Anki Deck Changes

Note 1: ETH::DiskMat

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

modified

Before

Front

In a Euclidean domain every element can be factored uniquely into irreducible elements (up to associates)

Back

In a Euclidean domain every element can be factored uniquely into irreducible elements (up to associates)

$a, b$ associates ($a \sim b$) if $a = ub$ for some unit $u$.

After

Front

In a Euclidean domain every element can be factored uniquely into irreducible elements (up to associates)

Back

In a Euclidean domain every element can be factored uniquely into irreducible elements (up to associates)

$a, b$ associates ($a \sim b$) if $a = ub$ for some unit $u$.

Proof sketch:

Consider a nonzero, nonunit $a \in R$.
If a is irreducible, we are done.
Otherwise, $a = bc$ with both $b,c$ nonunits.
By the Euclidean property, we may assume
$\delta(b), \delta(c) < \delta(a)$.
If either factor is reducible, factor it further.
This process must terminate, since $\delta$ takes values in $\mathbb{N}$ and strictly decreases.

Field-by-field Comparison

Field	Before	After
Extra	$a, b$ associates ($a \sim b$) if $a = ub$ for some unit $u$.	$a, b$ associates ($a \sim b$) if $a = ub$ for some unit $u$.<br><br><b>Proof sketch:<br></b><div><ol><li> <div>Consider a nonzero, nonunit $a \in R$.</div> </li><li> <div>If a is irreducible, we are done.</div> </li><li> <div>Otherwise, $a = bc$ with both $b,c$ nonunits.</div> </li><li> <div>By the Euclidean property, we may assume </div>$\delta(b), \delta(c) < \delta(a)$.</li><li> <div>If either factor is reducible, factor it further.</div> </li><li> <div>This process <b>must terminate</b>, since $\delta$ takes values in $\mathbb{N}$ and strictly decreases.</div></li></ol></div>

Tags: ETH::1._Semester::DiskMat::5._Algebra::6._Polynomials_over_a_Field::3._Analogies_Between_Z_and_F[x],_Euclidean_Domains_*

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Mqpm#lUsGl

modified

Before

Front

A function $f: A \rightarrow B$ has a right inverse if and only if $f$ is surjective.

Back

A function $f: A \rightarrow B$ has a right inverse if and only if $f$ is surjective.

We can create a function $g$ that outputs a unique value in $A$ for every input $b$. We can then revert it with $f$. Therefore, $\forall (f \circ g) b = b \iff f \circ g = \text{id}_B$

After

Front

A function $f: A \rightarrow B$ has a right inverse if and only if $f$ is surjective (not in script).

Back

A function $f: A \rightarrow B$ has a right inverse if and only if $f$ is surjective (not in script).

We can create a function $g$ that outputs a unique value in $A$ for every input $b$. We can then revert it with $f$. Therefore, $\forall (f \circ g) b = b \iff f \circ g = \text{id}_B$

Field-by-field Comparison

Field	Before	After
Text	<p>A function $f: A \rightarrow B$ has a {{c1::right inverse}} if and only if $f$ is {{c2::surjective}}.</p>	<p>A function $f: A \rightarrow B$ has a {{c1::right inverse}} if and only if $f$ is {{c2::surjective}} (not in script).</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Qn4Vs7Ck2H

modified

Before

Front

An interpretation consists of {{c1::a set $\mathcal{Z} \subseteq \Lambda$ of $\Lambda$}}, {{c2::a domain (a set of possible values) for each symbol in $\mathcal{Z}$}}, and {{c3::a function that assigns to each symbol in $\mathcal{Z}$ a value in the associated domain}}.

Back

An interpretation consists of {{c1::a set $\mathcal{Z} \subseteq \Lambda$ of $\Lambda$}}, {{c2::a domain (a set of possible values) for each symbol in $\mathcal{Z}$}}, and {{c3::a function that assigns to each symbol in $\mathcal{Z}$ a value in the associated domain}}.

A set of symbols $\mathcal{Z} \subseteq \Lambda$
- $\Lambda$ is the "alphabet" or collection of all available symbols
- $\mathcal{Z}$ is the subset of symbols we're actually interpreting
A domain for each symbol
- For each symbol in $\mathcal{Z}$, there's a set of possible values it could take
- Often the domain is defined in terms of the universe $U$ where a symbol can be a function, predicate or element of $U$.
An assignment function
- For each symbol in $\mathcal{Z}$, the function picks one specific value from its domain
- This gives meaning to each symbol

After

Front

An interpretation consists of {{c1::a set $\mathcal{Z} \subseteq \Lambda$ of $\Lambda$}}, {{c2::a domain (a set of possible values) for each symbol in $\mathcal{Z}$}}, and {{c3::a function that assigns to each symbol in $\mathcal{Z}$ a value in the associated domain}}.

Back

An interpretation consists of {{c1::a set $\mathcal{Z} \subseteq \Lambda$ of $\Lambda$}}, {{c2::a domain (a set of possible values) for each symbol in $\mathcal{Z}$}}, and {{c3::a function that assigns to each symbol in $\mathcal{Z}$ a value in the associated domain}}.

A set of symbols $\mathcal{Z} \subseteq \Lambda$
- $\Lambda$ is the "alphabet" or collection of all available symbols
- $\mathcal{Z}$ is the subset of symbols we're actually interpreting
A domain for each symbol
- For each symbol in $\mathcal{Z}$, there's a set of possible values it could take
- Often the domain is defined in terms of the universe $U$ where a symbol can be a function, predicate or element of $U$.
An assignment function

For each symbol in $\mathcal{Z}$, the function picks one specific value from its domain
This gives meaning to each symbol

An interpretation can be described either as

one big assignment function over typed symbols, or
a structured tuple that spells out those assignments separately.

Field-by-field Comparison

Field	Before	After
Extra	<ol><li><~~strong~~>A set of symbols</~~strong~~> $\mathcal{Z} \subseteq \Lambda$<ul> <li>$\Lambda$ is the "alphabet" or collection of all available symbols </li> <li>$\mathcal{Z}$ is the subset of symbols we're actually interpreting </li> </ul> </li> <li><~~strong~~>A domain for each symbol</~~strong~~> <ul> <li>For each symbol in $\mathcal{Z}$, there's a set of possible values it could take </li> <li>Often the domain is defined in terms of the <i>universe</i> $U$ where a symbol can be a function, predicate or element of $U$.<br><ol></ol></li> </ul> </li> <li><~~strong~~>An assignment function</~~strong>~~ <ul> <li>For each symbol in $\mathcal{Z}$, the function picks one specific value from its domain </li> <li>This gives meaning to each symbol</li> </ul> </~~li>~~ <h2></h2></ol>	<ol><li><b>A set of symbols</b> $\mathcal{Z} \subseteq \Lambda$<ul> <li>$\Lambda$ is the "alphabet" or collection of all available symbols </li> <li>$\mathcal{Z}$ is the subset of symbols we're actually interpreting </li> </ul> </li> <li><b>A domain for each symbol</b> <ul> <li>For each symbol in $\mathcal{Z}$, there's a set of possible values it could take </li> <li>Often the domain is defined in terms of the <i>universe</i> $U$ where a symbol can be a function, predicate or element of $U$.<br><ol></ol></li> </ul> </li> <li><b>An assignment function</b> </li><ul> <li>For each symbol in $\mathcal{Z}$, the function picks one specific value from its domain </li> <li>This gives meaning to each symbol</li></ul></ol><b>An interpretation can be described either as</b><br><ul><li>one big assignment function over typed symbols,<b> or</b><br></li><li>a structured tuple that spells out those assignments separately.</li></ul><ol> <h2></h2></ol>

Tags: ETH::1._Semester::DiskMat::6._Logic::3._Elementary_General_Concepts_in_Logic::3._Semantics::Interpretation

Note 4: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: s*8T*K?3f=

modified

Before

Front

A function $f: A \rightarrow B$ has a left inverse if and only if $f$ is injective.

Back

A function $f: A \rightarrow B$ has a left inverse if and only if $f$ is injective.

After

Front

A function $f: A \rightarrow B$ has a left inverse if and only if $f$ is injective (not in script).

Back

A function $f: A \rightarrow B$ has a left inverse if and only if $f$ is injective (not in script).

Field-by-field Comparison

Field	Before	After
Text	<p>A function $f: A \rightarrow B$ has a {{c1::left inverse}} if and only if $f$ is {{c2::injective}}.</p>	<p>A function $f: A \rightarrow B$ has a {{c1::left inverse}} if and only if $f$ is {{c2::injective}} (not in script).</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups

Note 5: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: u${[$*iYrd

modified

Before

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.

After

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 $3 \times 3$ invertible 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
Back	<p><strong>No!</strong> The uniqueness of the neutral element does <strong>not</strong> imply commutativity.</p><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>	<p><strong>No!</strong> The uniqueness of the neutral element does <strong>not</strong> imply commutativity.</p><p><strong>Counterexample</strong>: The identity matrix $I_3$ is the unique neutral element for the group of $3 \times 3$ <i>invertible</i> 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>

Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups