Note 1: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: D2>~h
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
quue has the following operations:
- enqueue(k, S): append at the end of the queue
- dequeue(S): remove and return the first element of the queue
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
quue has the following operations:
- enqueue(k, S): append at the end of the queue
- dequeue(S): remove and return the first element of the queue
After
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
queue has the following operations:
- enqueue(k, S): append at the end of the queue
- dequeue(S): remove and return the first element of the queue
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
queue has the following operations:
- enqueue(k, S): append at the end of the queue
- dequeue(S): remove and return the first element of the queue
Field-by-field Comparison
| Field |
Before |
After |
| Text |
The ADT <b>quue</b> has the following operations:<br><ul><li><b>enqueue(k, S)</b>: {{c2:: append at the end of the queue}}</li><li><b>dequeue(S)</b>: {{c4:: remove and return the first element of the queue}}</li></ul> |
The ADT <b>queue</b> has the following operations:<br><ul><li><b>enqueue(k, S)</b>: {{c2:: append at the end of the queue}}</li><li><b>dequeue(S)</b>: {{c4:: remove and return the first element of the queue}}</li></ul> |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
Note 2: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: F^&OZQURkx
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
queue can be efficiently implemented using a
singly linked list with a pointer to the end:
- push: \(O(1)\) insert at the end, with pointer to the end
- pop: \(O(1)\) remove the first element like in a stack
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
queue can be efficiently implemented using a
singly linked list with a pointer to the end:
- push: \(O(1)\) insert at the end, with pointer to the end
- pop: \(O(1)\) remove the first element like in a stack
After
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
queue can be efficiently implemented using a
singly linked list with a pointer to the end:
- push: \(O(1)\) insert at the end, with pointer to the end
- pop: \(O(1)\) remove the first element like in a stack
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
The ADT
queue can be efficiently implemented using a
singly linked list with a pointer to the end:
- push: \(O(1)\) insert at the end, with pointer to the end
- pop: \(O(1)\) remove the first element like in a stack
Field-by-field Comparison
| Field |
Before |
After |
| Text |
The ADT <b>queue</b> can be efficiently implemented using a {{c1:: <b>singly linked list with a pointer to the end</b>}}:<br><ul><li><b>push</b>: {{c2:: \(O(1)\) insert at the end, with pointer to the end}}<br></li><li><b>pop</b>: {{c3:: \(O(1)\) remove the first element like in a stack}}</li></ul> |
The ADT <b>queue</b> can be efficiently implemented using a {{c1::<b>singly linked list with a pointer to the end</b>}}:<br><ul><li><b>push</b>: {{c2:: \(O(1)\) insert at the end, with pointer to the end}}<br></li><li><b>pop</b>: {{c3:: \(O(1)\) remove the first element like in a stack}}</li></ul> |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
Note 3: ETH::A&D
Deck: ETH::A&D
Note Type: Algorithms
GUID: NAHrkHd^ik
modified
Before
Front
ETH::1._Semester::A&D::04._Sorting_Algorithms::4._Merge_Sort
Back
ETH::1._Semester::A&D::04._Sorting_Algorithms::4._Merge_Sort
Runtime of Merge Sort?
Best Case: \(O(n \log n)\)
Worst Case: \(O(n \log n)\)
After
Front
ETH::1._Semester::A&D::04._Sorting_Algorithms::4._Merge_Sort
Back
ETH::1._Semester::A&D::04._Sorting_Algorithms::4._Merge_Sort
Runtime of Merge Sort?
Best Case: \(O(n \log n)\)
Worst Case: \(O(n \log n)\)
Field-by-field Comparison
| Field |
Before |
After |
| Attributes |
not in place, thus the space complexity is \(K(n)\). (can be made in place)<br><b>Stable</b> |
Not in-place, thus the space complexity is \(K(n)\). (Although it can be programmed to be in-place.)<br><b>Stable</b> |
Tags:
ETH::1._Semester::A&D::04._Sorting_Algorithms::4._Merge_Sort
Note 4: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Classic
GUID: Nl}2-%ar31
modified
Before
Front
ETH::1._Semester::A&D::04._Sorting_Algorithms::7._Heapsort
Back
ETH::1._Semester::A&D::04._Sorting_Algorithms::7._Heapsort
What is a maxHeap?
A datastructure that stores the values in a tree form, with the largest element always as the root.
After
Front
ETH::1._Semester::A&D::04._Sorting_Algorithms::7._Heapsort
Back
ETH::1._Semester::A&D::04._Sorting_Algorithms::7._Heapsort
What is a maxHeap?
A datastructure that stores values in a tree form, with the largest element always being the root.
Field-by-field Comparison
| Field |
Before |
After |
| Back |
A datastructure that stores the values in a tree form, with the largest element always as the root. |
A datastructure that stores values in a tree form, with the largest element always being the root. |
Tags:
ETH::1._Semester::A&D::04._Sorting_Algorithms::7._Heapsort
Note 5: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: t/(N7FzdP}
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::6._Priority_Queue
The ADT priorityQueue can be efficiently implemented using a MaxHeap. This guarantees \(O(\log n)\) for both operations.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::6._Priority_Queue
The ADT priorityQueue can be efficiently implemented using a MaxHeap. This guarantees \(O(\log n)\) for both operations.
After
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::6._Priority_Queue
The ADT priorityQueue can be efficiently implemented using a MaxHeap.
This guarantees \(O(\log n)\) for both operations.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::6._Priority_Queue
The ADT priorityQueue can be efficiently implemented using a MaxHeap.
This guarantees \(O(\log n)\) for both operations.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<div>The ADT <b>priorityQueue</b> can be efficiently implemented using a {{c1:: <b>MaxHeap</b>}}. This guarantees {{c2:: \(O(\log n)\)}} for both operations.</div> |
<div>The ADT <b>priorityQueue</b> can be efficiently implemented using a {{c1:: <b>MaxHeap</b>}}. </div><div><br></div><div>This guarantees {{c2::\(O(\log n)\)}} for both operations.</div> |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::6._Priority_Queue
Note 6: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Classic
GUID: xP1aIt[ejN
modified
Before
Front
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation
When \(f \geq \Omega(g)\), this means what exactly?
Back
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation
When \(f \geq \Omega(g)\), this means what exactly?
\(\exists C \ge 0 \quad \forall n \in \mathbb{N} \quad f(n) \ge C\cdot g(n)\)
\(f\) grows asymptotically faster than \(g\)
After
Front
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation
When \(f \geq \Omega(g)\), this means what exactly?
Back
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation
When \(f \geq \Omega(g)\), this means what exactly?
\(\exists C \ge 0 \quad \forall n \in \mathbb{N} \quad f(n) \ge C\cdot g(n)\)
\(f\) grows asymptotically faster than \(g\).
Field-by-field Comparison
| Field |
Before |
After |
| Back |
\(\exists C \ge 0 \quad \forall n \in \mathbb{N} \quad f(n) \ge C\cdot g(n)\)<br><br>\(f\) grows asymptotically <b>faster</b> than \(g\) |
\(\exists C \ge 0 \quad \forall n \in \mathbb{N} \quad f(n) \ge C\cdot g(n)\)<br><br>\(f\) grows asymptotically <b>faster</b> than \(g\). |
Tags:
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation
Note 7: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: yy3TxuBe~r
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
A queue is also called FIFO.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
A queue is also called FIFO.
After
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
A queue is also called FIFO.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
A queue is also called FIFO.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
A queue is also called {{c1:: FIFO}}. |
A queue is also called {{c1:: FIFO::/works under the principle of..}}. |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
Note 8: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: G|6fl[78G`
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::4._(Non-)minimality_of_the_Group_Axioms
To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:
- G1 (associativity)
- G2 (neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).
- G3 (inverse) G3' -> you only need to prove the existence of a right inverse
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::4._(Non-)minimality_of_the_Group_Axioms
To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:
- G1 (associativity)
- G2 (neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).
- G3 (inverse) G3' -> you only need to prove the existence of a right inverse
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::4._(Non-)minimality_of_the_Group_Axioms
To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:
- G1 (Associativity)
- G2 (Neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).
- G3 (Inverse) G3' -> you only need to prove the existence of a right inverse
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::4._(Non-)minimality_of_the_Group_Axioms
To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:
- G1 (Associativity)
- G2 (Neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).
- G3 (Inverse) G3' -> you only need to prove the existence of a right inverse
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:<br>
- {{c2::G1 (associativity)}}<br>
- {{c3::G2 (neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).}}<br>
- {{c4::G3 (inverse) G3' -> you only need to prove the existence of a right inverse}}</p> |
<p>To prove \(\langle G; * \rangle\) is a group, you need to prove three axioms:<br></p><ol><li>{{c2::G1 (Associativity)}}</li><li>{{c3::G2 (Neutral element) G2' -> you only need to prove existence of a right neutral element (not a full two-sided neutral).}}</li><li>{{c4::G3 (Inverse) G3' -> you only need to prove the existence of a right inverse}}</li></ol><p></p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::4._(Non-)minimality_of_the_Group_Axioms
Note 9: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: K,;}YIg:-h
modified
Before
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept
A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of \(A\times B\).
If \(A = B\), then \(\rho\) is called a relation on \(A\).
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept
A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of \(A\times B\).
If \(A = B\), then \(\rho\) is called a relation on \(A\).
After
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept
A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of \(A\times B\).
If \(A = B\), then \(\rho\) is called a relation on \(A\).
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept
A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of \(A\times B\).
If \(A = B\), then \(\rho\) is called a relation on \(A\).
Field-by-field Comparison
| Field |
Before |
After |
| Text |
A <b>relation </b>\(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a {{c1::subset}} of {{c1::\(A\times B\).}} <br><br>If \(A = B\), then \(\rho\) is called {{c1::a <i>relation on</i> \(A\).}} |
A <b>relation </b>\(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a subset of {{c1::\(A\times B\)}}.<br><br>If \(A = B\), then \(\rho\) is called a {{c1::<i>relation on</i> \(A\)}}. |
Tags:
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept
Note 10: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Od*A$z}#*`
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For two groups \(\langle G;*;\widehat{};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a group homomorphism if for all \(a\) and \(b\):
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For two groups \(\langle G;*;\widehat{};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a group homomorphism if for all \(a\) and \(b\):
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For two groups \(\langle G;*;\widehat{\hspace{0.5em}};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a group homomorphism if for all \(a\) and \(b\):
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For two groups \(\langle G;*;\widehat{\hspace{0.5em}};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a group homomorphism if for all \(a\) and \(b\):
\(\psi(a*b) = \psi(a)\star\psi(b)\).
If \(\psi\) is a bijection from \(G\) to \(H\), then it is called an isomorphism.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
For two groups \(\langle G;*;\widehat{};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a <i>group homomorphism</i> if for all \(a\) and \(b\):<br>{{c1::\(\psi(a*b) = \psi(a)\star\psi(b)\)}}.<br>If \(\psi\) is {{c2::a bijection from \(G\) to \(H\)}}, then it is called an <i>isomorphism</i>. |
For two groups \(\langle G;*;\widehat{\hspace{0.5em}};e\rangle\)and \(\langle H;\star;\sim;e'\rangle\), a function \(\psi: G\to H\) is called a <i>group homomorphism</i> if for all \(a\) and \(b\):<br>{{c1::\(\psi(a*b) = \psi(a)\star\psi(b)\)}}.<br><br>If \(\psi\) is {{c2::a bijection from \(G\) to \(H\)}}, then it is called an <i>isomorphism</i>. |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
Note 11: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: lx/&=nJI{d
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Neutral Element of a group:
- Addition \(0\).
- Multiplication \(1\).
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Neutral Element of a group:
- Addition \(0\).
- Multiplication \(1\).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Neutral Element of a group:
- For Addition: \(0\)
- For Multiplication: \(1\)
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Neutral Element of a group:
- For Addition: \(0\)
- For Multiplication: \(1\)
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>Neutral Element of a group:</p><ul><li><b>Addition</b> {{c1::\(0\)}}. </li><li><b>Multiplication</b> {{c2::\(1\)}}.</li></ul> |
<p>Neutral Element of a group:</p><ul><li><b>For Addition:</b> {{c1::\(0\)}}</li><li><b>For Multiplication:</b> {{c2::\(1\)}}</li></ul> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 12: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: oo(x.D7C(:
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
What is the left cancellation law in a group?
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
What is the left cancellation law in a group?
Left cancellation law: \(a * b = a * c \ \implies \ b = c\)
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
What is the left cancellation law in a group?
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
What is the left cancellation law in a group?
\(a * b = a * c \ \implies \ b = c\)
Field-by-field Comparison
| Field |
Before |
After |
| Back |
Left cancellation law: \(a * b = a * c \ \implies \ b = c\) |
\(a * b = a * c \ \implies \ b = c\) |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 13: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: qo:})x5a6`
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group, the equations \(a * x = b\) and \(x * a = b\) have unique solutions for all \(a, b\) (property G3(v)).
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group, the equations \(a * x = b\) and \(x * a = b\) have unique solutions for all \(a, b\) (property G3(v)).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group, the equations \(a * x = b\) and \(x * a = b\) have unique solutions for all \(a, b\).
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group, the equations \(a * x = b\) and \(x * a = b\) have unique solutions for all \(a, b\).
Property G3 (v)
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>In a group, the equations \({{c1::a * x = b}}\) and \({{c2::x * a = b}}\) have {{c3::unique solutions}} for all \(a, b\) (property G3(v)).</p> |
<p>In a group, the equations \({{c1::a * x = b}}\) and \({{c2::x * a = b}}\) have {{c3::unique solutions}} for all \(a, b\).</p> |
| Extra |
|
Property G3 (v) |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 14: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: rI[60?4iFu
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
A group has the following properties:
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
A group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
A
group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
A
group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
Field-by-field Comparison
| Field |
Before |
After |
| Text |
A <b>group</b> has the following properties: |
A <b>group</b> has the following properties:<br><ol><li>{{c1::Closure}}</li><li>{{c2::Associativity}}</li><li>{{c3::Identity}}</li><li>{{c4::Inverse}}</li></ol> |
| Extra |
<ul><li>Closure</li><li>Associativity</li><li>Identity</li><li>Inverse</li></ul> |
|
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 15: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: u${[$*iYrd
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups
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
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups
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
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups
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
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups
Does the uniqueness of the neutral element imply that a group is abelian (commutative)?
I.e. does \(a*e = e*a\) mean G is abelian?
No! The uniqueness of the neutral element does not imply commutativity.
Counterexample: The identity matrix \(I_3\) is the unique neutral element for the group of \(3 \times 3\) real matrices under multiplication. We have \(A \cdot I_3 = I_3 \cdot A\) for all matrices \(A\), even though matrix multiplication is not commutative in general.
Field-by-field Comparison
| Field |
Before |
After |
| Front |
<p>Does the uniqueness of the neutral element imply that a group is abelian (commutative)?</p><br><br>I.e. does a*e = e*a mean G is abelian? |
<p>Does the uniqueness of the neutral element imply that a group is abelian (commutative)?</p>I.e. does \(a*e = e*a\) mean G is abelian? |
| Back |
<p><strong>No!</strong> The uniqueness of the neutral element does <strong>not</strong> imply commutativity.</p><br><p><strong>Counterexample</strong>: The identity matrix \(I_3\) is the unique neutral element for the group of \(3 \times 3\) real matrices under multiplication. We have \(A \cdot I_3 = I_3 \cdot A\) for all matrices \(A\), even though matrix multiplication is <strong>not commutative</strong> in general.</p> |
<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> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups
Note 16: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: wV8Y&j0xY.
modified
Before
Front
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:
- unary operators (NOT)
- quantifiers (for all and exists)
- operators (AND, OR)
- Implication
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
Name the binding strengths of PL tokens in order:
- unary operators (NOT)
- quantifiers (for all and exists)
- operators (AND, OR)
- Implication
After
Front
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:
- Unary operators (NOT)
- Quantifiers (for all and exists)
- Operators (AND, OR)
- Implications
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
Name the binding strengths of PL tokens in order:
- Unary operators (NOT)
- Quantifiers (for all and exists)
- Operators (AND, OR)
- Implications
Field-by-field Comparison
| Field |
Before |
After |
| Text |
Name the binding strengths of PL tokens in order:<br><ol><li>{{c1:: unary operators (NOT)}}</li><li>{{c2:: quantifiers (for all and exists)}}</li><li>{{c3:: operators (AND, OR)}}</li><li>{{c4:: Implication}}</li></ol> |
Name the binding strengths of PL tokens in order:<br><ol><li>{{c1:: Unary operators (NOT)}}</li><li>{{c2:: Quantifiers (for all and exists)}}</li><li>{{c3:: Operators (AND, OR)}}</li><li>{{c4:: Implications}}</li></ol> |
Tags:
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 17: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: y4s0XCy@A
modified
Before
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
Consider the poset \((A; \preceq)\).
\(a \in A\) is the least (greatest) element of \(A\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
Consider the poset \((A; \preceq)\).
\(a \in A\) is the least (greatest) element of \(A\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
After
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
Consider the poset \((A; \preceq)\).
\(a \in A\) is the least (greatest) element of \(A\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
Consider the poset \((A; \preceq)\).
\(a \in A\) is the least (greatest) element of \(A\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
Note that a least or a greatest element need not exist. However, there can be at most one least element, as suggested by the word “the” in the definition.
This follows directly from the antisymmetry of \(\preceq\). If there were two least elements, they would be mutually comparable, and hence must
be equal.
Field-by-field Comparison
| Field |
Before |
After |
| Extra |
|
Note that a least or a greatest element need not exist. However, there can be at most one least element, as suggested by the word “the” in the definition. <br><br>This follows directly from the antisymmetry of \(\preceq\). If there were two least elements, they would be mutually comparable, and hence must<br>be equal. |
Tags:
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
Note 18: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: yAP=DE#~t<
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
An
abelian group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
- Commutative
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
An
abelian group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
- Commutative
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
An
abelian group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
- Commutativity
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
An
abelian group has the following properties:
- Closure
- Associativity
- Identity
- Inverse
- Commutativity
Field-by-field Comparison
| Field |
Before |
After |
| Text |
An <b>abelian group</b> has the following properties:<br><ol><li>{{c1::Closure}}</li><li>{{c2::Associativity}}</li><li>{{c3::Identity}}</li><li>{{c4::Inverse}}</li><li>{{c5::Commutative}}</li></ol> |
An <b>abelian group</b> has the following properties:<br><ol><li>{{c1::Closure}}</li><li>{{c2::Associativity}}</li><li>{{c3::Identity}}</li><li>{{c4::Inverse}}</li><li>{{c5::Commutativity}}</li></ol> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
Note 19: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: ztTfjE7<>>
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Inverse in a group:
- Addition \(-a\)
- Multiplication {{c2::\(a^{-1}\)}}.
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Inverse in a group:
- Addition \(-a\)
- Multiplication {{c2::\(a^{-1}\)}}.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Inverse in a group:
- For Addition: \(-a\)
- For Multiplication: {{c2::\(a^{-1}\)}}.
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Inverse in a group:
- For Addition: \(-a\)
- For Multiplication: {{c2::\(a^{-1}\)}}.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>Inverse in a group:</p><ul><li><b>Addition </b>{{c1::\(-a\)}}</li><li><b>Multiplication </b>{{c2::\(a^{-1}\)}}.</li></ul> |
<p>Inverse in a group:</p><ul><li><b>For Addition: </b>{{c1::\(-a\)}}</li><li><b>For Multiplication: </b>{{c2::\(a^{-1}\)}}.</li></ul> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 20: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: eUCQYkiYf@
modified
Before
Front
ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals
What is a property that always hold for linear transformations?
Back
ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals
What is a property that always hold for linear transformations?
for a linear transformation \(T(X)\): \(T(0) = 0\)
After
Front
ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals
What is a property that always hold for a linear transformation \(T(X)\)?
Back
ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals
What is a property that always hold for a linear transformation \(T(X)\)?
\(T(0) = 0\)
Field-by-field Comparison
| Field |
Before |
After |
| Front |
What is a property that always hold for linear transformations? |
What is a property that always hold for a linear transformation \(T(X)\)? |
| Back |
for a linear transformation \(T(X)\): \(T(0) = 0\) |
\(T(0) = 0\) |
Tags:
ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals