Anki Deck Changes

Commit: 386f00be - fixed all close that bish

Author: obrhubr <obrhubr@gmail.com>

Date: 2025-12-18T17:41:55+01:00

Changes: 59 note(s) changed (5 added, 54 modified, 0 deleted)

ℹ️ Cosmetic Changes Hidden: 28 note(s) had formatting-only changes and are not shown below • 1 HTML formatting changes

Note 1: ETH::A&D

Deck: ETH::A&D
Note Type: Algorithms
GUID: F^nhYx>fXl
modified

Before

Front

Back

ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm PlsFix::AD_Algo
Runtime of
Boruvka

Runtime:

Approach: Add all vertices to the set of components (so every vertex has its own component). As long as the size of the components set is greater than 1, connect each component to the one with the cheapest edge.

Uses: Find MST in weighted, undirected graph
?

After

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Back

ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm PlsFix::AD_Algo
Runtime of
Boruvka

Runtime:

Approach: Add all vertices to the set of components (so every vertex has its own component). As long as the size of the components set is greater than 1, connect each component to the one with the cheapest edge.

Uses: Find MST in weighted, undirected graph
?

Field-by-field Comparison
Field Before After
Approach <ol><li>For Boruvka, we start with the set of edges&nbsp;\(F = \emptyset\). We treat each of the&nbsp;<em>isolated vertices</em>&nbsp;of the graph as it’s&nbsp;<em>own connected component</em>.</li><li>Each vertex marks it’s cheapest outgoing edge as a&nbsp;<em>safe edge</em>&nbsp;(making use of the cut property). We add these to&nbsp;\(F\).</li></ol><ul><li>Note that some of the edges might be chosen by both adjacent vertices, we still only add them once.<br><img src="paste-053ccf0acc6d560628bd8518b928a0d6c2687cb1.jpg"></li></ul><ol><li>Now, repeat by finding the cheapest outgoing edge for each component. Do this until all are connected.</li><li>\(F\)&nbsp;constitutes the edges of the MST.</li></ol>
Tags: ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm PlsFix::AD_Algo

Note 2: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: mMw-t6k&nH
modified

Before

Front

DUPLICATE ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm
Describe the steps of Boruvka's Algorithm:

Back

DUPLICATE ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm
Describe the steps of Boruvka's Algorithm:

  1. For Boruvka, we start with the set of edges \(F = \emptyset\). We treat each of the isolated vertices of the graph as it’s own connected component.
  2. Each vertex marks it’s cheapest outgoing edge as a safe edge (making use of the cut property). We add these to \(F\).
  • Note that some of the edges might be chosen by both adjacent vertices, we still only add them once._
  1. Now, repeat by finding the cheapest outgoing edge for each component. Do this until all are connected.
  2. \(F\) constitutes the edges of the MST.

After

Front

ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm PlsFix::DUPLICATE
Describe the steps of Boruvka's Algorithm:

Back

ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm PlsFix::DUPLICATE
Describe the steps of Boruvka's Algorithm:

  1. For Boruvka, we start with the set of edges \(F = \emptyset\). We treat each of the isolated vertices of the graph as it’s own connected component.
  2. Each vertex marks it’s cheapest outgoing edge as a safe edge (making use of the cut property). We add these to \(F\).
  • Note that some of the edges might be chosen by both adjacent vertices, we still only add them once.
  1. Now, repeat by finding the cheapest outgoing edge for each component. Do this until all are connected.
  2. \(F\) constitutes the edges of the MST.
Field-by-field Comparison
Field Before After
Back <ol><br><li>For Boruvka, we start with the set of edges&nbsp;\(F = \emptyset\). We treat each of the <em>isolated vertices</em> of the graph as it’s <em>own connected component</em>.</li><li>Each vertex marks it’s cheapest outgoing edge as a <em>safe edge</em> (making use of the cut property). We add these to&nbsp;\(F\).</li></ol><ul><li>Note that some of the edges might be chosen by both adjacent vertices, we still only add them once._<br><img src="paste-053ccf0acc6d560628bd8518b928a0d6c2687cb1.jpg"></li></ul><ol><li>Now, repeat by finding the cheapest outgoing edge for each component. Do this until all are connected.</li><li>\(F\)&nbsp;constitutes the edges of the MST.</li></ol> <ol><br><li>For Boruvka, we start with the set of edges&nbsp;\(F = \emptyset\). We treat each of the <em>isolated vertices</em> of the graph as it’s <em>own connected component</em>.</li><li>Each vertex marks it’s cheapest outgoing edge as a <em>safe edge</em> (making use of the cut property). We add these to&nbsp;\(F\).</li></ol><ul><li>Note that some of the edges might be chosen by both adjacent vertices, we still only add them once.<br><img src="paste-053ccf0acc6d560628bd8518b928a0d6c2687cb1.jpg"></li></ul><ol><li>Now, repeat by finding the cheapest outgoing edge for each component. Do this until all are connected.</li><li>\(F\)&nbsp;constitutes the edges of the MST.</li></ol>
Tags: DUPLICATE ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm PlsFix::DUPLICATE

Note 3: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: N@-]h|e+xG
modified

Before

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs PlsFix::ClozeThatBish
How can we represent a graph and what are the runtimes of common operations in them?

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs PlsFix::ClozeThatBish
How can we represent a graph and what are the runtimes of common operations in them?
1. Adjacency matrix
check if \(uv \in E \rightarrow O(1)\)
given a vertex \(u\) we can find all adjacent vertices in \(O(n)\) 
2. Adjacency lists
check if \(uv \in E \rightarrow O(1 + \min\{\text{deg}(u), \text{deg}(v) \})\) (we have to check the smaller of the two adjacency lists
given a vertex \(u\) we can find all adjacent vertices in \(O(1+\text{deg}(u) )\)

After

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Runtime: Operations in an Adjacency List:

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Runtime: Operations in an Adjacency List:
1. Check if \(uv \in E \): \(O(1 + \min\{\text{deg}(u), \text{deg}(v) \})\) (we have to check the smaller of the two adjacency lists
2. Vertex \(u\), find all adjacent vertices: \(O(1+\text{deg}(u) )\)
Field-by-field Comparison
Field Before After
Front How can we represent a graph and what are the runtimes of common operations in them? <b>Runtime</b>: Operations in an Adjacency&nbsp;<b>List</b>:
Back <b>1. Adjacency matrix</b><br>check if&nbsp;\(uv \in E \rightarrow O(1)\)<br>given a vertex&nbsp;\(u\)&nbsp;we can find all adjacent vertices in&nbsp;\(O(n)\)&nbsp;<br><b>2. Adjacency lists<br></b>check if&nbsp;\(uv \in E \rightarrow O(1 + \min\{\text{deg}(u), \text{deg}(v) \})\)&nbsp;(we have to check the smaller of the two adjacency lists<br>given a vertex&nbsp;\(u\)&nbsp;we can find all adjacent vertices in&nbsp;\(O(1+\text{deg}(u) )\) 1. Check if&nbsp;\(uv \in E \):&nbsp;\(O(1 + \min\{\text{deg}(u), \text{deg}(v) \})\)&nbsp;(we have to check the smaller of the two adjacency lists<br>2. Vertex&nbsp;\(u\), find all adjacent vertices:&nbsp;\(O(1+\text{deg}(u) )\)
Tags: PlsFix::ClozeThatBish ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 4: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: mabS@W|F#G
modified

Before

Front

ETH::1._Semester::A&D::08._Directed_Graphs::2._Topological_Sorting PlsFix::ClozeThatBish
Intuitively describe a topological sorting of a directed graph.

Back

ETH::1._Semester::A&D::08._Directed_Graphs::2._Topological_Sorting PlsFix::ClozeThatBish
Intuitively describe a topological sorting of a directed graph.
it is a sorting of the vertices of a directed graph, which we can build from the back by always finding a vertex which has no succeeding vertices, remove it from the graph and add it to the front of our topologically sorted list.

This is not possible if there is a directed cycle in the graph.

The resulting list can be useful for example when we interpret all preceeding nodes as requirements for the succeeding node, then the topological sort is a valid way to fulfill these requirements

After

Front

ETH::1._Semester::A&D::08._Directed_Graphs::2._Topological_Sorting
Explain how to find a topological order (high-level):

Back

ETH::1._Semester::A&D::08._Directed_Graphs::2._Topological_Sorting
Explain how to find a topological order (high-level):
it is a sorting of the vertices of a directed graph, which we can build from the back by always finding a vertex which has no succeeding vertices, remove it from the graph and add it to the front of our topologically sorted list.

This is not possible if there is a directed cycle in the graph.
Field-by-field Comparison
Field Before After
Front Intuitively describe a topological sorting of a directed graph. Explain how to find a topological order (high-level):
Back it is a sorting of the vertices of a directed graph, which we can build from the back by always finding a vertex which has no succeeding vertices, remove it from the graph and add it to the front of our topologically sorted list.<br><br>This is not possible if there is a directed cycle in the graph.<br><br>The resulting list can be useful for example when we interpret all preceeding nodes as requirements for the succeeding node, then the topological sort is a valid way to fulfill these requirements it is a sorting of the vertices of a directed graph, which we can build from the back by always finding a vertex which has no succeeding vertices, remove it from the graph and add it to the front of our topologically sorted list.<br><br>This is not possible if there is a directed cycle in the graph.
Tags: PlsFix::ClozeThatBish ETH::1._Semester::A&D::08._Directed_Graphs::2._Topological_Sorting

Note 5: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: M?qP8s.,s4
added

Previous

Note did not exist

New Note

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
How can we represent a graph?

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
How can we represent a graph?
1. Adjacency matrix
2. Adjacency lists
Field-by-field Comparison
Field Before After
Front How can we represent a graph?
Back <b>1. </b>Adjacency<b> matrix<br>2. </b>Adjacency<b> lists</b>
Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 6: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: ra?I5x|G>*
added

Previous

Note did not exist

New Note

Front

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Runtime: Operations in an Adjacency Matrix:

Back

ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs
Runtime: Operations in an Adjacency Matrix:
1. check if \(uv \in E\): \(O(1)\)
2. Vertex \(u\) , find all adjacent vertices in:  \(O(n)\)
Field-by-field Comparison
Field Before After
Front <b>Runtime</b>: Operations in an Adjacency <b>Matrix</b>:
Back 1. check if&nbsp;\(uv \in E\):&nbsp;\(O(1)\)<br>2. Vertex&nbsp;\(u\)&nbsp;, find all adjacent vertices in:&nbsp;&nbsp;\(O(n)\)
Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 7: ETH::DiskMat

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

Before

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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 PlsFix::ClozeThatBish
What does it mean for two formulas \(F\) and \(G\) to be logically equivalent (\(F \equiv G\))?

Back

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 PlsFix::ClozeThatBish
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 \(G\).

After

Front

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
\(F \equiv G\) 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

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
\(F \equiv G\) 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
Text What does it mean for two formulas \(F\) and \(G\) to be logically equivalent (\(F \equiv G\))? {{c2::\(F \equiv G\)}}&nbsp;means {{c1:: they correspond to the same function}}, i.e., {{c3:: their truth values are equal for&nbsp;<strong>all</strong>&nbsp;truth assignments to the propositional symbols appearing in&nbsp;\(F\)&nbsp;or&nbsp;\(G\)}}.
Extra \(F \equiv G\) 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 \(G\).
Tags: PlsFix::ClozeThatBish 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

Note 8: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: lNUw/[p~+9
modified

Before

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
What is a tautology in propositional logic?

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
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 \(\top\).

After

Front

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
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 \(\top\).

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
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
Field Before After
Text What is a tautology in propositional logic? A formula&nbsp;\(F\)&nbsp;is a {{c1:: tautology (or valid)}} if it {{c2:: is true for&nbsp;<strong>all</strong>&nbsp;truth assignments of the involved propositional symbols}}. Denoted as {{c3::&nbsp;\(\models F\)&nbsp;or&nbsp;\(\top\)}}.
Extra 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 \(\top\).
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability

Note 9: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: dNOrR*l4!S
modified

Before

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
What does it mean for a formula to be satisfiable?

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
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.

After

Front

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
A formula \(F\) is satisfiable if it is true for at least one truth assignment of the involved propositional symbols.

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
A formula \(F\) is satisfiable if it is true for at least one truth assignment of the involved propositional symbols.
Field-by-field Comparison
Field Before After
Text What does it mean for a formula to be satisfiable? A formula&nbsp;\(F\)&nbsp;is {{c1:: satisfiable}} if it {{c2:: is true for&nbsp;<strong>at least one</strong>&nbsp;truth assignment of the involved propositional symbols}}.
Extra A formula \(F\) is satisfiable if it is true for <strong>at least one</strong> truth assignment of the involved propositional symbols.
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability

Note 10: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Q;AJBWzP3u
modified

Before

Front

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
What does it mean for a formula to be unsatisfiable?

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
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 \(\perp\).

After

Front

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
A formula is unsatisfiable if it is never true under any truth assignment. Denoted as \(\perp\).

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
A formula is unsatisfiable if it is never true under any truth assignment. Denoted as \(\perp\).
Field-by-field Comparison
Field Before After
Text What does it mean for a formula to be unsatisfiable? A formula is {{c1:: unsatisfiable}} if it {{c2:: is&nbsp;<strong>never</strong>&nbsp;true under any truth assignment. Denoted as&nbsp;\(\perp\)}}.
Extra A formula is unsatisfiable if it is <strong>never</strong> true under any truth assignment. Denoted as \(\perp\).
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability

Note 11: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Pi%FwZEpJz
modified

Before

Front

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
How are tautologies related to logical consequence (implication)?

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability PlsFix::ClozeThatBish
How are tautologies related to logical consequence (implication)?
For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology if and only if \(F \models G\).

After

Front

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology if and only if \(F \models G\).

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability
For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology if and only if \(F \models G\).
Field-by-field Comparison
Field Before After
Text How are tautologies related to logical consequence (implication)? For any formulas&nbsp;\(F\)&nbsp;and&nbsp;\(G\), {{c1::\(F \rightarrow G\)}}&nbsp;is a tautology&nbsp;<strong>if and only if</strong>&nbsp;{{c2::\(F \models G\)}}.
Extra For any formulas \(F\) and \(G\), \(F \rightarrow G\) is a tautology <strong>if and only if</strong> \(F \models G\).
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::6._Tautologies_and_Satisfiability

Note 12: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: HTIS
modified

Before

Front

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations PlsFix::ClozeThatBish
What are the two trivial equivalence relations on a set \(A\)?

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations PlsFix::ClozeThatBish
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 \(\{a\}\)

After

Front

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations
What are the two trivial equivalence relations on a set \(A\)?

1.  Complete relation \(A \times A\) → single equivalence class \(A\)
2. {{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\)}}

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations
What are the two trivial equivalence relations on a set \(A\)?

1.  Complete relation \(A \times A\) → single equivalence class \(A\)
2. {{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\)}}
Field-by-field Comparison
Field Before After
Text What are the two trivial equivalence relations on a set \(A\)? What are the two trivial equivalence relations on a set \(A\)?<br><br>1. {{c1::&nbsp;<strong>Complete relation</strong>&nbsp;\(A \times A\)&nbsp;→ single equivalence class&nbsp;\(A\)}}<br>2.&nbsp;{{c2::&nbsp;<strong>Identity relation</strong>&nbsp;→ equivalence classes are all singletons&nbsp;\(\{a\}\)}}
Extra 1. <strong>Complete relation</strong> \(A \times A\) → single equivalence class \(A\) 2. <strong>Identity relation</strong> → equivalence classes are all singletons \(\{a\}\)
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations

Note 13: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: qnpI?yoaky
modified

Before

Front

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::1._Definition PlsFix::ClozeThatBish
What three properties must a relation have to be a partial order?

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::1._Definition PlsFix::ClozeThatBish
What three properties must a relation have to be a partial order?
1. Reflexive 2. Antisymmetric 3. Transitive

After

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::1._Definition
What three properties must a relation have to be a partial order:
1.  Reflexive
2.  Antisymmetric
3.  Transitive

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::1._Definition
What three properties must a relation have to be a partial order:
1.  Reflexive
2.  Antisymmetric
3.  Transitive
Field-by-field Comparison
Field Before After
Text What three properties must a relation have to be a partial order? What three properties must a relation have to be a partial order:<br>1. {{c1::&nbsp;<b>Reflexive</b>}}<br>2. {{c2::&nbsp;<b>Antisymmetric</b>}}<br>3. {{c3::&nbsp;<b>Transitive</b>}}
Extra 1. <strong>Reflexive</strong> 2. <strong>Antisymmetric</strong> 3. <strong>Transitive</strong>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::1._Definition

Note 14: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: v
modified

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::ClozeThatBish
What two properties must a relation \(f: A \to B\) have to be a function?

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::ClozeThatBish
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: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
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: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
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: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
Field-by-field Comparison
Field Before After
Text What two properties must a relation \(f: A \to B\) have to be a function? What two properties must a relation \(f: A \to B\) have to be a function?<br><br>1. {{c1::&nbsp;<strong>Totally defined</strong>:&nbsp;\(\forall a \in A \ \exists b \in B : a \ f \ b\)&nbsp;}}<br>2.&nbsp;{{c2::&nbsp;<strong>Well-defined</strong>:&nbsp;\(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)}}
Extra 1. <strong>Totally defined</strong>: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 2. <strong>Well-defined</strong>: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)<br><div><br></div>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions

Note 15: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: G2]R~8h{q4
modified

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::7._Countable_and_Uncountable_Sets::3._Important_Countable_Sets PlsFix::ClozeThatBish
What operations preserve countability?

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::7._Countable_and_Uncountable_Sets::3._Important_Countable_Sets PlsFix::ClozeThatBish
What operations preserve countability?
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

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::7._Countable_and_Uncountable_Sets::3._Important_Countable_Sets
What operations preserve countability?

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

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::7._Countable_and_Uncountable_Sets::3._Important_Countable_Sets
What operations preserve countability?

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
Text What operations preserve countability? What operations preserve countability?<br><br>Let&nbsp;\(A\)&nbsp;and&nbsp;\(A_i\)&nbsp;for&nbsp;\(i \in \mathbb{N}\)&nbsp;be countable sets. Then:&nbsp;<div>&nbsp;- (i) {{c1::\(A^n\)&nbsp;(\(n\)-tuples) is countable }}</div><div>&nbsp;- (ii) {{c2::\(\bigcup_{i\in \mathbb{N}} A_i\)&nbsp;(countable union) is countable }}</div><div>&nbsp;- (iii) {{c3::\(A^*\)&nbsp;(finite sequences) is countable}}</div>
Extra Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then:&nbsp;<div>&nbsp;- (i) \(A^n\) (\(n\)-tuples) is countable&nbsp;</div><div>&nbsp;- (ii) \(\bigcup_{i\in \mathbb{N}} A_i\) (countable union) is countable&nbsp;</div><div>&nbsp;- (iii) \(A^*\) (finite sequences) is countable</div>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::7._Countable_and_Uncountable_Sets::3._Important_Countable_Sets

Note 16: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: pjd-vCXMX,
modified

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets PlsFix::ClozeThatBish
In the poset \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), identify the minimal and maximal elements.

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets PlsFix::ClozeThatBish
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)

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
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:  \(5, 6, 7, 8, 9\)
  • Least or greatest element  There is none (not all elements comparable)

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
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:  \(5, 6, 7, 8, 9\)
  • Least or greatest element  There is none (not all elements comparable)
Field-by-field Comparison
Field Before After
Text In the poset \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), identify the minimal and maximal elements. In the poset \((\{2, 3, 4, 5, 6, 7, 8, 9\}; |)\), identify the minimal and maximal elements.<br><ul><li><strong>Minimal elements</strong>: {{c1::&nbsp;\(2, 3, 5, 7\)&nbsp;(primes)}}</li><li><strong>Maximal elements</strong>: {{c2::&nbsp;\(5, 6, 7, 8, 9\)}}</li><li><strong>Least or greatest element</strong>&nbsp;{{c3:: There is none (not all elements comparable)}}</li></ul><div><img src="paste-1d2f8dcd3adedbac7c91aff60842c9ece732a3a8.jpg"></div>
Extra <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><div><img src="paste-1d2f8dcd3adedbac7c91aff60842c9ece732a3a8.jpg"><br></div>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets

Note 17: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: pL:[)Gqs`_
modified

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::ClozeThatBish
For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are:
1. The range of \(f\)
2. The preimage of \([4, 9]\)

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::ClozeThatBish
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 \([4, 9]\): \([-3, -2] \cup [2, 3]\)

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are:
1. Range: {{c1::\(\mathbb{R}^{\geq 0}\) (non-negative reals)}}
2. Preimage of \([4, 9]\): \([-3, -2] \cup [2, 3]\)

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are:
1. Range: {{c1::\(\mathbb{R}^{\geq 0}\) (non-negative reals)}}
2. Preimage of \([4, 9]\): \([-3, -2] \cup [2, 3]\)
Field-by-field Comparison
Field Before After
Text 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 \([4, 9]\) For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are: <br>1.&nbsp;<strong>Range</strong>: {{c1::\(\mathbb{R}^{\geq 0}\)&nbsp;(non-negative reals)}}<br>2.&nbsp;<strong>Preimage of&nbsp;\([4, 9]\)</strong>: {{c2::\([-3, -2] \cup [2, 3]\)}}
Extra 1. <strong>Range</strong>: \(\mathbb{R}^{\geq 0}\) (non-negative reals) <br>2. <strong>Preimage of \([4, 9]\)</strong>: \([-3, -2] \cup [2, 3]\)
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions

Note 18: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: IH=>J8$0Y%
modified

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::2._Representations_of_Relations PlsFix::ClozeThatBish
What are the three ways to represent a relation on finite sets?

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::2._Representations_of_Relations PlsFix::ClozeThatBish
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)

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ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::2._Representations_of_Relations
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)

Back

ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::2._Representations_of_Relations
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)
Field-by-field Comparison
Field Before After
Text What are the three ways to represent a relation on finite sets? What are the three ways to represent a relation on finite sets?<br><br>1. {{c1::&nbsp;<strong>Set notation</strong>&nbsp;(subset of&nbsp;\(A \times B\))}}<br>2. {{c2::&nbsp;<strong>Boolean matrix</strong>&nbsp;(1 if&nbsp;\((a,b) \in \rho\), 0 otherwise)}}<br>3. {{c3::&nbsp;<strong>Directed graph</strong>&nbsp;(nodes are elements, edges are relations)}}
Extra 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)
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::2._Representations_of_Relations

Note 19: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: dK0`$S[9VD
modified

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic PlsFix::ClozeThatBish
List all three pairs of related but distinct logical symbols and their types.

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic PlsFix::ClozeThatBish
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)

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic
List all types of symbols meaning equivalence:

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic
List all types of symbols meaning equivalence:
Equivalences
  • \(\equiv\)  (formula→statement)
  • \(\leftrightarrow\) (formula→formula)
  • \(\Leftrightarrow\) (statement→statement)
Field-by-field Comparison
Field Before After
Front List all three pairs of related but distinct logical symbols and their types. List all types of symbols meaning equivalence:
Back 1. \(\equiv\) (formula→statement), \(\leftrightarrow\) (formula→formula), \(\Leftrightarrow\) (statement→statement) 2. \(\models\) (formula→statement), \(\rightarrow\) (formula→formula), \(\Rightarrow\) (statement→statement) <b>Equivalences</b><br><ul><li>\(\equiv\)&nbsp; (formula→statement)</li><li>\(\leftrightarrow\) (formula→formula)</li><li>\(\Leftrightarrow\) (statement→statement)</li></ul>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic

Note 20: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: wV8Y&j0xY.
modified

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::1._Logical_Constants,_Operators,_and_Formulas PlsFix::ClozeThatBish
Name the binding strengths of PL tokens in order.

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::1._Logical_Constants,_Operators,_and_Formulas PlsFix::ClozeThatBish
Name the binding strengths of PL tokens in order.
 - unary operators (NOT)
 - quantifiers (for all and exists)
 - operators (AND, OR)
 - Implication

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::1._Logical_Constants,_Operators,_and_Formulas
Name the binding strengths of PL tokens in order:
  1. unary operators (NOT)
  2.  quantifiers (for all and exists)
  3.  operators (AND, OR)
  4.  Implication

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::1._Logical_Constants,_Operators,_and_Formulas
Name the binding strengths of PL tokens in order:
  1. unary operators (NOT)
  2.  quantifiers (for all and exists)
  3.  operators (AND, OR)
  4.  Implication
Field-by-field Comparison
Field Before After
Text Name the binding strengths of PL tokens in order. Name the binding strengths of PL tokens in order:<br><ol><li>{{c1:: unary operators (NOT)}}</li><li>{{c2::&nbsp;quantifiers (for all and exists)}}</li><li>{{c3::&nbsp;operators (AND, OR)}}</li><li>{{c4::&nbsp;Implication}}</li></ol>
Extra &nbsp;- unary operators (NOT)<br>&nbsp;- quantifiers (for all and exists)<br>&nbsp;- operators (AND, OR)<br>&nbsp;- Implication
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic::1._Logical_Constants,_Operators,_and_Formulas

Note 21: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: ujCuoEmotl
modified

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ETH::1._Semester::DiskMat::4._Number_Theory::3._Factorization_into_Primes::2._Proof_of_the_Fundamental_Theorem_of_Arithmetic_* PlsFix::ClozeThatBish
If a prime divides a product, what can we conclude? (Lemma 4.7)

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ETH::1._Semester::DiskMat::4._Number_Theory::3._Factorization_into_Primes::2._Proof_of_the_Fundamental_Theorem_of_Arithmetic_* PlsFix::ClozeThatBish
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: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]

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ETH::1._Semester::DiskMat::4._Number_Theory::3._Factorization_into_Primes::2._Proof_of_the_Fundamental_Theorem_of_Arithmetic_*
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: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]

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ETH::1._Semester::DiskMat::4._Number_Theory::3._Factorization_into_Primes::2._Proof_of_the_Fundamental_Theorem_of_Arithmetic_*
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
Text If a prime divides a product, what can we conclude? (Lemma 4.7) If&nbsp;\(p\)&nbsp;is a prime which divides the product&nbsp;\(x_1 x_2 \dots x_n\)&nbsp;of some integers, then&nbsp;\(p\)&nbsp;{{c1:: divides at least one of them:&nbsp;\[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]}}<br>
Extra 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: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::4._Number_Theory::3._Factorization_into_Primes::2._Proof_of_the_Fundamental_Theorem_of_Arithmetic_*

Note 22: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: nizWAJt?$u
modified

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ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::2._Modular_Arithmetic PlsFix::ClozeThatBish
What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16)

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ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::2._Modular_Arithmetic PlsFix::ClozeThatBish
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)

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ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::2._Modular_Arithmetic
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)

Back

ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::2._Modular_Arithmetic
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)
Field-by-field Comparison
Field Before After
Text What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16) What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16)<br><br><strong>(i)</strong>&nbsp;{{c1::&nbsp;\(a \equiv_m R_m(a)\)&nbsp;(the remainder represents the equivalence class)}}<br><b>(ii)</b>&nbsp;{{c2::&nbsp;\(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\)&nbsp;(congruence iff same remainder)}}
Extra <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)
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::2._Modular_Arithmetic

Note 23: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: xWhw%ncc|4
modified

Before

Front

ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::1._Modular_Congruences PlsFix::ClozeThatBish
Verify that \(\equiv_m\) is reflexive, symmetric, and transitive.

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ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::1._Modular_Congruences PlsFix::ClozeThatBish
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 \(m | (a-b+b-c) = (a-c)\) ✓

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ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::1._Modular_Congruences
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 \(m | (a-b+b-c) = (a-c)\) ✓

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ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::1._Modular_Congruences
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
Text Verify that \(\equiv_m\) is reflexive, symmetric, and transitive. Verify that \(\equiv_m\) is reflexive, symmetric, and transitive.<br><ul><li><strong>Reflexive</strong>: {{c1::&nbsp;\(a \equiv_m a\)&nbsp;since&nbsp;\(m | (a - a) = 0\)&nbsp;✓}}</li><li><strong>Symmetric</strong>:&nbsp;{{c2:: \(a \equiv_m b \Rightarrow m | (a-b) \Rightarrow m | (b-a) \Rightarrow b \equiv_m a\)&nbsp;✓}}</li><li><strong>Transitive</strong>: {{c3:: If&nbsp;\(m | (a-b)\)&nbsp;and&nbsp;\(m | (b-c)\), then&nbsp;\(m | (a-b+b-c) = (a-c)\)&nbsp;✓}}</li></ul>
Extra <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>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::4._Number_Theory::5._Congruences_and_Modular_Arithmetic::1._Modular_Congruences

Note 24: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: O?~Mb}~!3:
modified

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::04._Modus_Ponens
Proof method: "Modus Ponens"

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::04._Modus_Ponens
Proof method: "Modus Ponens"
1. Find a suitable statement \(R\)
2. Prove \(R\)
3. Prove \(R \implies S\)

After

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::04._Modus_Ponens
Proof method: "Modus Ponens"
1. Find a suitable statement \(R\)
2.  Prove \(R\)
3.  Prove \(R \implies S\)

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::04._Modus_Ponens
Proof method: "Modus Ponens"
1. Find a suitable statement \(R\)
2.  Prove \(R\)
3.  Prove \(R \implies S\)
Field-by-field Comparison
Field Before After
Text Proof method: "Modus Ponens" Proof method: "Modus Ponens"<br>1. {{c1:: Find a suitable statement&nbsp;\(R\)}}<div>2. {{c2::&nbsp;Prove&nbsp;\(R\)}}</div><div>3. {{c3::&nbsp;Prove&nbsp;\(R \implies S\)}}</div>
Extra 1. Find a suitable statement&nbsp;\(R\)<div>2. Prove&nbsp;\(R\)</div><div>3. Prove&nbsp;\(R \implies S\)</div>
Tags: ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::04._Modus_Ponens

Note 25: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: xDDC{82KOB
modified

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::06._Proofs_by_Contradiction
Proof method: Proof by Contradiction

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::06._Proofs_by_Contradiction
Proof method: Proof by Contradiction
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)

After

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::06._Proofs_by_Contradiction
Proof method: Proof by Contradiction

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::06._Proofs_by_Contradiction
Proof method: Proof by Contradiction
1.
2.
3.
Field-by-field Comparison
Field Before After
Extra 1. Find a suitable statement&nbsp;\( T\)<div>2. Prove that&nbsp;\( T \) is false</div><div>3. Assume that&nbsp;\( S \) is false and prove that&nbsp;\( T \) is true (-&gt; contradiction)</div> 1. {{c1:: Find a suitable statement&nbsp;\( T\)}}<div>2. {{c2::&nbsp;Prove that&nbsp;\( T \) is false}}</div><div>3. {{c3::&nbsp;Assume that&nbsp;\( S \) is false and prove that&nbsp;\( T \) is true (-&gt; contradiction)}}</div>
Tags: ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::06._Proofs_by_Contradiction

Note 26: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: vi7xPhAi#`
modified

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ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements PlsFix::ClozeThatBish

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

ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements PlsFix::ClozeThatBish

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.

After

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ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements

If \(e * a = a * e = a\) for all \(a \in S\), then \(e\) is simply called a neutral element or identity element.

Back

ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements

If \(e * a = a * e = a\) for all \(a \in S\), then \(e\) is simply called a neutral element or identity 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> <p>If {{c2::\(e * a = a * e = a\)}} for all \(a \in S\), then \(e\) is simply called a {{c1::neutral element or identity element}}.</p>
Tags: PlsFix::ClozeThatBish ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements

Note 27: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: j]Gy^>$7h+
modified

Before

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ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups

For \(H\) to be a subgroup, the neutral element must be in \(H\): \(e \in H\).

and be mapped to from the neutral: \(\varphi(e) = e'\).

Back

ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups

For \(H\) to be a subgroup, the neutral element must be in \(H\): \(e \in H\).

and be mapped to from the neutral: \(\varphi(e) = e'\).

After

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ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups

For \(H\) to be a subgroup, the neutral element must be in \(H\): \(e \in H\).

Back

ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups

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> and {{c1::be mapped to from the neutral: \(\varphi(e) = e'\)}}. <p>For \(H\) to be a subgroup, the {{c1::neutral element}} must be in \(H\): {{c1::\(e \in H\)}}.</p>
Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups

Note 28: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: HR>/;ZN2^c
added

Previous

Note did not exist

New Note

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ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic
List all types of symbols meaning implication:

Back

ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic
List all types of symbols meaning implication:
Implications
  • \(\models\) (formula→statement)
  • \(\rightarrow\) (formula→formula)
  • \(\Rightarrow\) (statement→statement)
Field-by-field Comparison
Field Before After
Front List all types of symbols meaning implication:
Back <b>Implications</b><br><ul><li>\(\models\) (formula→statement)</li><li>\(\rightarrow\) (formula→formula)</li><li>\(\Rightarrow\) (statement→statement)</li></ul>
Tags: ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::3._A_First_Introduction_to_Propositional_Logic

Note 29: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: xelDz|Gp*?
added

Previous

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ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
A right (left) neutral element  is an elements such that for all \(a \in G\),  \(a*e = a\) (\(e*a = a\)).

Back

ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
A right (left) neutral element  is an elements such that for all \(a \in G\),  \(a*e = a\) (\(e*a = a\)).
Field-by-field Comparison
Field Before After
Text <div>A {{c1::right (left) neutral element}}&nbsp; is an elements such that for all&nbsp;\(a \in G\), {{c2::&nbsp;\(a*e = a\)&nbsp;(\(e*a = a\))}}.</div>
Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements

Note 30: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: j0~h}Ph2E;
modified

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ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
What does the linearity axiom say and how can it be interpreted?

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ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
What does the linearity axiom say and how can it be interpreted?
for a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\)

the following is true: \(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)

this can be interpreted: 
i) \(T(x+x') = T(x) + T(x')\) and
ii) \(T(\lambda x) = \lambda T(x)\)

After

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ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
What does the linearity axiom say and how can it be interpreted for a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\):

i)  \(T(x+x') = T(x) + T(x')\)
ii)  \(T(\lambda x) = \lambda T(x)\)

Back

ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
What does the linearity axiom say and how can it be interpreted for a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\):

i)  \(T(x+x') = T(x) + T(x')\)
ii)  \(T(\lambda x) = \lambda T(x)\)
Field-by-field Comparison
Field Before After
Text What does the linearity axiom say and how can it be interpreted? What does the linearity axiom say and how can it be interpreted for a function&nbsp;\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\):<br><br>i) {{c1::&nbsp;\(T(x+x') = T(x) + T(x')\)}}<br>ii) {{c2::&nbsp;\(T(\lambda x) = \lambda T(x)\)}}
Extra for a function&nbsp;\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\)<br><br>the following is true:&nbsp;\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)<br><br>this can be interpreted:&nbsp;<br>i)&nbsp;\(T(x+x') = T(x) + T(x')\)&nbsp;and<br>ii)&nbsp;\(T(\lambda x) = \lambda T(x)\)
Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations

Note 31: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: e`7dX^~/:X
added

Previous

Note did not exist

New Note

Front

ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
For a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\) the linearity axiom is:

Back

ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
For a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\) the linearity axiom is:
\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)
Field-by-field Comparison
Field Before After
Front For a function&nbsp;\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\)&nbsp;the linearity axiom is:
Back \(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)
Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
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