Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Av3Ww9Kn5Y

modified

Before

Front

For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, then {{c1::\(P_i^{(k)}(t_1, \dots, t_k)\) is a formula}}, called an atomic formula.

Back

For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, then {{c1::\(P_i^{(k)}(t_1, \dots, t_k)\) is a formula}}, called an atomic formula.

After

Front

For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, then {{c1::\(P_i^{(k)}(t_1, \dots, t_k)\) is a formula}}, called an atomic formula.

Back

For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, then {{c1::\(P_i^{(k)}(t_1, \dots, t_k)\) is a formula}}, called an atomic formula.

A formula in 1st order logic with no logical connectives (like \(\lnot, \land, \lor, \rightarrow \)) and no quantifiers (\(\forall, \exists\))

Field-by-field Comparison

Field	Before	After
Extra		A formula in 1st order logic with <b>no logical connectives</b> (like \(\lnot, \land, \lor, \rightarrow \)) and <b>no quantifiers</b> (\(\forall, \exists\))

Tags: ETH::1._Semester::DiskMat::6._Logic::6._Predicate_Logic_(First-order_Logic)::1._Syntax

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: CX)J6e_z}-

modified

Before

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.

After

Front

\(F[x]_{m(x)} =\) {{c2::\(F[x]_{m(x)}^* \cup \{0\}\)}} iff \(m(x)\) is irreducible.

Back

\(F[x]_{m(x)} =\) {{c2::\(F[x]_{m(x)}^* \cup \{0\}\)}} 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>	<p>\(F[x]_{m(x)} =\) {{c2::\(F[x]_{m(x)}^* \cup \{0\}\)}} iff {{c1:: \(m(x)\) is irreducible}}.</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::8._Finite_Fields::1._The_Ring_F[x]ₘ₍ₓ₎

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: G&Y|dtr7^k

modified

Before

Front

If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?

Back

If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?

\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]

After

Front

If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?

Back

If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?

\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]
Explanation:
For primes: a number is not coprime to \(p^e\) \(\iff\) it contains a factor of \(p\). These are \(p, 2p, 3p, \ldots, p^{e-1}p\) (exactly \(p^{e-1}\) numbers) \(\implies\)\(\varphi(p^e) = p^e - p^{e - 1} = (p-1)p^{e-1}\)
For all numbers: \(\varphi(mn)=\varphi(m)\varphi(n)\) if \(m\) and \(n\) are coprime, so we end up w/ a product.

Field-by-field Comparison

Field	Before	After
Back	\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]	\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]<br><b>Explanation:<br></b><i>For primes:</i> a number is not coprime to \(p^e\) \(\iff\) it contains a factor of \(p\). These are \(p, 2p, 3p, \ldots, p^{e-1}p\) (exactly \(p^{e-1}\) numbers) \(\implies\)\(\varphi(p^e) = p^e - p^{e - 1} = (p-1)p^{e-1}\)<br><i>For all numbers:</i> \(\varphi(mn)=\varphi(m)\varphi(n)\) if \(m\) and \(n\) are coprime, so we end up w/ a product.<br>

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function

Note 4: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: hAzQO,E_+E

modified

Before

Front

What property does the order of elements in finite groups have?

Back

What property does the order of elements in finite groups have?

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.)

After

Front

What property does the order of elements in finite groups have?

Back

What property does the order of elements in finite groups have?

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.)

Proof: Since the order is finite, elements must repeat. That means, there exist \(m > n \geq 0\) s.t. \(g^m = g^n\)
\(\implies g^{m-n} = e\)

Field-by-field Comparison

Field	Before	After
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>	<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><p><b>Proof:</b> Since the order is finite, elements must repeat. That means, there exist \(m > n \geq 0\) s.t. \(g^m = g^n\)<br>\(\implies g^{m-n} = e\)<br></p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::4._The_Order_of_Group_Elements_and_of_a_Group

Note 5: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: q|}rXYFly~

modified

Before

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 \(\mathbb{Z}_m^*\): \[\varphi(m) = |\mathbb{Z}_m^*|\]}}

After

Front

Euler's totient function \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) is defined as {{c2::the cardinality of \(\mathbb{Z}_m^*\): \[\varphi(m) = |\mathbb{Z}_m^*|\]}}

Back

Euler's totient 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	~~The~~ {{c1::Euler function}} \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) is defined as {{c2::the cardinality of \(\mathbb{Z}_m^\): \[\varphi(m) = \|\mathbb{Z}_m^\|\]}}	{{c1::Euler's totient function}} \(\varphi: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\) is defined as {{c2::the cardinality of \(\mathbb{Z}_m^\): \[\varphi(m) = \|\mathbb{Z}_m^\|\]}}

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function

Note 6: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: vx[#sC8q?V

modified

Before

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.

Note that even though q is not prime thus not every integer is comprime, GF(q) is not Z_q.

After

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.

Note that even though q is not prime thus not every integer is coprime, GF(q) is not Z_q.

Field-by-field Comparison

Field	Before	After
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><p><i>Note that even though q is not prime thus not every integer is comprime, GF(q) is <b>not</b> Z_q.</i></p>	<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><p><i>Note that even though q is not prime thus not every integer is coprime, GF(q) is <b>not</b> Z_q.</i></p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::8._Finite_Fields::3._Some_Facts_About_Finite_Fields_*

Note 7: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: ykM`*q&]Lu

modified

Before

Front

\(F \equiv G\) 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 \(G\).

Back

\(F \equiv G\) 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 \(G\).

After

Front

\(F \equiv G\) means F and G are equivalent, i.e., their truth values are equal for all truth assignments to the propositional symbols appearing in \(F\) or \(G\).

Back

\(F \equiv G\) means F and G are equivalent, 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
Text	{{c2::\(F \equiv G\)}} means {{c1:: ~~they correspond to the same function~~}}, i.e., {{c3:: their truth values are equal for <strong>all</strong> truth assignments to the propositional symbols appearing in \(F\) or \(G\)}}.	{{c2::\(F \equiv G\)}} means {{c1::F and G are equivalent}}, i.e., {{c3:: their truth values are equal for <strong>all</strong> truth assignments to the propositional symbols appearing in \(F\) or \(G\)}}.

Tags: ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::3._Logical_Equivalence_and_some_Basic_Laws