Anki Deck Changes

Commit: c3092b18 - dailies

Author: lhorva <lhorva@student.ethz.ch>

Date: 2026-01-09T02:42:19+01:00

Changes: 14 note(s) changed (0 added, 14 modified, 0 deleted)

ℹ️ Cosmetic Changes Hidden: 4 note(s) had formatting-only changes and are not shown below • 1 HTML formatting changes • 3 mixed cosmetic changes

Note 1: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Reverso
GUID: A1y[0:/g)f
modified

Before

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Closed Walk

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Closed Walk
Graph Theory:

Zyklus

After

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Closed Walk

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Closed Walk
Zyklus
Field-by-field Comparison
Field Before After
Front Graph Theory:<br><br>Closed Walk Closed Walk
Back Graph Theory:<br><br>Zyklus Zyklus
Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 2: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Reverso
GUID: B+m&Yt;~bM
modified

Before

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Path

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Path
Graph Theory:

Pfad

After

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Path

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Path
Pfad
Field-by-field Comparison
Field Before After
Front Graph Theory:<br><br>Path Path
Back Graph Theory:<br><br>Pfad Pfad
Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 3: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: C}:U@+B*;Q
modified

Before

Front

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i^3\)}}  \(\leq\) \(O(n^4)\)  (Sum)

Back

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i^3\)}}  \(\leq\) \(O(n^4)\)  (Sum)

After

Front

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i^3\)::Sum}}  \(\leq\) \(O(n^4)\)

Back

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i^3\)::Sum}}  \(\leq\) \(O(n^4)\)
Field-by-field Comparison
Field Before After
Text {{c1:: \(\sum_{i = 1}^{n} i^3\)}}&nbsp; \(\leq\)&nbsp;{{c2::\(O(n^4)\)&nbsp;&nbsp;(Sum)}} {{c1:: \(\sum_{i = 1}^{n} i^3\)::Sum}}&nbsp; \(\leq\)&nbsp;{{c2::\(O(n^4)\)}}
Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums

Note 4: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: NU;6ob<^n3
modified

Before

Front

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)}}  \(=\) {{c2::  \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\) (Sum)}} 

Back

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)}}  \(=\) {{c2::  \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\) (Sum)}} 
inner loop depends on outer

After

Front

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)::Sum}}  \(=\) {{c2::  \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\)}} 

Back

ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)::Sum}}  \(=\) {{c2::  \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\)}} 
inner loop depends on outer
Field-by-field Comparison
Field Before After
Text {{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)}}&nbsp; \(=\)&nbsp;{{c2::&nbsp; \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\)&nbsp;(Sum)}}&nbsp; {{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)::Sum}}&nbsp; \(=\)&nbsp;{{c2::&nbsp; \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\)}}&nbsp;
Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums

Note 5: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Reverso
GUID: OW(TL-EP^P
modified

Before

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Cycle

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Cycle
Graph Theory:

Kreis

After

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Cycle

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Cycle
Kreis
Field-by-field Comparison
Field Before After
Front Graph Theory:<br><br>Cycle Cycle
Back Graph Theory:<br><br>Kreis Kreis
Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 6: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Reverso
GUID: r8fq,6tx}{
modified

Before

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Walk

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Graph Theory:

Walk
Graph Theory:

Weg

After

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Walk

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Walk
Weg
Field-by-field Comparison
Field Before After
Front Graph Theory:<br><br>Walk Walk
Back Graph Theory:<br><br>Weg Weg
Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 7: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: C18gm]huq&
modified

Before

Front

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

In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.

Back

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

In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.

After

Front

ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group:

{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} 

Back

ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
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> In a group: <br><br>{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}}&nbsp;
Extra This is a property from a Lemma.
Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups

Note 8: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: k}1~03snwg
modified

Before

Front

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::3._Divisors

In a ring, \(d\) is a gcd of \(a\) and \(b\) if:

Back

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::3._Divisors

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: \[d \ | \ a \ \land \ d \ | \ b \ \land \ \forall c ((c \ | \ a \ \land \ c \ | \ b) \rightarrow c \ | \ d)\]

After

Front

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::3._Divisors

In a ring, \(d\) is a gcd of \(a\) and \(b\) if:

Back

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::3._Divisors

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 \(b\)
- Every common divisor of \(a\) and \(b\) divides \(d\) 

Formally:\[d \ | \ a \ \land \ d \ | \ b \ \land \ \forall c ((c \ | \ a \ \land \ c \ | \ b) \rightarrow c \ | \ d)\]

Field-by-field Comparison
Field Before After
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> <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&nbsp;\(b\)<br> - Every common divisor of \(a\)&nbsp;and&nbsp;\(b\)&nbsp;divides&nbsp;\(d\)&nbsp;</p><p>Formally:\[d \ | \ a \ \land \ d \ | \ b \ \land \ \forall c ((c \ | \ a \ \land \ c \ | \ b) \rightarrow c \ | \ d)\]<br></p>
Tags: ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::3._Divisors

Note 9: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: qAyHDnFN7L
modified

Before

Front

ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
What is the number of generators of \(\mathbb{Z}_n^*\)?

Back

ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
What is the number of generators of \(\mathbb{Z}_n^*\)?
1. Verify that \(\mathbb{Z}_n^*\)is cyclic (iff n = 2, 4, \(p^e\), \(2p^e\), with \(e \ge 1\) and \(p\) is an odd prime)
2. If \(\mathbb{Z}_n^*\) is cyclic then it is isomorphic to \(\mathbb{Z}_{\varphi(n)}^+\) (by lemma) 
3. The number of generators of \(\mathbb{Z}_{\varphi(n)}^+\) is \(\varphi(\varphi(n))\) as it is the number of coprime elements of the group.

After

Front

ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
What is the number of generators of \(\mathbb{Z}_n^*\)?

Back

ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
What is the number of generators of \(\mathbb{Z}_n^*\)?
1. Verify that \(\mathbb{Z}_n^*\)is cyclic (iff n = 2, 4, \(p^e\), \(2p^e\), with \(e \ge 1\) and \(p\) is an odd prime)
2. If \(\mathbb{Z}_n^*\) is cyclic then it is isomorphic to \(\mathbb{Z}_{\varphi(n)}^+\) (by Lemma) 
3. The number of generators of \(\mathbb{Z}_{\varphi(n)}^+\) is \(\varphi(\varphi(n))\) as it is the number of elements coprime to the group order.
Field-by-field Comparison
Field Before After
Back 1. Verify that&nbsp;\(\mathbb{Z}_n^*\)is cyclic (iff n = 2, 4,&nbsp;\(p^e\),&nbsp;\(2p^e\), with&nbsp;\(e \ge 1\)&nbsp;and&nbsp;\(p\)&nbsp;is an odd prime)<br>2. If&nbsp;\(\mathbb{Z}_n^*\)&nbsp;is cyclic then it is isomorphic to&nbsp;\(\mathbb{Z}_{\varphi(n)}^+\)&nbsp;(by lemma)&nbsp;<br>3. The number of generators of&nbsp;\(\mathbb{Z}_{\varphi(n)}^+\)&nbsp;is&nbsp;\(\varphi(\varphi(n))\)&nbsp;as it is the number of coprime elements of the group. 1. Verify that&nbsp;\(\mathbb{Z}_n^*\)is cyclic (iff n = 2, 4,&nbsp;\(p^e\),&nbsp;\(2p^e\), with&nbsp;\(e \ge 1\)&nbsp;and&nbsp;\(p\)&nbsp;is an odd prime)<br>2. If&nbsp;\(\mathbb{Z}_n^*\)&nbsp;is cyclic then it is isomorphic to&nbsp;\(\mathbb{Z}_{\varphi(n)}^+\)&nbsp;(by Lemma)&nbsp;<br>3. The number of generators of&nbsp;\(\mathbb{Z}_{\varphi(n)}^+\)&nbsp;is&nbsp;\(\varphi(\varphi(n))\)&nbsp;as it is the number of elements coprime to the group order.
Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups

Note 10: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: wY#5P^[
modified

Before

Front

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains

State Lemma 5.20 about division in integral domains: (The quotient has what property?)

Back

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains

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 \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).

After

Front

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains

State Lemma 5.20 about division in integral domains: (The quotient has what property?)

Back

ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains

State Lemma 5.20 about division in integral domains: (The quotient has what property?)

Lemma 5.20: In an integral domain, if \(a \mid b\) (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 \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).

Field-by-field Comparison
Field Before After
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 \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).</p> <p><strong>Lemma 5.20</strong>: In an integral domain, if \(a \mid b\) (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 \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).</p>
Tags: ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains
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