardinality of \(F[x]_{m(x)}\)?</p> | |
| Back | <p><strong>Lemma 5.34</strong>: Let \(F\) be a finite field with \(q\) elements and let \(m(x)\) be a polynomial of degree \(d\) over \(F\). Then: \[|F[x]_{m(x)}| = q^d\]</p> <p><strong>Explanation</strong>: Each polynomial has \(d\) coefficients, and each coefficient can be any of \(q\) elements from \(F\).</p> |
Note 470: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
E3nG0q}H>n
Previous
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New Note
Front
Is \(F[x]_{m(x)}\) a monoid, group, ring, field?
Back
Is \(F[x]_{m(x)}\) a monoid, group, ring, field?
Lemma 5.35: \(F[x]_{m(x)}\) is a ring with respect to addition and multiplication modulo \(m(x)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Is \(F[x]_{m(x)}\) a monoid, group, ring, field?</p> | |
| Back | <p><strong>Lemma 5.35</strong>: \(F[x]_{m(x)}\) is a <strong>ring</strong> with respect to addition and multiplication modulo \(m(x)\).</p> |
Note 471: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
jyS*[vJ/iH
Previous
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New Note
Front
When does an element of \(F[x]_{m(x)}\) have an inverse?
Back
When does an element of \(F[x]_{m(x)}\) have an inverse?
Lemma 5.36: The congruence equation \[a(x)b(x) \equiv_{m(x)} 1\] for a given \(a(x)\) has a solution \(b(x) \in F[x]_{m(x)}\) if and only if \(\gcd(a(x), m(x)) = 1\). The solution is unique.
In other words: \[ F[x]_{m(x)}^* = \{a(x) \in F[x]_{m(x)} \ | \ \gcd(a(x), m(x)) = 1\} \]
This is analogous to \(\mathbb{Z}_m^*\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>When does an element of \(F[x]_{m(x)}\) have an inverse?</p> | |
| Back | <p><strong>Lemma 5.36</strong>: The congruence equation \[a(x)b(x) \equiv_{m(x)} 1\] for a given \(a(x)\) has a solution \(b(x) \in F[x]_{m(x)}\) <strong>if and only if</strong> \(\gcd(a(x), m(x)) = 1\). The solution is <strong>unique</strong>.</p> <p>In other words: \[ F[x]_{m(x)}^* = \{a(x) \in F[x]_{m(x)} \ | \ \gcd(a(x), m(x)) = 1\} \]</p> <p>This is analogous to \(\mathbb{Z}_m^*\).</p> |
Note 472: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
CX)J6e_z}-
Previous
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New Note
Front
\(F[x]_{m(x)}\) is equal {{c2::to \(F[x]_{m(x)}^* \setminus \{0\}\)}} iff \(m(x)\) is irreducible.
Back
\(F[x]_{m(x)}\) is equal {{c2::to \(F[x]_{m(x)}^* \setminus \{0\}\)}} iff \(m(x)\) is irreducible.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>\(F[x]_{m(x)}\) is equal {{c2::to \(F[x]_{m(x)}^* \setminus \{0\}\)}} iff {{c1:: \(m(x)\) is irreducible}}.</p> |
Note 473: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
nIuNPsEb_k
Previous
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New Note
Front
State Theorem 5.37 about when \(F[x]_{m(x)}\) is a field.
Back
State Theorem 5.37 about when \(F[x]_{m(x)}\) is a field.
Theorem 5.37: The ring \(F[x]_{m(x)}\) is a field if and only if \(m(x)\) is irreducible.
Explanation: If \(m(x)\) is irreducible, then \(\gcd(a(x), m(x)) = 1\) for all non-zero \(a(x)\) in \(F[x]_{m(x)}\), so all elements (except \(0\)) are units.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>State Theorem 5.37 about when \(F[x]_{m(x)}\) is a field.</p> | |
| Back | <p><strong>Theorem 5.37</strong>: The ring \(F[x]_{m(x)}\) is a field <strong>if and only if</strong> \(m(x)\) is <strong>irreducible</strong>.</p> <p><strong>Explanation</strong>: If \(m(x)\) is irreducible, then \(\gcd(a(x), m(x)) = 1\) for all non-zero \(a(x)\) in \(F[x]_{m(x)}\), so all elements (except \(0\)) are units.</p> |
Note 474: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
wJ,ON3lFCv
Previous
Note did not exist
New Note
Front
Give an example of an extension field constructed from polynomials.
Back
Give an example of an extension field constructed from polynomials.
\(\mathbb{R}[x]_{x^2+1}\) is a field, isomorphic to \(\mathbb{C\) (the complex numbers).
Explanation: \(x^2 + 1\) is irreducible over \(\mathbb{R}\) (no real roots). Elements of \(\mathbb{R}[x]_{x^2+1}\) are of the form \(a + bx\) where \(x^2 = -1\), which corresponds exactly to complex numbers \(a + bi\).
There are no other extension fields on \(\mathbb{R}\) that aren't isomorphic to \(\mathbb{C}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>Give an example of an extension field constructed from polynomials.</p> | |
| Back | <p>\(\mathbb{R}[x]_{x^2+1}\) is a field, <strong>isomorphic to \(\mathbb{C\)</strong> (the complex numbers).</p> <p><strong>Explanation</strong>: \(x^2 + 1\) is irreducible over \(\mathbb{R}\) (no real roots). Elements of \(\mathbb{R}[x]_{x^2+1}\) are of the form \(a + bx\) where \(x^2 = -1\), which corresponds exactly to complex numbers \(a + bi\).</p> <p>There are no other extension fields on \(\mathbb{R}\) that aren't isomorphic to \(\mathbb{C}\).</p> |
Note 475: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
o)+^3Q.5-H
Previous
Note did not exist
New Note
Front
When does an irreducible polynomial exist in \(\text{GF}(p)[x]\)?
Back
When does an irreducible polynomial exist in \(\text{GF}(p)[x]\)?
For every prime \(p\) and every \(d > 1\), there exists an irreducible polynomial of degree \(d\) in \(\text{GF}(p)[x]\).
In particular, there exists a finite field with \(p^d\) elements.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>When does an irreducible polynomial exist in \(\text{GF}(p)[x]\)?</p> | |
| Back | <p>For every prime \(p\) and every \(d > 1\), there exists an <strong>irreducible polynomial</strong> of degree \(d\) in \(\text{GF}(p)[x]\).</p> <p>In particular, there exists a <strong>finite field</strong> with \(p^d\) elements.</p> |
Note 476: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
u7I279IY#p
Previous
Note did not exist
New Note
Front
When is there a finite field with \(q\) elements?
Back
When is there a finite field with \(q\) elements?
\(\text{GF}(q)\) is only finite if and only if \(q\) is a power of a prime, i.e. \(q = p^k\) for \(p\) prime.
Any two fields of the same size \(q\) are isomorphic.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>When is there a finite field with \(q\) elements?</p> | |
| Back | <p>\(\text{GF}(q)\) is only finite <em>if and only if</em> \(q\) is a <em>power</em> of a prime, i.e. \(q = p^k\) for \(p\) prime.</p> <p>Any two fields of the same size \(q\) are isomorphic.</p> |
Note 477: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
vx[#sC8q?V
Previous
Note did not exist
New Note
Front
What property does every finite field \(\text{GF}(q)\) have (and what does \(q\) satisfy)?
Back
What property does every finite field \(\text{GF}(q)\) have (and what does \(q\) satisfy)?
Theorem 5.40: The multiplicative group of every finite field \(\text{GF}(q)\) is cyclic (as \(q\) is a power of a prime, if \(\text{GF}(q)\) is cyclic).
This group has order \(q - 1\) and \(\varphi(q-1)\) generators.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What property does every finite field \(\text{GF}(q)\) have (and what does \(q\) satisfy)?</p> | |
| Back | <p><strong>Theorem 5.40</strong>: The multiplicative group of every finite field \(\text{GF}(q)\) is cyclic (as \(q\) is a power of a prime, if \(\text{GF}(q)\) is cyclic).</p> <p>This group has order \(q - 1\) and \(\varphi(q-1)\) generators.</p> |
Note 478: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
NJ8=)_1qP|
Previous
Note did not exist
New Note
Front
An \((n,k)\)-encoding function \(E\) is an {{c2::injective function \(E: \mathcal{A}^k \rightarrow \mathcal{A}^n\) where \(n > k\)}}.
Back
An \((n,k)\)-encoding function \(E\) is an {{c2::injective function \(E: \mathcal{A}^k \rightarrow \mathcal{A}^n\) where \(n > k\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>An {{c1::\((n,k)\)-encoding function}} \(E\) is an {{c2::injective function \(E: \mathcal{A}^k \rightarrow \mathcal{A}^n\) where \(n > k\)}}.</p> |
Note 479: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
jTx~;>i=Aw
Previous
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New Note
Front
An encoding function maps \(k\) information symbols to $n encoded symbols.
Back
An encoding function maps \(k\) information symbols to $n encoded symbols.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>An encoding function maps {{c1::\(k\) information symbols}} to ${{c3::n encoded symbols}}.</p> |
Note 480: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
z)u:>M_Ael
Previous
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New Note
Front
The {{c2::output \((c_0, \dots, c_{n-1})\)}} of an encoding function is called a codeword.
Back
The {{c2::output \((c_0, \dots, c_{n-1})\)}} of an encoding function is called a codeword.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c2::output \((c_0, \dots, c_{n-1})\)}} of an encoding function is called a {{c1::codeword}}.</p> |
Note 481: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
dU/F
Previous
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New Note
Front
The set \(\mathcal{C} = {{c1::\text{Im}(E)}}\) is called the set of codewords.
Back
The set \(\mathcal{C} = {{c1::\text{Im}(E)}}\) is called the set of codewords.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The set \(\mathcal{C} = {{c1::\text{Im}(E)}}\) is called the {{c2::set of codewords}}.</p> |
Note 482: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
do;Stqp
Previous
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New Note
Front
An \((n,k)\)-error-correcting code over the alphabet \(\mathcal{A}\) with \(|\mathcal{A}| = q\) is a subset of \(\mathcal{A}^n\) of cardinality \(q^k\).
Back
An \((n,k)\)-error-correcting code over the alphabet \(\mathcal{A}\) with \(|\mathcal{A}| = q\) is a subset of \(\mathcal{A}^n\) of cardinality \(q^k\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>An \((n,k)\)-error-correcting code over the alphabet \(\mathcal{A}\) with \(|\mathcal{A}| = q\) is a subset of \(\mathcal{A}^n\) of cardinality {{c1::\(q^k\)}}.</p> |
Note 483: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
A9?srsv3Y:
Previous
Note did not exist
New Note
Front
What is the \((n, k)\)-error correcting code over the alphabet \(\mathcal{A}\) with $|\mathcal{A}| = q?
Back
What is the \((n, k)\)-error correcting code over the alphabet \(\mathcal{A}\) with $|\mathcal{A}| = q?
It's a subset of \(\mathcal{A}^n\) of cardinality \(q^k\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the \((n, k)\)-error correcting code over the alphabet \(\mathcal{A}\) with $|\mathcal{A}| = q?</p> | |
| Back | <p>It's a subset of \(\mathcal{A}^n\) of cardinality \(q^k\).</p> |
Note 484: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
mT06zMA!$%
Previous
Note did not exist
New Note
Front
The Hamming distance between two strings of equal length over a finite alphabet \(\mathcal{A}\) is the number of positions at which the two strings differ.
Back
The Hamming distance between two strings of equal length over a finite alphabet \(\mathcal{A}\) is the number of positions at which the two strings differ.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c1::Hamming distance}} between two strings of equal length over a finite alphabet \(\mathcal{A}\) is the {{c2::number of positions at which the two strings differ}}.</p> |
Note 485: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
bx_roOuYn/
Previous
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New Note
Front
The Hamming weight of a string in a finite alphabet \(\mathcal{A}\) is the number of positions at which the string is non-zero.
Back
The Hamming weight of a string in a finite alphabet \(\mathcal{A}\) is the number of positions at which the string is non-zero.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The {{c1::Hamming weight}} of a string in a finite alphabet \(\mathcal{A}\) is the {{c2::number of positions at which the string is non-zero}}.</p> |
Note 486: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
np*2077JVj
Previous
Note did not exist
New Note
Front
The minimum distance of an error-correcting code \(\mathcal{C}\), denoted \(d_{\min}(\mathcal{C})\), is the minimum of the Hamming distance between any two codewords.
Back
The minimum distance of an error-correcting code \(\mathcal{C}\), denoted \(d_{\min}(\mathcal{C})\), is the minimum of the Hamming distance between any two codewords.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>The minimum distance of an error-correcting code \(\mathcal{C}\), denoted \(d_{\min}(\mathcal{C})\), is the {{c3::minimum of the Hamming distance}} between any two codewords.</p> |
Note 487: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
jYfc^7cMcd
Previous
Note did not exist
New Note
Front
When is a decoding function \(t\)-error correcting?
Back
When is a decoding function \(t\)-error correcting?
A decoding function \(D\) is \(t\)-error-correcting for encoding function \(E\) if for any \((a_0, \dots, a_{k-1})\): \[D((r_0, \dots, r_{n-1})) = (a_0, \dots, a_{k-1})\] for any \((r_0, \dots, r_{n-1})\) with Hamming distance at most \(t\) from \(E((a_0, \dots, a_{k-1}))\).
In other words, every codeword with a maximum of \(t\) errors, is correctly decoded.
A code is \(t\)-error-correcting if there exists \(D\)0 and \(D\)1 with \(D\)2 where \(D\)3 is \(D\)4-error-correcting.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>When is a decoding function \(t\)-error correcting?</p> | |
| Back | <p>A decoding function \(D\) is \(t\)-error-correcting for encoding function \(E\) if for any \((a_0, \dots, a_{k-1})\): \[D((r_0, \dots, r_{n-1})) = (a_0, \dots, a_{k-1})\] for any \((r_0, \dots, r_{n-1})\) with Hamming distance at most \(t\) from \(E((a_0, \dots, a_{k-1}))\).</p> <p><em>In other words</em>, every codeword with a maximum of \(t\) errors, is correctly decoded.</p> <p>A code is \(t\)-error-correcting if there exists \(D\)0 and \(D\)1 with \(D\)2 where \(D\)3 is \(D\)4-error-correcting.</p> |
Note 488: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
h3KTs;Sad%
Previous
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New Note
Front
A code \(\mathcal{C}\) with minimum distance \(d\) is \(t\)-error correcting if and only if:
Back
A code \(\mathcal{C}\) with minimum distance \(d\) is \(t\)-error correcting if and only if:
\(d \geq 2t + 1\).
Intuition: To correct \(t\) errors, codewords must be at least \(2t + 1\) apart (so that even with \(t\) errors, the received word is closer to the correct codeword than to any other).
If they were only \(2t\) apart for each codeword, then there would be a tie.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>A code \(\mathcal{C}\) with minimum distance \(d\) is \(t\)-error correcting if and only if:</p> | |
| Back | <p>\(d \geq 2t + 1\).</p> <p><strong>Intuition</strong>: To correct \(t\) errors, codewords must be at least \(2t + 1\) apart (so that even with \(t\) errors, the received word is closer to the correct codeword than to any other).</p> <p>If they were only \(2t\) apart for each codeword, then there would be a <strong>tie</strong>.</p> |
Note 489: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
u%LA!tL]Sb
Previous
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New Note
Front
What is a polynomial based encoding function?
Back
What is a polynomial based encoding function?
Theorem 5.42: Let \(\mathcal{A} = \text{GF}(q)\) and let \(\alpha_0, \dots, \alpha_{n-1}\) be arbitrary distinct elements of \(\text{GF}(q)\). Encode by polynomial evaluation: \[E((a_0, \dots, a_{k-1})) = (a(\alpha_0), \dots, a(\alpha_{n-1}))\] where \(a(x)\) is the polynomial with coefficients \((a_0, \dots, a_{k-1})\).
The code has minimum distance \(d_{\min} = n - k + 1\).
Key property: The polynomial can be interpolated from any \(k\) values by Lagrangian interpolation. Two codewords cannot agree at \(k\) positions (else they'd be equal), so they disagree in at least \(n - k + 1\) positions.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is a polynomial based encoding function?</p> | |
| Back | <p><strong>Theorem 5.42</strong>: Let \(\mathcal{A} = \text{GF}(q)\) and let \(\alpha_0, \dots, \alpha_{n-1}\) be arbitrary distinct elements of \(\text{GF}(q)\). Encode by polynomial evaluation: \[E((a_0, \dots, a_{k-1})) = (a(\alpha_0), \dots, a(\alpha_{n-1}))\] where \(a(x)\) is the polynomial with coefficients \((a_0, \dots, a_{k-1})\).</p> <p>The code has minimum distance \(d_{\min} = n - k + 1\).</p> <p><strong>Key property</strong>: The polynomial can be interpolated from any \(k\) values by Lagrangian interpolation. Two codewords cannot agree at \(k\) positions (else they'd be equal), so they disagree in at least \(n - k + 1\) positions.</p> |
Note 490: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic (with MathJax)
GUID:
added
Note Type: Basic (with MathJax)
GUID:
n|SJ_Z2SP!
Previous
Note did not exist
New Note
Front
What is the minimum distance of two codewords in a polynomial code?
Back
What is the minimum distance of two codewords in a polynomial code?
The code has minimum distance \(d_{\min} = n - k + 1\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <p>What is the minimum distance of two codewords in a polynomial code?</p> | |
| Back | <p>The code has minimum distance \(d_{\min} = n - k + 1\).</p> |
Note 491: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
yK:4+{Du_V
Previous
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New Note
Front
A codeword \(c\) of length \(n\) in a polynomial code with degree \(k-1\) can be interpolated from any \(k\) values by Lagrangian interpolation.
Back
A codeword \(c\) of length \(n\) in a polynomial code with degree \(k-1\) can be interpolated from any \(k\) values by Lagrangian interpolation.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>A codeword \(c\) of length \(n\) in a <em>polynomial code</em> with degree \(k-1\) can be interpolated from {{c1:: <em>any \(k\) values</em> by Lagrangian interpolation}}.</p> |
Note 492: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze (with MathJax)
GUID:
added
Note Type: Cloze (with MathJax)
GUID:
GJW/OqN_%q
Previous
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New Note
Front
Two codewords in a polynomial code with degree \(k-1\) cannot agree at \(k\) positions (else they'd be equal), so they disagree in at least \(n - k + 1\) positions.
Back
Two codewords in a polynomial code with degree \(k-1\) cannot agree at \(k\) positions (else they'd be equal), so they disagree in at least \(n - k + 1\) positions.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <p>Two codewords in a <em>polynomial code</em> with degree \(k-1\) cannot agree at {{c1:: \(k\) positions (else they'd be equal)}}, so they disagree in {{c2:: at least \(n - k + 1\) positions}}.</p> |
Note 493: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
oo(x.D7C(:
Previous
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New Note
Front
What is the left cancellation law in a group?
Back
What is the left cancellation law in a group?
Left cancellation law: \(a * b = a * c \ \implies \ b = c\)
Left cancellation law: \(a * b = a * c \ \implies \ b = c\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is the left cancellation law in a group? | |
| Back | Left cancellation law: \(a * b = a * c \ \implies \ b = c\) |
Note 494: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
zDSrp9w@De
Previous
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New Note
Front
In a group, the equation \(a * x = b\) has a unique solution \(x\) for any \(a\) and \(b\) (So does the equation \(x * a = b\)).
Back
In a group, the equation \(a * x = b\) has a unique solution \(x\) for any \(a\) and \(b\) (So does the equation \(x * a = b\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In a group, the equation \(a * x = b\) has {{c1:: a unique solution \(x\)}} for any \(a\) and \(b\) {{c1:: (So does the equation \(x * a = b\))}}. |
Note 495: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
Fl3HSpM`6f
Previous
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New Note
Front
When proving \(H\) is a subgroup, we have to prove the closure of \(H\).
Back
When proving \(H\) is a subgroup, we have to prove the closure of \(H\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | When proving \(H\) is {{c2:: a subgroup}}, we have to prove the {{c1:: <b>closure</b> of \(H\)}}. |
Note 496: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
Ak;RI/ADAm
Previous
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New Note
Front
Steps to proving an isomorphism \(\phi: G \rightarrow H\):
Back
Steps to proving an isomorphism \(\phi: G \rightarrow H\):
We have to prove the map is:
We have to prove the map is:
- well-defined
- The image of \(\phi\) lies entirely within \(H\)
- homomorphism-property \(\phi(g_1 \cdot g_2) = \phi(g_1) \cdot \phi(g_2)\)
- injectivity
- surjectivity
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Steps to proving an isomorphism \(\phi: G \rightarrow H\): | |
| Back | We have to prove the map is:<br><ul><li>well-defined</li><li>The image of \(\phi\) lies entirely within \(H\)</li><li>homomorphism-property \(\phi(g_1 \cdot g_2) = \phi(g_1) \cdot \phi(g_2)\)</li><li>injectivity</li><li>surjectivity</li></ul> |
Note 497: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
.@%aS+kuV
Previous
Note did not exist
New Note
Front
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \(a0 =\) \(0a = 0\).
Back
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \(a0 =\) \(0a = 0\).
The zero (neutral of additive group) pulls all other elements to 0 by multiplication.
\(0a=(0+0)a=0a+0a\) and thus \(0a - 0a = 0a \implies 0 = 0a\)
The zero (neutral of additive group) pulls all other elements to 0 by multiplication.
\(0a=(0+0)a=0a+0a\) and thus \(0a - 0a = 0a \implies 0 = 0a\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \(a0 =\) {{c1::\(0a = 0\)}}. | |
| Back Extra | The zero (neutral of additive group) pulls all other elements to 0 by multiplication.<br><br>\(0a=(0+0)a=0a+0a\) and thus \(0a - 0a = 0a \implies 0 = 0a\) |
Note 498: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
c/L6mH(n[?
Previous
Note did not exist
New Note
Front
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)b =\) \(-(ab)\).
Back
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)b =\) \(-(ab)\).
Proof: \(ab+(−a)b=(a+(−a))b=0⋅b=0\)
Proof: \(ab+(−a)b=(a+(−a))b=0⋅b=0\)
Since \((−a)b\) satisfies \(ab+(−a)b=0\), we have \((−a)b=−(ab\)).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)b =\) {{c1::\(-(ab)\)}}. | |
| Back Extra | Proof: \(ab+(−a)b=(a+(−a))b=0⋅b=0\)<br><br><div>Since \((−a)b\) satisfies \(ab+(−a)b=0\), we have \((−a)b=−(ab\)). </div> |
Note 499: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
Pmsd]lM3W/
Previous
Note did not exist
New Note
Front
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)(-b) = \) \(ab\).
Back
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)(-b) = \) \(ab\).
\((−a)(−b)=−(a(−b))=−(−(ab))=ab\)
\((−a)(−b)=−(a(−b))=−(−(ab))=ab\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)(-b) = \) {{c1::\(ab\)}}. | |
| Back Extra | \((−a)(−b)=−(a(−b))=−(−(ab))=ab\) |
Note 500: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
px2&&RIh%e
Previous
Note did not exist
New Note
Front
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), if it's non trivial (more than one element) \(1 \neq 0\)
Back
In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), if it's non trivial (more than one element) \(1 \neq 0\)
If \(1=0\), then for all \(a \in R\) : \(a=1⋅a=0⋅a=0\)
So the ring would be trivial (only contains 0).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), if it's non trivial (more than one element) {{c1:: \(1 \neq 0\)}} | |
| Back Extra | <div>If \(1=0\), then for all \(a \in R\) : \(a=1⋅a=0⋅a=0\)</div><div><br></div><div>So the ring would be trivial (only contains 0). </div> |
Note 501: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
tNw~Zi`Up.
Previous
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New Note
Front
In any commutative ring: If \(a \ | \ b\) and \(b \ | \ c\) then \(a \ | \ c\), i.e. the relation | is transitive.
Back
In any commutative ring: If \(a \ | \ b\) and \(b \ | \ c\) then \(a \ | \ c\), i.e. the relation | is transitive.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <div>In any commutative ring: If \(a \ | \ b\) and \(b \ | \ c\) then {{c1:: \(a \ | \ c\), i.e. the relation | is transitive}}.</div> |
Note 502: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
rU5OFOfB=/
Previous
Note did not exist
New Note
Front
In any commutative ring, if \(a \ | \ b\) then \(a \ | \ bc\) for all \(c\).
Back
In any commutative ring, if \(a \ | \ b\) then \(a \ | \ bc\) for all \(c\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | <div>In any commutative ring, if \(a \ | \ b\) then {{c1:: \(a \ | \ bc\)}} for all \(c\).</div> |
Note 503: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
pVXp%#QpPg
Previous
Note did not exist
New Note
Front
In any commutative ring, if \(a \ | \ b\) and \(a \ | \ c\), then \(a \ | \ (b + c)\).
Back
In any commutative ring, if \(a \ | \ b\) and \(a \ | \ c\), then \(a \ | \ (b + c)\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In any commutative ring, if \(a \ | \ b\) and \(a \ | \ c\), then {{c1:: \(a \ | \ (b + c)\)}}. |
Note 504: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
wN=e)I[rpJ
Previous
Note did not exist
New Note
Front
Every polynomial of degree 1 is irreducible.
Back
Every polynomial of degree 1 is irreducible.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Every polynomial of degree {{c1:: 1}} is {{c2:: irreducible}}. |
Note 505: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
l^=&ux},*9
Previous
Note did not exist
New Note
Front
Every polynomial of degree 2 is either irreducible or the product of two polynomials degree 1.
Back
Every polynomial of degree 2 is either irreducible or the product of two polynomials degree 1.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Every polynomial of degree {{c1:: 2}} is either {{c2:: irreducible or the product of two polynomials degree 1}}. |
Note 506: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
o>U!Pt@-U1
Previous
Note did not exist
New Note
Front
Every polynomial of degree 3 is either irreducible, or it has at least a factor of degree 1.
Back
Every polynomial of degree 3 is either irreducible, or it has at least a factor of degree 1.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Every polynomial of degree {{c1:: 3}} is {{c2:: either irreducible, or it has at least a factor of degree 1}}. |
Note 507: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Cloze-4da8d
GUID:
added
Note Type: Cloze-4da8d
GUID:
tpZRO@D#F:
Previous
Note did not exist
New Note
Front
Every polynomial of degree 4 is either irreducible or it has a factor of degree 1 or irreducible factor of degree 2.
Back
Every polynomial of degree 4 is either irreducible or it has a factor of degree 1 or irreducible factor of degree 2.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Every polynomial of degree {{c1:: 4}} is {{c2:: either irreducible or it has a factor of degree 1 or irreducible factor of degree 2}}. |
Note 508: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Basic-e0c2b
GUID:
added
Note Type: Basic-e0c2b
GUID:
cl$26mU5(,
Previous
Note did not exist
New Note
Front
What is a zerodivisor and in which structure do they exist?
Back
What is a zerodivisor and in which structure do they exist?
A zerodivisor is an element \(a \neq 0\) in a commutative ring for which there exists a \(b \neq 0\) such that \(ab = 0\).
This is commonly encountered for the polynomial rings formed over \(\text{GF}[x]_{m(x)}\) with \(m(x)\) not irreducible (i.e. it's not a field).
A zerodivisor is an element \(a \neq 0\) in a commutative ring for which there exists a \(b \neq 0\) such that \(ab = 0\).
This is commonly encountered for the polynomial rings formed over \(\text{GF}[x]_{m(x)}\) with \(m(x)\) not irreducible (i.e. it's not a field).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is a zerodivisor and in which structure do they exist? | |
| Back | A <b>zerodivisor</b> is an element \(a \neq 0\) in a <b>commutative ring</b> for which there exists a \(b \neq 0\) such that \(ab = 0\).<br><br>This is commonly encountered for the polynomial rings formed over \(\text{GF}[x]_{m(x)}\) with \(m(x)\) not irreducible (i.e. it's not a field). |