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Cardinal numbers and BFS/DFS.

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Joshua Potter 2024-12-24 16:57:28 -07:00
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"Basic": [

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title: "2024-12-24"
---
- [ ] Anki Flashcards
- [ ] KoL
- [x] Anki Flashcards
- [x] KoL
- [ ] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
- [ ] Korean (Read 1 Story)
* Basic identities regarding cardinal numbers.
* Additional notes on BFS and DFS as well as DFS edge classifications.

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@ -1,7 +1,7 @@
---
title: Breadth-First Search
TARGET DECK: Obsidian::STEM
FILE TAGS: algorithm data_structure::graph
FILE TAGS: algorithm::bfs data_structure::graph
tags:
- bfs
- graph
@ -13,6 +13,8 @@ Bread-first search operates on a graph $G = \langle V, E \rangle$ and a **source
![[bfs.gif]]
To keep track of progress, BFS colors each vertex white, gray, or black. All vertices start out white. They are colored gray upon discovery. They are painted black once all edges have been explored.
%%ANKI
Basic
What is BFS an acronym for?
@ -28,13 +30,6 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1727042295717-->
END%%
%%ANKI
Cloze
The {1:source} of breadth-first {2:search} is the {2:root} of the breadth-first {1:tree}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727042295723-->
END%%
%%ANKI
Basic
Which of undirected and directed graphs is BFS applicable to?
@ -43,38 +38,6 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1727042295728-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, what does the predecessor of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727042295733-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, what does the parent of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727042295739-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, the predecessor of a node is also known as what?
Back: The parent of the node.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727044035958-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, the parent of a node is also known as what?
Back: The predecessor of the node.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727044035963-->
END%%
%%ANKI
Basic
What ADT is typically used to manage the set of most recently discovered BFS vertices?
@ -97,12 +60,13 @@ What basic graph algorithm is the following a demonstration of?
![[bfs.gif]]
Back: Breadth-first search.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: algorithm::dfs
<!--ID: 1727044035969-->
END%%
%%ANKI
Basic
In BFS, what happens to the nodes found within the internal queue?
What happens to nodes found within the internal queue?
Back: Undiscovered nodes adjacent to those in the queue are enqueued.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727044035972-->
@ -113,6 +77,7 @@ Basic
Which of BFS or DFS is used to find shortest paths?
Back: BFS.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: algorithm::dfs
<!--ID: 1727044035975-->
END%%
@ -180,6 +145,114 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1727044184066-->
END%%
%%ANKI
Basic
What does a white vertex typically represent?
Back: A vertex that has not been discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
When is a white vertex painted gray?
Back: Upon discovery.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
When is a white vertex painted black?
Back: N/A. It must be painted gray before it's painted black.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What does a gray vertex typically represent?
Back: A vertex that is in the queue, i.e. the frontier discovery happens against.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
When is a gray vertex painted white?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
When is a gray vertex painted black?
Back: After all of its edges have been examined.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What does a black vertex typically represent?
Back: A vertex whose edges have all been explored.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
## Breadth-First Forests
To color an entire graph black, BFS may need to be invoked multiple times. After each invocation of BFS, a new invocation can be run with any remaining white vertex as the source. Each invocation yields a **breadth-first tree**. Multiple invocations yield a **breadth-first forest**.
%%ANKI
Basic
When might white vertices remain after BFS is invoked?
Back: When there exist vertices unreachable from the last used source vertex.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276213-->
END%%
%%ANKI
Basic
How many invocations of BFS are required to color a graph black?
Back: One or more.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276214-->
END%%
%%ANKI
Cloze
The {1:source} of a breadth-first {2:search} is the {2:root} of a breadth-first {1:tree}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727042295723-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, what does the predecessor of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727042295733-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, what does the parent of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727042295739-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, the predecessor of a node is also known as what?
Back: The parent of the node.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727044035958-->
END%%
%%ANKI
Basic
With respect to breadth-first trees, the parent of a node is also known as what?
Back: The predecessor of the node.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1727044035963-->
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

244
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@ -1,7 +1,7 @@
---
title: Depth-First Search
TARGET DECK: Obsidian::STEM
FILE TAGS: algorithm data_structure::graph
FILE TAGS: algorithm::dfs data_structure::graph
tags:
- dfs
- graph
@ -13,6 +13,10 @@ Depth-first search operates on a graph $G = \langle V, E \rangle$ and a **source
![[dfs.gif]]
To keep track of progress, DFS colors each vertex white, gray, or black. All vertices start out white. They are colored gray upon discovery. They are painted black once all edges have been explored.
Vertices also typically have two timestamps recorded: on discovery and on finish.
%%ANKI
Basic
What is DFS an acronym for?
@ -36,14 +40,6 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1729641729231-->
END%%
%%ANKI
Basic
With respect to depth-first trees, what does the predecessor of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1729641729235-->
END%%
%%ANKI
Basic
What ADT is typically used to manage the set of most recently discovered DFS vertices?
@ -56,6 +52,7 @@ END%%
Cloze
A {1:queue} is to {2:BFS} whereas a {2:stack} is to {1:DFS}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: algorithm::bfs
<!--ID: 1729641729242-->
END%%
@ -73,6 +70,7 @@ What basic graph algorithm is the following a demonstration of?
![[dfs.gif]]
Back: Depth-first search.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: algorithm::bfs
<!--ID: 1729641729249-->
END%%
@ -124,6 +122,234 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1729641729272-->
END%%
%%ANKI
Basic
What does a white vertex typically represent?
Back: A vertex that has not been discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276185-->
END%%
%%ANKI
Basic
When is a white vertex painted gray?
Back: Upon discovery.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276187-->
END%%
%%ANKI
Basic
When is a white vertex painted black?
Back: N/A. It must be painted gray before it's painted black.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276188-->
END%%
%%ANKI
Basic
What does a gray vertex typically represent?
Back: A vertex that is in the queue, i.e. the frontier discovery happens against.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276189-->
END%%
%%ANKI
Basic
When is a gray vertex painted white?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276191-->
END%%
%%ANKI
Basic
When is a gray vertex painted black?
Back: After all of its edges have been examined.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276193-->
END%%
%%ANKI
Basic
What does a black vertex typically represent?
Back: A vertex whose edges have all been explored.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276195-->
END%%
%%ANKI
Basic
What two timestamps are associated with each vertex?
Back: A timestamp on discovery and a timestamp when finished processing.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276196-->
END%%
%%ANKI
Basic
What range of values can a timestamp $t$ take on?
Back: $1 \leq t \leq 2\lvert V \rvert$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276198-->
END%%
%%ANKI
Basic
Suppose vertex $v$ has discovery time $d$ and finish time $f$. When was $v$ colored white?
Back: At timestamps $< f$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276199-->
END%%
%%ANKI
Basic
Suppose vertex $v$ has discovery time $d$ and finish time $f$. When was $v$ colored gray?
Back: At timestamps $\geq d$ and $< f$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276200-->
END%%
%%ANKI
Basic
Suppose vertex $v$ has discovery time $d$ and finish time $t$. When was $v$ colored black?
Back: At timestamps $\geq f$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276201-->
END%%
%%ANKI
Basic
What aspect of DFS has parenthesis structure?
Back: The discovery and finish timestamps of vertices.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276202-->
END%%
## Depth-First Forests
To color an entire graph black, BFS may need to be invoked multiple times. After each invocation of BFS, a new invocation can be run with any remaining white vertex as the source. Each invocation yields a **depth-first tree**. Multiple invocations yield a **depth-first forest**.
%%ANKI
Basic
With respect to depth-first trees, what does the predecessor of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1729641729235-->
END%%
%%ANKI
Basic
When might white vertices remain after DFS is invoked?
Back: When there exist vertices unreachable from the last used source vertex.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276203-->
END%%
%%ANKI
Basic
How many invocations of DFS are required to color a graph black?
Back: One or more.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276205-->
END%%
%%ANKI
Cloze
The {1:source} of a depth-first {2:search} is the {2:root} of a depth-first {1:tree}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276206-->
END%%
%%ANKI
Basic
With respect to depth-first trees, what does the predecessor of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
With respect to depth-first trees, what does the parent of a node $N$ refer to?
Back: The node from which $N$ was discovered.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276207-->
END%%
%%ANKI
Basic
With respect to depth-first trees, the predecessor of a node is also known as what?
Back: The parent of the node.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276209-->
END%%
%%ANKI
Basic
With respect to depth-first trees, the parent of a node is also known as what?
Back: The predecessor of the node.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1735081276210-->
END%%
### Edge Classification
A depth-first forest can contain four different types of edges:
1. A **tree edge** is an edge $\langle u, v \rangle$ such that $v$ was first discovered by exploring edge $\langle u, v \rangle$.
2. A **back edge** is an edge $\langle u, v \rangle$ connecting vertex $u$ to an ancestor $v$.
1. Self-loops are considered back edges.
3. A **forward edge** is a non-tree edge $\langle u, v \rangle$ connecting vertex $u$ to a proper descendant $v$.
4. A **cross edge** is any other edge.
%%ANKI
Basic
How many edge types might there exist in a depth-first forest?
Back: Four.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What are the different edge types possibly found in a depth-first forest?
Back: Tree, forward, back, and cross.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Let $\langle u, v \rangle$ be an edge in a depth-first forest. When is $\langle u, v \rangle$ a tree edge?
Back: When $v$ was first discovered along edge $\langle u, v \rangle$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Let $\langle u, v \rangle$ be an edge in a depth-first forest. When is $\langle u, v \rangle$ a back edge?
Back: When $v$ is an ancestor of edge $u$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Let $\langle u, v \rangle$ be an edge in a depth-first forest. When is $\langle u, v \rangle$ a forward edge?
Back: When $\langle u, v \rangle$ is a non-tree edge such that $v$ is a proper descendant of $u$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Let $\langle u, v \rangle$ be an edge in a depth-first forest. When is $\langle u, v \rangle$ a cross edge?
Back: When $\langle u, v \rangle$ is not a tree, forward, or back edge.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What kind of edge is a self-loop in a depth-first forest classified as?
Back: A back edge.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

157
notes/set/cardinality.md
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@ -703,6 +703,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1733675522751-->
END%%
%%ANKI
Basic
Suppose sets $A$ and $B$ are finite. When is $^BA$ infinite?
Back: N/A. The set of functions from one finite set to another is always finite.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143693-->
END%%
### Addition
Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa + \lambda = \mathop{\text{card}}(K \cup L)$, where $K$ and $L$ are any disjoint sets of cardinality $\kappa$ and $\lambda$, respectively.
@ -818,6 +826,38 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1734520487333-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. Does $\kappa + \lambda = \lambda + \kappa$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143694-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. *Why* does $\kappa + \lambda = \lambda + \kappa$?
Back: Because the union of sets is commutative.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143695-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. Does $\kappa + (\lambda + \mu) = (\kappa + \lambda) + \mu$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143696-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. *Why* does $\kappa + (\lambda + \mu) = (\kappa + \lambda) + \mu$?
Back: Because the union of sets is associative.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143697-->
END%%
### Multiplication
Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa \cdot \lambda = \mathop{\text{card}}(K \times L)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively.
@ -956,6 +996,62 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. Does $\kappa \cdot \lambda = \lambda \cdot \kappa$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143698-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. *Why* does $\kappa \cdot \lambda = \lambda \cdot \kappa$?
Back: Because $K \times L \approx L \times K$ for any sets $K$ and $L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143699-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. Does $\kappa \cdot (\lambda \cdot \mu) = (\kappa \cdot \lambda) \cdot \mu$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143700-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. *Why* does $\kappa \cdot (\lambda \cdot \mu) = (\kappa \cdot \lambda) \cdot \mu$?
Back: Because $K \times (L \times M) \approx (K \times L) \times M$ for any sets $K$, $L$, and $M$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143701-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. What does the distributive property state?
Back: $\kappa \cdot (\lambda + \mu) = (\kappa \cdot \lambda) + (\kappa \cdot \mu)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143702-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. *Why* does $\kappa \cdot (\lambda + \mu) = (\kappa \cdot \lambda) + (\kappa \cdot \mu)$?
Back: Because the Cartesian product distributes over the union operation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143703-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. What does $\kappa \cdot (\lambda + 1)$ evaluate to?
Back: $\kappa \cdot \lambda + \kappa$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143704-->
END%%
### Exponentiation
Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa^\lambda = \mathop{\text{card}}(^LK)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively.
@ -1079,6 +1175,67 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1734520487388-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. Does $\kappa ^ \lambda = \lambda ^ \kappa$?
Back: Not necessarily.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143705-->
END%%
%%ANKI
Cloze
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. Then $\kappa^{\lambda + \mu} =$ {$\kappa^\lambda \cdot \kappa^\mu$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143706-->
END%%
%%ANKI
Cloze
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. Then $(\kappa \cdot \lambda)^\mu =$ {$\kappa^\mu \cdot \lambda^\mu$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143707-->
END%%
%%ANKI
Cloze
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. Then $(\kappa^\lambda)^\mu =$ {$\kappa^{\lambda \cdot \mu}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143708-->
END%%
%%ANKI
Basic
Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. What $\lambda$-calculus concept does $(\kappa^\lambda)^\mu = \kappa^{\lambda \cdot \mu}$ embody?
Back: Currying.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143709-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. Rewrite $\kappa^{\lambda + 1}$ without using addition.
Back: $\kappa^\lambda \cdot \kappa$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143710-->
END%%
%%ANKI
Basic
Let $\kappa$ be a cardinal number. How is the factorial of $\kappa$ denoted?
Back: $\kappa !$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143711-->
END%%
%%ANKI
Basic
Let $\kappa$ be a cardinal number. How is the factorial of $\kappa$ defined?
Back: As $\mathop{\text{card}} \{ f \mid f \text{ is a permutation of } K\}$ for some $\mathop{\text{card}} K = \kappa$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143712-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

8
notes/set/functions.md
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@ -252,6 +252,14 @@ Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [htt
<!--ID: 1729804914213-->
END%%
%%ANKI
Basic
How is a permutation defined using the concept of functions?
Back: A permutation of a set $A$ is a one-to-one function from $A$ to $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735074143690-->
END%%
## Injections
A function is **injective** or **one-to-one** if each element of the codomain is mapped to by at most one element of the domain.