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
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
After
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
Field-by-field Comparison
| Field |
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| Text |
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> |
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> |
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: Horvath Cloze
GUID: M,?u9cw(S%
modified
Before
Front
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\)
Back
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\)
After
Front
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\) (O-notation)
Back
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\) (O-notation)
Field-by-field Comparison
| Field |
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| Text |
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(O(n \log(n))\):: (O-notation)}} |
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(O(n \log(n))\) (O-notation)}} |
Tags:
ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Sums
Note 4: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: M11/nZaHIu
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::2._ADT_Dictionary
ADT-Dictionary:
|
search |
insert |
delete |
| unsorted Array |
\(O(n)\) |
\(O(1)\) |
\(O(n)\) |
| sorted Array |
\(O(\log n)\) |
\(O(n)\) |
\(O(n)\) |
| DLL |
\(O(n)\) (no bin. search possible) |
\(O(1)\) |
\(O(n)\) |
Back
ETH::1._Semester::A&D::05._Data_Structures::2._ADT_Dictionary
ADT-Dictionary:
|
search |
insert |
delete |
| unsorted Array |
\(O(n)\) |
\(O(1)\) |
\(O(n)\) |
| sorted Array |
\(O(\log n)\) |
\(O(n)\) |
\(O(n)\) |
| DLL |
\(O(n)\) (no bin. search possible) |
\(O(1)\) |
\(O(n)\) |
After
Front
ETH::1._Semester::A&D::05._Data_Structures::2._ADT_Dictionary
|
search |
insert |
delete |
| unsorted Array |
\(O(n)\) |
\(O(1)\) |
\(O(n)\) |
| sorted Array |
\(O(\log n)\) |
\(O(n)\) |
\(O(n)\) |
| DLL |
\(O(n)\) (no bin. search possible) |
\(O(1)\) |
\(O(n)\) |
Back
ETH::1._Semester::A&D::05._Data_Structures::2._ADT_Dictionary
|
search |
insert |
delete |
| unsorted Array |
\(O(n)\) |
\(O(1)\) |
\(O(n)\) |
| sorted Array |
\(O(\log n)\) |
\(O(n)\) |
\(O(n)\) |
| DLL |
\(O(n)\) (no bin. search possible) |
\(O(1)\) |
\(O(n)\) |
Field-by-field Comparison
| Field |
Before |
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| Text |
<b></b><b></b><b></b><b></b>
ADT-Dictionary:
<table>
<tbody><tr>
<td></td>
<td><b>search</b></td>
<td><b>insert</b></td>
<td><b>delete</b></td>
</tr>
<tr>
<td>unsorted Array</td>
<td>{{c1::\(O(n)\)}}</td>
<td>{{c2::\(O(1)\)}}</td>
<td>{{c3::\(O(n)\)}}</td>
</tr>
<tr>
<td>sorted Array</td>
<td>{{c4::\(O(\log n)\)}}</td>
<td>{{c5::\(O(n)\)}}</td>
<td>{{c6::\(O(n)\)}}</td>
</tr>
<tr>
<td>DLL</td>
<td>{{c7::\(O(n)\) (no bin. search possible)}}</td>
<td>{{c8::\(O(1)\)}}</td>
<td>{{c9::\(O(n)\)}}</td>
</tr>
</tbody></table> |
<b></b><b></b><b></b><b></b><table>
<tbody><tr>
<td></td>
<td><b>search</b></td>
<td><b>insert</b></td>
<td><b>delete</b></td>
</tr>
<tr>
<td>unsorted Array</td>
<td>{{c1::\(O(n)\)}}</td>
<td>{{c2::\(O(1)\)}}</td>
<td>{{c3::\(O(n)\)}}</td>
</tr>
<tr>
<td>sorted Array</td>
<td>{{c4::\(O(\log n)\)}}</td>
<td>{{c5::\(O(n)\)}}</td>
<td>{{c6::\(O(n)\)}}</td>
</tr>
<tr>
<td>DLL</td>
<td>{{c7::\(O(n)\) (no bin. search possible)}}</td>
<td>{{c8::\(O(1)\)}}</td>
<td>{{c9::\(O(n)\)}}</td>
</tr>
</tbody></table> |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::2._ADT_Dictionary
Note 5: 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 \(K(n)\). (Although it can be programmed to be in-place.)<br><b>Stable</b> |
not in place, thus the space complexity is \(K(n)\). (can be made in place)<br><b>Stable</b> |
Tags:
ETH::1._Semester::A&D::04._Sorting_Algorithms::4._Merge_Sort
Note 6: 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 values in a tree form, with the largest element always being 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 the values in a tree form, with the largest element always as the root.
Field-by-field Comparison
| Field |
Before |
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| Back |
A datastructure that stores values in a tree form, with the largest element always being the root. |
A datastructure that stores the values in a tree form, with the largest element always as the root. |
Tags:
ETH::1._Semester::A&D::04._Sorting_Algorithms::7._Heapsort
Note 7: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: Pf|C9|^n[w
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::2._Linked_List
In a
singly and
doubly linked list, the operation:
- Insert is \(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Theta(l)\).
- Get is \(\Theta(i)\) very slow as we need to traverse the entire list up to i
- insertAfter is \(O(1)\) if we get the memory address of the element to insert after.
- delete is:
SLL: \(\Theta(l)\) as we need to find the previous element and change it's pointer.
DLL: \(O(1)\) we know the address of the previous element and then just edit it's pointer.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::2._Linked_List
In a
singly and
doubly linked list, the operation:
- Insert is \(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Theta(l)\).
- Get is \(\Theta(i)\) very slow as we need to traverse the entire list up to i
- insertAfter is \(O(1)\) if we get the memory address of the element to insert after.
- delete is:
SLL: \(\Theta(l)\) as we need to find the previous element and change it's pointer.
DLL: \(O(1)\) we know the address of the previous element and then just edit it's pointer.
After
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::2._Linked_List
In a
singly and
doubly linked list, the operation:
- Insert is \(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Theta(l)\).
- Get is \(\Theta(i)\) very slow as we need to traverse the entire list up to i
- insertAfter is \(O(1)\) if we get the memory address of the element to insert after.
- delete is:
SLL: {{c4::\(\Theta(l)\) as we need to find the previous element and change it's pointer}.}
DLL: \(O(1)\) we know the address of the previous element and then just edit it's pointer.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::2._Linked_List
In a
singly and
doubly linked list, the operation:
- Insert is \(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Theta(l)\).
- Get is \(\Theta(i)\) very slow as we need to traverse the entire list up to i
- insertAfter is \(O(1)\) if we get the memory address of the element to insert after.
- delete is:
SLL: {{c4::\(\Theta(l)\) as we need to find the previous element and change it's pointer}.}
DLL: \(O(1)\) we know the address of the previous element and then just edit it's pointer.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
In a <b>singly</b> and <b>doubly linked list</b>, the operation:<br><ul><li><b>Insert</b> is {{c1::\(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Theta(l)\). }}<br></li><li><b>Get</b> is {{c2::\(\Theta(i)\) very slow as we need to traverse the entire list up to <b>i</b>}}<br></li><li><b>insertAfter</b> is {{c3:: \(O(1)\) if we get the memory address of the element to insert after.}}<br></li><li><b>delete</b> is:<br> SLL: {{c4::\(\Theta(l)\) as we need to find the previous element and change it's pointer.}}<br> DLL: {{c5:: \(O(1)\) we know the address of the previous element and then just edit it's pointer.}}</li></ul> |
In a <b>singly</b> and <b>doubly linked list</b>, the operation:<br><ul><li><b>Insert</b> is {{c1::\(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Theta(l)\). }}<br></li><li><b>Get</b> is {{c2::\(\Theta(i)\) very slow as we need to traverse the entire list up to <b>i</b>}}<br></li><li><b>insertAfter</b> is {{c3:: \(O(1)\) if we get the memory address of the element to insert after.}}<br></li><li><b>delete</b> is:<br> SLL: {{c4::\(\Theta(l)\) as we need to find the previous element and change it's pointer}.}<br> DLL: {{c5:: \(O(1)\) we know the address of the previous element and then just edit it's pointer.}}</li></ul> |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::2._Linked_List
Note 8: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: nK{)v6I%zc
modified
Before
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::4._Stack
The ADT
stack can be efficiently implemented using a
singly linked list:
- push: \(\Theta(1)\)
- pop: \(\Theta(1)\) as this removes the first element. This means we just copy the pointer next from the first element to be the first pointer of the list.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::4._Stack
The ADT
stack can be efficiently implemented using a
singly linked list:
- push: \(\Theta(1)\)
- pop: \(\Theta(1)\) as this removes the first element. This means we just copy the pointer next from the first element to be the first pointer of the list.
After
Front
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::4._Stack
The ADT
stack can be efficiently implemented using a
singly linked list:
- push: \(\Theta(1)\)
- pop: \(\Theta(1)\) as this removes the first element. This means we just copy the pointer next from the first element to be the first pointer of the list.
Back
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::4._Stack
The ADT
stack can be efficiently implemented using a
singly linked list:
- push: \(\Theta(1)\)
- pop: \(\Theta(1)\) as this removes the first element. This means we just copy the pointer next from the first element to be the first pointer of the list.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
The ADT <b>stack</b> can be efficiently implemented using a {{c1::<b>singly linked list</b>}}:<br><ul><li><b>push</b>: {{c2:: \(\Theta(1)\)}}<br></li><li><b>pop</b>: {{c3:: \(\Theta(1)\) as this removes the first element. This means we just copy the pointer next from the first element to be the first pointer of the list.}}<br></li></ul> |
The ADT <b>stack</b> can be efficiently implemented using a {{c1:: <b>singly linked list</b>}}:<br><ul><li><b>push</b>: {{c2:: \(\Theta(1)\)}}<br></li><li><b>pop</b>: {{c3:: \(\Theta(1)\) as this removes the first element. This means we just copy the pointer next from the first element to be the first pointer of the list.}}<br></li></ul> |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::4._Stack
Note 9: 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>}}. </div><div><br></div><div>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>}}. 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 10: 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 11: 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::/works under the principle of..}}. |
A queue is also called {{c1:: FIFO}}. |
Tags:
ETH::1._Semester::A&D::05._Data_Structures::1._ADT_List::5._Queue
Note 12: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: BA!Uj{h&4e
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{im} h\)}} is the set of all elements in \(H\) that are mapped to by some element in \(G\).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{im} h\)}} is the set of all elements in \(H\) that are mapped to by some element in \(G\).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{Im} (h)\)}} is the set of all elements in \(H\) that are mapped to by some element in \(G\).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{Im} (h)\)}} is the set of all elements in \(H\) that are mapped to by some element in \(G\).
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{im} h\)}} is the {{c2::set of all elements in \(H\) that are mapped to by some element in \(G\)}}.</p> |
<p>For a homomorphism \(h: G \rightarrow H\), the {{c1::image \(\text{Im} (h)\)}} is the {{c2::set of all elements in \(H\) that are mapped to by some element in \(G\)}}.</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
Note 13: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: C18gm]huq&
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}}
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}}
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a Group:
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
In a Group:
<br><br>{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} |
<p>In a Group:<br>
{{c1:: \(\widehat{a * b}\) }} = {{c2:: \(\widehat{b} * \widehat{a}\)}} This is a property from a Lemma.</p> |
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 Classic
GUID: D0k7OAp
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
What happens if there is a left and right neutral element in a group?
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
What happens if there is a left and right neutral element in a group?
Lemma 5.1: If \(\langle S; * \rangle\) has both a left and a right neutral element, then they are equal.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
What happens if there is a left and right neutral element in a group?
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
What happens if there is a left and right neutral element in a group?
Lemma 5.1: If \(\langle S; * \rangle\) has both a left and a right neutral element, then they are equal.
\(e * e' = e\) (\(e'\) right inverse)
\(e * e' = e'\) (\(e\) left inverse)
Field-by-field Comparison
| Field |
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| Back |
<p><strong>Lemma 5.1</strong>: If \(\langle S; * \rangle\) has both a left and a right neutral element, then they are <strong>equal</strong>.</p> |
<p><strong>Lemma 5.1</strong>: If \(\langle S; * \rangle\) has both a left and a right neutral element, then they are <strong>equal</strong>.</p><p>\(e * e' = e\) (\(e'\) right inverse)</p><p>\(e * e' = e'\) (\(e\) left inverse)</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::1._Neutral_Elements
Note 15: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: FSUY[I=V>]
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
Corollary 5.14 (Fermat's Little Theorem): For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]
In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]
Proof: This follows from Corollary 5.10 (\(a^{|G|} = e\)) since the order of \(\mathbb{Z}_m^*\) is \[a^{p-1} \equiv_p 1\]0 (and \[a^{p-1} \equiv_p 1\]1 for prime \[a^{p-1} \equiv_p 1\]2).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
Corollary 5.14 (Fermat's Little Theorem): For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]
In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]
Proof: This follows from Corollary 5.10 (\(a^{|G|} = e\)). Since \(\gcd(a, m)=1\), it is an element of \(\mathbb{Z}_m^*\) and thus an element of a group. \(\langle a \rangle\) therefore must be a subgroup with an order that divides \(\mathbb{Z}_m^* = \varphi(m)\)\(\iff \varphi(m) = \operatorname{ord}(a) \cdot k\) (Lagrange's) .
\(\implies (a^{\operatorname{ord}(a)})^k = a^{\varphi(m)} \equiv_m 1\)
Field-by-field Comparison
| Field |
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| Back |
<p><strong>Corollary 5.14 (Fermat's Little Theorem)</strong>: For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]</p>
<p>In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]</p>
<p><strong>Proof</strong>: This follows from Corollary 5.10 (\(a^{|G|} = e\)) since the order of \(\mathbb{Z}_m^*\) is \[a^{p-1} \equiv_p 1\]0 (and \[a^{p-1} \equiv_p 1\]1 for prime \[a^{p-1} \equiv_p 1\]2).</p> |
<p><strong>Corollary 5.14 (Fermat's Little Theorem)</strong>: For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]</p>
<p>In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]</p>
<p><strong>Proof</strong>: This follows from Corollary 5.10 (\(a^{|G|} = e\)). Since \(\gcd(a, m)=1\), it is an element of \(\mathbb{Z}_m^*\) and thus an element of a group. \(\langle a \rangle\) therefore must be a subgroup with an order that divides \(\mathbb{Z}_m^* = \varphi(m)\)\(\iff \varphi(m) = \operatorname{ord}(a) \cdot k\) (Lagrange's) .</p><p>\(\implies (a^{\operatorname{ord}(a)})^k = a^{\varphi(m)} \equiv_m 1\)</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Note 16: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: G3,dI)){d{
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
Which elements generate \(\mathbb{Z}_n\)? Also proof
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
Which elements generate \(\mathbb{Z}_n\)? Also proof
\(\mathbb{Z}_n\) is generated by all \(a \in \mathbb{Z}_n\) for which \(\gcd(n, a) = 1\) (all elements coprime to \(n\)).
Proof idea:
- If \(\mathbb{Z}_n = \langle a \rangle\), then \(1 \in \langle a \rangle\)
- This means \(au \equiv_n 1\) for some \(u\)
- By Bézout's identity, this implies \(\gcd(a, n) = 1\)
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
Which elements generate \(\mathbb{Z}_n\)? Also proof
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
Which elements generate \(\mathbb{Z}_n\)? Also proof
\(\mathbb{Z}_n\) is generated by all \(a \in \mathbb{Z}_n\) for which \(\gcd(n, a) = 1\) (all elements coprime to \(n\)).
Proof idea:
- If \(\mathbb{Z}_n = \langle a \rangle\), then \(1 \in \langle a \rangle\)
- This means \(au \equiv_n 1\) for some \(u\)
- By Bézout's identity, this implies \(\gcd(a, n) = 1\) (since \(\gcd\) must divide both \(au-qn\) and 1).
Field-by-field Comparison
| Field |
Before |
After |
| Back |
<p>\(\mathbb{Z}_n\) is generated by <strong>all \(a \in \mathbb{Z}_n\) for which \(\gcd(n, a) = 1\)</strong> (all elements coprime to \(n\)).</p>
<p><strong>Proof idea</strong>:<br>
- If \(\mathbb{Z}_n = \langle a \rangle\), then \(1 \in \langle a \rangle\)<br>
- This means \(au \equiv_n 1\) for some \(u\)<br>
- By Bézout's identity, this implies \(\gcd(a, n) = 1\)</p> |
<p>\(\mathbb{Z}_n\) is generated by <strong>all \(a \in \mathbb{Z}_n\) for which \(\gcd(n, a) = 1\)</strong> (all elements coprime to \(n\)).</p>
<p><strong>Proof idea</strong>:<br>
- If \(\mathbb{Z}_n = \langle a \rangle\), then \(1 \in \langle a \rangle\)<br>
- This means \(au \equiv_n 1\) for some \(u\)<br>
- By Bézout's identity, this implies \(\gcd(a, n) = 1\) (since \(\gcd\) must divide both \(au-qn\) and 1).</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
Note 17: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: GE_=q.pKz`
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Group axiom G1 states that the operation \(*\) is associative: .
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Group axiom G1 states that the operation \(*\) is associative: .
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Group axiom G1 states that the operation \(*\) is associative: \(a * (b * c) = (a * b) * c\) for all \(a, b, c\) in \(G\).
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Group axiom G1 states that the operation \(*\) is associative: \(a * (b * c) = (a * b) * c\) for all \(a, b, c\) in \(G\).
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>Group axiom <strong>G1</strong> states that the operation \(*\) is {{c1::associative: {{c2::\(a * (b * c) = (a * b) * c\)}} for all \(a, b, c\) in \(G\)}}.</p> |
<p>Group axiom <strong>G1</strong> states that the operation \(*\) is {{c1::associative}}: {{c2::\(a * (b * c) = (a * b) * c\)}} for all \(a, b, c\) in \(G\).</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 18: 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></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> |
<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> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::4._(Non-)minimality_of_the_Group_Axioms
Note 19: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: J!)tsK,]3<
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::6._Fields
A field (Körper) is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
Back
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::6._Fields
A field (Körper) is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)
After
Front
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::6._Fields
A field (Körper) is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
Back
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::6._Fields
A field (Körper) is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)
non-trivial: {0} is not a field. In particular, 1 = 0 (neutral element of mult. = neutral element of add.) causes trouble.
Field-by-field Comparison
| Field |
Before |
After |
| Extra |
Example: \(\mathbb{R}\), but not \(\mathbb{Z}\) |
Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)<br><br>non-trivial: {0} is not a field. In particular, 1 = 0 (neutral element of mult. = neutral element of add.) causes trouble. |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::6._Fields
Note 20: 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 subset of {{c1::\(A\times B\)}}.<br><br>If \(A = B\), then \(\rho\) is called a {{c1::<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 {{c1::subset}} of {{c1::\(A\times B\).}} <br><br>If \(A = B\), then \(\rho\) is called {{c1::a <i>relation on</i> \(A\).}} |
Tags:
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept
Note 21: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Mz.H046~kk
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
To verify the homomorphism property, check that: \(\phi(g_1 \cdot g_2) = \phi(g_1) + \phi(g_2)\) for all \(g_1, g_2\) in \(G\).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
To verify the homomorphism property, check that: \(\phi(g_1 \cdot g_2) = \phi(g_1) + \phi(g_2)\) for all \(g_1, g_2\) in \(G\).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
To verify the homomorphism property, check that: \(\psi(g_1 \cdot g_2) = \psi(g_1) + \psi(g_2)\) for all \(g_1, g_2\) in \(G\).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
To verify the homomorphism property, check that: \(\psi(g_1 \cdot g_2) = \psi(g_1) + \psi(g_2)\) for all \(g_1, g_2\) in \(G\).
Field-by-field Comparison
| Field |
Before |
After |
| Text |
<p>To verify the {{c1::homomorphism property}}, check that: {{c2::\(\phi(g_1 \cdot g_2) = \phi(g_1) + \phi(g_2)\)}} for all \(g_1, g_2\) in \(G\).</p> |
<p>To verify the {{c1::homomorphism property}}, check that: {{c2::\(\psi(g_1 \cdot g_2) = \psi(g_1) + \psi(g_2)\)}} for all \(g_1, g_2\) in \(G\).</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
Note 22: 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 \(\langle G;*;\widehat{\hspace{0.5em}};e\rangle\)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.
After
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.
Field-by-field Comparison
| Field |
Before |
After |
| Text |
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>. |
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>. |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::2._Group_Homomorphisms
Note 23: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: PjfIvXynOi
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?
|
\(\mathbb{Z}_m\) |
\(\mathbb{Z}_m^*\) |
| \(\oplus\) |
Yes (forms a group) |
No |
| \(\odot\) |
No |
Yes (forms a group) |
Key point: \(\mathbb{Z}_m\) with addition \(\oplus\) is a group. \(\mathbb{Z}_m^*\) (coprime elements) with multiplication \(\odot\) is a group.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Which group operations work for \(\mathbb{Z}_m\) and \(\mathbb{Z}_m^*\)?
|
\(\mathbb{Z}_m\) |
\(\mathbb{Z}_m^*\) |
| \(\oplus\) |
Yes (forms a group) |
No |
| \(\odot\) |
No |
Yes (forms a group) |
Key point: \(\mathbb{Z}_m\) with addition \(\oplus\) is a group. \(\mathbb{Z}_m^*\) (coprime elements) with multiplication \(\odot\) is a group.
\(\mathbb{Z}_m^*\) is not a group under addition b/c it doesn't contain the neutral element 0.
Field-by-field Comparison
| Field |
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| Back |
<table>
<thead>
<tr>
<th></th>
<th>\(\mathbb{Z}_m\)</th>
<th>\(\mathbb{Z}_m^*\)</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(\oplus\)</td>
<td><strong>Yes</strong> (forms a group)</td>
<td>No</td>
</tr>
<tr>
<td>\(\odot\)</td>
<td>No</td>
<td><strong>Yes</strong> (forms a group)</td>
</tr>
</tbody>
</table>
<p><strong>Key point</strong>: \(\mathbb{Z}_m\) with addition \(\oplus\) is a group. \(\mathbb{Z}_m^*\) (coprime elements) with multiplication \(\odot\) is a group.</p> |
<table>
<thead>
<tr>
<th></th>
<th>\(\mathbb{Z}_m\)</th>
<th>\(\mathbb{Z}_m^*\)</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(\oplus\)</td>
<td><strong>Yes</strong> (forms a group)</td>
<td>No</td>
</tr>
<tr>
<td>\(\odot\)</td>
<td>No</td>
<td><strong>Yes</strong> (forms a group)</td>
</tr>
</tbody>
</table>
<p><strong>Key point</strong>: \(\mathbb{Z}_m\) with addition \(\oplus\) is a group. \(\mathbb{Z}_m^*\) (coprime elements) with multiplication \(\odot\) is a group.</p><p>\(\mathbb{Z}_m^*\) is not a group under addition b/c it doesn't contain the neutral element 0.</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Note 24: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: c
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
For what order is every group cyclic?
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
For what order is every group cyclic?
If the order of the group is prime, it is cyclic!
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
For what order is every group cyclic?
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
For what order is every group cyclic?
If the order of the group is prime, it is cyclic!
Every element has order 1 or \(|G|\) (Lagrange). Therefore, it is either the neutral element or a generator of the entire group.
Field-by-field Comparison
| Field |
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| Back |
<p>If the <strong>order of the group</strong> is <strong>prime</strong>, it is cyclic!</p> |
<p>If the <strong>order of the group</strong> is <strong>prime</strong>, it is cyclic!</p><p><br></p><p>Every element has order 1 or \(|G|\) (Lagrange). Therefore, it is either the neutral element or a generator of the entire group.</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
Note 25: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: eJwT]j&5OY
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
\(\mathbb{Z}_m\) (with \(\oplus\)) is not a group with respect to multiplication modulo \(m\) because elements that are not coprime to \(m\) don't have a multiplicative inverse.
For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).
Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(\oplus\)0).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
\(\mathbb{Z}_m\) (with \(\oplus\)) is not a group with respect to multiplication modulo \(m\) because elements that are not coprime to \(m\) don't have a multiplicative inverse.
For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).
Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(m\)).
Field-by-field Comparison
| Field |
Before |
After |
| Back |
<p>\(\mathbb{Z}_m\) (with \(\oplus\)) is <strong>not a group</strong> with respect to multiplication modulo \(m\) because elements that are <strong>not coprime</strong> to \(m\) don't have a <strong>multiplicative inverse</strong>.</p>
<p>For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).</p>
<p>Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(\oplus\)0).</p> |
<p>\(\mathbb{Z}_m\) (with \(\oplus\)) is <strong>not a group</strong> with respect to multiplication modulo \(m\) because elements that are <strong>not coprime</strong> to \(m\) don't have a <strong>multiplicative inverse</strong>.</p>
<p>For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).</p>
<p>Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(m\)).</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function
Note 26: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: jngIBgkHz<
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
State Corollary 5.11 about groups of prime order (what property, what does each element satisfy).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
State Corollary 5.11 about groups of prime order (what property, what does each element satisfy).
Corollary 5.11: Every group of prime order is cyclic, and in such a group every element except the neutral element is a generator.
Proof: Only \(1 | p\) and \(p | p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group).
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
State Corollary 5.11 about groups of prime order (what property, what does each element satisfy).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
State Corollary 5.11 about groups of prime order (what property, what does each element satisfy).
Corollary 5.11: Every group of prime order is cyclic, and in such a group every element except the neutral element is a generator.
Proof: Only \(1 | p\) and \(p | p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group; Lagrange).
Field-by-field Comparison
| Field |
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| Back |
<p><strong>Corollary 5.11</strong>: Every group of <strong>prime order</strong> is cyclic, and in such a group <strong>every element except the neutral element is a generator</strong>.</p>
<p><strong>Proof</strong>: Only \(1 | p\) and \(p | p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group).</p> |
<p><strong>Corollary 5.11</strong>: Every group of <strong>prime order</strong> is cyclic, and in such a group <strong>every element except the neutral element is a generator</strong>.</p>
<p><strong>Proof</strong>: Only \(1 | p\) and \(p | p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group; Lagrange).</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::7._The_Order_of_Subgroups
Note 27: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: l];xKGd{%I
modified
Before
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
A function \( f: A \rightarrow B\) is surjective (or onto) if \( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken.
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
A function \( f: A \rightarrow B\) is surjective (or onto) if \( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken.
After
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
A function \( f: A \rightarrow B\) is surjective (or onto) if \( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
A function \( f: A \rightarrow B\) is surjective (or onto) if \( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken
Field-by-field Comparison
| Field |
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| Text |
A function \( f: A \rightarrow B\) is {{c1::surjective (or onto)}} if {{c2::\( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken}}. |
A function \( f: A \rightarrow B\) is {{c1::surjective (or onto)}} if {{c2::\( \forall b \ \exists a \ , b = f(a)\), i.e. every value is taken}} |
Tags:
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
Note 28: 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:
- 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: \(1\)
After
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 \(1\).
Field-by-field Comparison
| Field |
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| Text |
<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> |
<p>Neutral Element of a group:</p><ul><li><b>Addition</b> {{c1::\(0\)}}. </li><li><b>Multiplication</b> {{c2::\(1\)}}.</li></ul> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 29: 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?
\(a * b = a * c \ \implies \ b = c\)
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?
Left cancellation law: \(a * b = a * c \ \implies \ b = c\)
Field-by-field Comparison
| Field |
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| Back |
\(a * b = a * c \ \implies \ b = c\) |
Left cancellation law: \(a * b = a * c \ \implies \ b = c\) |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 30: 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 \(a, b\).
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 \(a, b\) (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)).
Field-by-field Comparison
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| Text |
<p>In a group, the equations \({{c1::a * x = b}}\) and \({{c2::x * a = b}}\) have {{c3::unique solutions}} for all \(a, b\).</p> |
<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> |
| Extra |
Property G3 (v) |
|
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 31: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
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:
After
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
Field-by-field Comparison
| Field |
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| Front |
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> |
A <b>group</b> has the following properties: |
| Back |
|
<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 32: 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>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><br><br>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><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><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> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::5._Some_Examples_of_Groups
Note 33: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: uaVso1SVrk
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::4._The_Order_of_Group_Elements_and_of_a_Group
If no \(m\) exists, such that \(a^m = e\) in a group, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::4._The_Order_of_Group_Elements_and_of_a_Group
If no \(m\) exists, such that \(a^m = e\) in a group, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::4._The_Order_of_Group_Elements_and_of_a_Group
If no \(m>0\) exists, such that \(a^m = e\) in a group, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::4._The_Order_of_Group_Elements_and_of_a_Group
If no \(m>0\) exists, such that \(a^m = e\) in a group, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.
Field-by-field Comparison
| Field |
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| Text |
<p>If {{c2:: no \(m\) exists, such that \(a^m = e\) in a group}}, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.</p> |
<p>If {{c2:: no \(m>0\) exists, such that \(a^m = e\) in a group}}, the order \(\text{ord}(a)\) of \(a\) is {{c1:: said to be infinite, written \(\text{ord}(a) = \infty\)}}.</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::4._The_Order_of_Group_Elements_and_of_a_Group
Note 34: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: vt:Wqzxx@@
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups
A subgroup \(H\) of a group \(G\) is a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, inverses exist).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups
A subgroup \(H\) of a group \(G\) is a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, inverses exist).
Trivial subgroups: \(\{e\}, G\)
After
Front
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups
A subgroup \(H\) of a group \(G\) is a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, includes neutral element, inverses exist).
Back
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups
A subgroup \(H\) of a group \(G\) is a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, includes neutral element, inverses exist).
Trivial subgroups: \(\{e\}, G\)
Field-by-field Comparison
| Field |
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| Text |
A subgroup \(H\) of a group \(G\) is {{c1::a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, <b>inverses</b> exist).}} |
A subgroup \(H\) of a group \(G\) is {{c1::a subset \(H \subseteq G\) which is a group in itself (closed with respect to all operations, includes neutral element, <b>inverses</b> exist).}} |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups
Note 35: 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)
- 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
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)
- 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
Field-by-field Comparison
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| 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:: Implications}}</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:: Implication}}</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 36: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: wY#5P^[
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains
State Lemma 5.20 about division in integral domains: (The quotient has what property?)
Back
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains
State Lemma 5.20 about division in integral domains: (The quotient has what property?)
Lemma 5.20: In an integral domain, if \(a | b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is unique and is denoted by \(c = b/a\) (the quotient).
Explanation: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\), so \(b = ac\)0.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains
State Lemma 5.20 about division in integral domains: (The quotient has what property?)
Back
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains
State Lemma 5.20 about division in integral domains: (The quotient has what property?)
Lemma 5.20: In an integral domain, if \(a | b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is unique and is denoted by \(c = b/a\) (the quotient).
Explanation: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).
Field-by-field Comparison
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<p><strong>Lemma 5.20</strong>: In an integral domain, if \(a | b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is <strong>unique</strong> and is denoted by \(c = b/a\) (the quotient).</p>
<p><strong>Explanation</strong>: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\), so \(b = ac\)0.</p> |
<p><strong>Lemma 5.20</strong>: In an integral domain, if \(a | b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is <strong>unique</strong> and is denoted by \(c = b/a\) (the quotient).</p>
<p><strong>Explanation</strong>: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).</p> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains
Note 37: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: xH`d$W-97_
modified
Before
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group:
\(\widehat{(\widehat{a})} =\) \(a\)
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a group:
\(\widehat{(\widehat{a})} =\) \(a\)
Inverse of inverse is the original element.
This is a property from Lemma 5.3.
After
Front
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a Group:
\(\widehat{(\widehat{a})} =\) \(a\) (inverse of inverse is the original element). This is a property from Lemma 5.3.
Back
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
In a Group:
\(\widehat{(\widehat{a})} =\) \(a\) (inverse of inverse is the original element). This is a property from Lemma 5.3.
Field-by-field Comparison
| Field |
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| Text |
In a group:
<br><br>\(\widehat{(\widehat{a})} =\){{c1:: \(a\) }} |
<p>In a Group:<br>
\(\widehat{(\widehat{a})} =\){{c1:: \(a\) }} {{c1:: (inverse of inverse is the original element)}}. This is a property from Lemma 5.3.</p> |
| Extra |
Inverse of inverse is the original element.<br><br>This is a property from Lemma 5.3. |
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Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 38: 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 \(b \in A\)
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.
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 \(b \in A\)
Field-by-field Comparison
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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. |
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Tags:
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets
Note 39: 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:
- 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::\(a^{-1}\)}}.
After
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::\(a^{-1}\)}}.
Field-by-field Comparison
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<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> |
<p>Inverse in a group:</p><ul><li><b>Addition </b>{{c1::\(-a\)}}</li><li><b>Multiplication </b>{{c2::\(a^{-1}\)}}.</li></ul> |
Tags:
ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
Note 40: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: Dd-0>Kd049
modified
Before
Front
ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication
The columns of \(A\) are independent if and only if \(x = 0\) is the only vector for which \(Ax = 0\).
Back
ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication
The columns of \(A\) are independent if and only if \(x = 0\) is the only vector for which \(Ax = 0\).
After
Front
ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication
The columns of \(A\) are independent if and only if \(x = 0\) is the only vector for which \(Ax = 0\) (Linear combination view).
Back
ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication
The columns of \(A\) are independent if and only if \(x = 0\) is the only vector for which \(Ax = 0\) (Linear combination view).
Field-by-field Comparison
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The columns of \(A\) are independent if and only if {{c1::\(x = 0\) is the only vector for which \(Ax = 0\)::(Linear combination view)}}. |
The columns of \(A\) are independent if and only if {{c1::\(x = 0\) is the only vector for which \(Ax = 0\)}} (Linear combination view). |
Tags:
ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication
Note 41: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: col^b$YzMt
modified
Before
Front
ETH::1._Semester::LinAlg::1._Vectors::2._Scalar_products,_lengths_and_angles::2._Euclidean_norm
\(v^\top v = \) \(||v||^2\)
Back
ETH::1._Semester::LinAlg::1._Vectors::2._Scalar_products,_lengths_and_angles::2._Euclidean_norm
\(v^\top v = \) \(||v||^2\)
as \(||v|| = \sqrt{v^\top v}\).
After
Front
ETH::1._Semester::LinAlg::1._Vectors::2._Scalar_products,_lengths_and_angles::2._Euclidean_norm
\(v^\top v = \) \(||v||^2\) (in terms of norm)
Back
ETH::1._Semester::LinAlg::1._Vectors::2._Scalar_products,_lengths_and_angles::2._Euclidean_norm
\(v^\top v = \) \(||v||^2\) (in terms of norm)
as \(||v|| = \sqrt{v^\top v}\).
Field-by-field Comparison
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\(v^\top v = \){{c1:: \(||v||^2\)::(in terms of norm)}} |
\(v^\top v = \){{c1:: \(||v||^2\)}} (in terms of norm) |
Tags:
ETH::1._Semester::LinAlg::1._Vectors::2._Scalar_products,_lengths_and_angles::2._Euclidean_norm
Note 42: 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 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 \(T(X)\)?
\(T(0) = 0\)
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 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 \(T(X)\): \(T(0) = 0\)
Field-by-field Comparison
| Field |
Before |
After |
| Front |
What is a property that always hold for a linear transformation \(T(X)\)? |
What is a property that always hold for linear transformations? |
| Back |
\(T(0) = 0\) |
for a linear transformation \(T(X)\): \(T(0) = 0\) |
Tags:
ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals
Note 43: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: gU%jisb2z3
modified
Before
Front
ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties
Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:
- \(v \cdot w = w \cdot v\) (symmetry / commutativity)
- \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
- \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
- \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness)
Back
ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties
Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:
- \(v \cdot w = w \cdot v\) (symmetry / commutativity)
- \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
- \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
- \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness)
After
Front
ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties: \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar.
- \(v \cdot w = w \cdot v\) (symmetry / commutatitivity
- \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
- \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
- \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness
Back
ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties: \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar.
- \(v \cdot w = w \cdot v\) (symmetry / commutatitivity
- \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
- \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
- \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness
Field-by-field Comparison
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Scalar product properties<br><br>Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:<br><ol><li>{{c1::\(v \cdot w = w \cdot v\) (symmetry / commutativity)}}</li><li>{{c2:: \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)}}</li><li>{{c3:: \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)}}</li><li>{{c4:: \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness)}}</li></ol> |
Scalar product properties: \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar.<br><ol><li>{{c1::\(v \cdot w = w \cdot v\) (symmetry / commutatitivity}}</li><li>{{c2:: \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)}}</li><li>{{c3:: \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)}}</li><li>{{c4:: \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness}}</li></ol> |
Tags:
ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Note 44: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Q{f*UJkPmo
deleted
Deleted Note
Front
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::5._Composition_of_Relations ETH::1._Semester::DiskMat::Exams::3._Relations::FS24
Relation and function composition is associative.
Back
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::5._Composition_of_Relations ETH::1._Semester::DiskMat::Exams::3._Relations::FS24
Relation and function composition is associative.
This is important as \((\rho \circ \tau) \circ (\rho \circ \tau) = \rho \circ (\tau \circ \rho) \circ \tau\) is really useful in some exercises.
Current
Note has been deleted
Field-by-field Comparison
| Field |
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Relation and function composition is {{c1::associative}}. |
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| Extra |
This is important as \((\rho \circ \tau) \circ (\rho \circ \tau) = \rho \circ (\tau \circ \rho) \circ \tau\) is really useful in some exercises. |
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Tags:
ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::5._Composition_of_Relations
ETH::1._Semester::DiskMat::Exams::3._Relations::FS24
Note 45: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: yAP=DE#~t<
deleted
Deleted Note
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
Current
Note has been deleted
Field-by-field Comparison
| Field |
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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 46: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: Ie1sVs`1ap
deleted
Deleted Note
Front
ETH::1._Semester::LinAlg::7._The_determinant::1._2x2_Matrices
Give me the determinant of \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\):
Back
ETH::1._Semester::LinAlg::7._The_determinant::1._2x2_Matrices
Give me the determinant of \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\):
\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}\]
Current
Note has been deleted
Field-by-field Comparison
| Field |
Before |
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| Front |
Give me the determinant of \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\): |
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| Back |
\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}\] |
|
Tags:
ETH::1._Semester::LinAlg::7._The_determinant::1._2x2_Matrices