1272 lines
41 KiB
Markdown
1272 lines
41 KiB
Markdown
---
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title: Functions
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::function
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tags:
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- function
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- set
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---
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## Overview
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A **function** $F$ is a single-valued [[relations|relation]]. We say $F$ **maps $A$ into $B$**, denoted $F \colon A \rightarrow B$, if and only if $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F \subseteq B$.
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%%ANKI
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Basic
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Which of relations or functions is the more general concept?
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Back: Relations.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443345-->
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END%%
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%%ANKI
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Basic
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What *is* a function?
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Back: A relation $F$ such that for each $x \in \mathop{\text{dom}}F$, there exists a unique $y$ such that $xFy$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443366-->
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END%%
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%%ANKI
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Basic
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For function $F$ and $x \in \mathop{\text{dom}}F$, what name is given to $F(x)$?
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Back: The value of $F$ at $x$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443370-->
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END%%
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%%ANKI
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Basic
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Who introduced the function notation $F(x)$?
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Back: Leonhard Euler.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443374-->
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END%%
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%%ANKI
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Basic
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Let $F$ be a function and $\langle x, y \rangle \in F$. Rewrite the membership as an expression excluding $y$.
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Back: $\langle x, F(x) \rangle \in F$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443379-->
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END%%
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%%ANKI
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Basic
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Let $F$ be a function and $\langle x, y \rangle \in F$. Rewrite the membership as an expression excluding $x$.
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443384-->
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END%%
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%%ANKI
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Basic
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Consider notation $F(x)$. What assumption is $F$ assumed to satisfy?
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Back: It is assumed to be a function.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443388-->
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END%%
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%%ANKI
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Basic
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Consider notation $F(x)$. What assumption is $x$ assumed to satisfy?
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Back: It is assumed to be in the domain of $F$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443393-->
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END%%
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%%ANKI
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Cloze
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A function is a {single-valued} relation.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443397-->
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END%%
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%%ANKI
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Basic
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How is $F \colon A \rightarrow B$ pronounced?
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Back: $F$ maps $A$ into $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443401-->
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END%%
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%%ANKI
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Basic
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What three conditions hold iff $F$ maps $A$ into $B$?
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Back: $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F \subseteq B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443404-->
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END%%
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%%ANKI
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Basic
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Consider function $F \colon A \rightarrow B$. What term is used to refer to $A$?
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Back: The domain.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718464126872-->
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END%%
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%%ANKI
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Basic
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Consider function $F \colon A \rightarrow B$. What term is used to refer to $B$?
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Back: The codomain.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718464126879-->
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END%%
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%%ANKI
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Basic
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How does the range of a function compare to its codomain?
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Back: The range is a subset of the codomain.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718464126883-->
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END%%
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%%ANKI
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Basic
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Is $\varnothing$ a function?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913529-->
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END%%
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## Injections
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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.
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%%ANKI
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Basic
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What does it mean for a function to be injective?
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Back: Each element of the codomain is mapped to by at most one element of the domain.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718464126887-->
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END%%
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%%ANKI
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Basic
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What does it mean for a function to be one-to-one?
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Back: Each element of the codomain is mapped to by at most one element of the domain.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870487-->
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END%%
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%%ANKI
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Basic
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Each element of an injection's codomain is mapped to by how many elements of the domain?
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Back: At most one.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718464498595-->
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END%%
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%%ANKI
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Basic
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Suppose `Function.Injective f` for $f \colon A \rightarrow B$. What predicate logical formula describes $f$?
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Back: $\forall a_1, a_2 \in A, (f(a_1) = f(a_2) \Rightarrow a_1 = a_2$)
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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Tags: lean logic::predicate
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<!--ID: 1718464498603-->
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END%%
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%%ANKI
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Basic
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Does the following depict an injection?
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![[function-bijective.png]]
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Back: Yes.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718465870490-->
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END%%
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%%ANKI
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Basic
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Does the following depict a one-to-one function?
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![[function-injective.png]]
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Back: Yes.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718465870493-->
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END%%
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%%ANKI
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Basic
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Does the following depict a one-to-one function?
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![[function-surjective.png]]
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Back: No.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718465870497-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the following an injection?
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![[function-general.png]]
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Back: Both $1 \mapsto d$ and $2 \mapsto d$.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718465870505-->
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END%%
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%%ANKI
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Basic
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Is a single-valued set a function?
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Back: Not necessarily.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443358-->
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END%%
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%%ANKI
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Basic
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Is a single-valued relation a function?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443362-->
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END%%
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%%ANKI
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Basic
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Is a single-rooted set a function?
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Back: Not necessarily.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870509-->
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END%%
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%%ANKI
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Basic
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Is a single-rooted relation a function?
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Back: Not necessarily.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870519-->
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END%%
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%%ANKI
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Cloze
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{One-to-one} is to functions whereas {single-rooted} is to relations.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870525-->
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END%%
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%%ANKI
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Basic
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Is a one-to-one function a single-rooted relation?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870531-->
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END%%
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%%ANKI
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Basic
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Is a single-rooted relation a one-to-one function?
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Back: Not necessarily.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870536-->
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END%%
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%%ANKI
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Basic
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Is a single-rooted function a one-to-one function?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870541-->
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END%%
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### Left Inverses
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Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a **left inverse**) such that $G \circ F = I_A$ if and only if $F$ is one-to-one.
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%%ANKI
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Basic
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What is the most specific mathematical object that describes a left inverse?
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Back: A function.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719683931406-->
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END%%
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%%ANKI
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Basic
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How is a left inverse of $F \colon A \rightarrow B$ defined?
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Back: As a function $G \colon B \rightarrow A$ such that $G \circ F = I_A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719684322548-->
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END%%
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%%ANKI
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Basic
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How is a left inverse of set $A$ defined?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719684322553-->
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END%%
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%%ANKI
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Basic
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Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its domain?
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Back: $B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719680660507-->
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END%%
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%%ANKI
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Basic
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What does $I_A$ usually denote?
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Back: The identity function on set $A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913532-->
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END%%
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%%ANKI
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Basic
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How is the identity function on set $B$ denoted?
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Back: $I_B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719683703723-->
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END%%
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%%ANKI
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Basic
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Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its codomain?
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Back: $A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719680660511-->
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END%%
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%%ANKI
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Basic
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Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more compactly write $G \circ F$?
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Back: $I_A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913534-->
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END%%
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%%ANKI
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Basic
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Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more compactly write $F \circ G$?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913535-->
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END%%
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%%ANKI
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Basic
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Let $F$ be a left inverse of function $G$. How do they interestingly compose?
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Back: As $F \circ G$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913538-->
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END%%
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%%ANKI
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Cloze
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Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:left} inverse iff $F$ is {3:one-to-one}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913540-->
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END%%
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%%ANKI
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Basic
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Does proving "left inverses iff injective" rely on AoC?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913542-->
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END%%
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%%ANKI
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Basic
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What are the hypotheses of "left inverses iff injective"?
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Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719683703726-->
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END%%
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%%ANKI
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Basic
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Let $F \colon A \rightarrow B$. *Why* does "left inverses iff injective" assume $A \neq \varnothing$?
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Back: Because a mapping from nonempty $B$ to $\varnothing$ cannot be a function.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719683703729-->
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END%%
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%%ANKI
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Basic
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Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "left inverses iff injective" require AoC?
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Back: It doesn't.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719683703730-->
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END%%
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## Surjections
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A function is **surjective** or **onto** if each element of the codomain is mapped to by at least one element of the domain. That is, **$F$ maps $A$ onto $B$** if and only if $F$ is a function, $\mathop{\text{dom}}A$, and $\mathop{\text{ran}}F = B$.
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%%ANKI
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Basic
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What does it mean for function to be surjective?
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Back: Each element of the codomain is mapped to by at least one element of the domain.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718464126891-->
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END%%
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%%ANKI
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Basic
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What does it mean for a function to be onto?
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Back: Each element of the codomain is mapped to by at least one element of the domain.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870546-->
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END%%
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%%ANKI
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Basic
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Each element of a surjection's codomain is mapped to by how many elements of the domain?
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Back: At least one.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718464498606-->
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END%%
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%%ANKI
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Basic
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Suppose `Function.Surjective f` for $f \colon A \rightarrow B$. What predicate logical formula describes $f$?
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Back: $\forall b \in B, \exists a \in A, f(a) = b$
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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Tags: lean logic::predicate
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<!--ID: 1718464498615-->
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END%%
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%%ANKI
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Cloze
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{1:Injective} is to {2:one-to-one} as {2:surjective} is to {1:onto}.
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Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
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<!--ID: 1718464126897-->
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END%%
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%%ANKI
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Basic
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What three conditions hold iff $F$ maps $A$ onto $B$?
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Back: $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F = B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443408-->
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END%%
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%%ANKI
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Basic
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Let $F$ map $A$ into $B$. Does $F$ map $A$ onto $B$?
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Back: Not necessarily.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443412-->
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END%%
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%%ANKI
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Basic
|
|
Let $F$ map $A$ onto $B$. Does $F$ map $A$ into $B$?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718427443415-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Let $F$ be a function. Then $F$ maps {$\mathop{\text{dom} }F$} onto {$\mathop{\text{ran} }F$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718427443419-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does the following depict a surjection?
|
|
![[function-bijective.png]]
|
|
Back: Yes.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870552-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does the following depict an onto function?
|
|
![[function-injective.png]]
|
|
Back: No.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870558-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does the following depict an onto function?
|
|
![[function-surjective.png]]
|
|
Back: Yes.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870565-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't the following a surjection?
|
|
![[function-general.png]]
|
|
Back: No element of $X$ maps to $a$ or $b$.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870573-->
|
|
END%%
|
|
|
|
### Right Inverses
|
|
|
|
Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a right inverse) such that $F \circ G = I_B$ if and only if $F$ maps $A$ onto $B$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the most specific mathematical object that describes a right inverse?
|
|
Back: A function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719683931410-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is a right inverse of $F \colon A \rightarrow B$ defined?
|
|
Back: As a function $G \colon B \rightarrow A$ such that $F \circ G = I_B$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719684322557-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is a right inverse of set $A$ defined?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719684322561-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
{1:Left} inverses are to {2:injections} whereas {2:right} inverses are to {1:surjections}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913533-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its domain?
|
|
Back: $B$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719680660514-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its codomain?
|
|
Back: $A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719680660513-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more compactly write $G \circ F$?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913536-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more compactly write $F \circ G$?
|
|
Back: $I_B$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913537-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ be a right inverse of function $G$. How do they interestingly compose?
|
|
Back: As $G \circ F$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913539-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:right} inverse iff $F$ is {3:onto $B$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913541-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does proving "right inverses iff surjective" rely on AoC?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913543-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What are the hypotheses of "right inverses iff surjective"?
|
|
Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719683703732-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$. *Why* does "right inverses iff surjective" assume $A \neq \varnothing$?
|
|
Back: Because a mapping from nonempty $B$ to $\varnothing$ cannot be a function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719683703734-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "right inverses iff surjective" require AoC?
|
|
Back: There is no other mechanism for choosing an $x \in A$ for *each* $y \in B$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719683703736-->
|
|
END%%
|
|
|
|
## Bijections
|
|
|
|
A function is **bijective** or a **one-to-one correspondence** if each element of the codomain is mapped to by exactly one element of the domain.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does it mean for a function to be bijective?
|
|
Back: It is both injective and surjective.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718464728903-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Each element of a bijection's codomain is mapped to by how many elements of the domain?
|
|
Back: Exactly one.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718464728907-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
{1:Injective} is to {2:one-to-one} as {2:bijective} is to {1:one-to-one correspondence}.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718464728899-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
{1:Surjective} is to {2:onto} as {2:bijective} is to {1:one-to-one correspondence}.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870579-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does the following depict a bijection?
|
|
![[function-bijective.png]]
|
|
Back: Yes.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870585-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't the following a one-to-one correspondence?
|
|
![[function-injective.png]]
|
|
Back: The function does not map onto $Y$.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870592-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't the following a one-to-one correspondence?
|
|
![[function-surjective.png]]
|
|
Back: The function is not one-to-one.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870599-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What distinguishes a one-to-one function from a one-to-one correspondence?
|
|
Back: The former is not necessarily surjective.
|
|
Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
<!--ID: 1718465870605-->
|
|
END%%
|
|
|
|
## Inverses
|
|
|
|
Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}$$
|
|
%%ANKI
|
|
Basic
|
|
What is the most specific mathematical object that describes an inverse?
|
|
Back: A relation.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719683931414-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the inverse operation apply to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770704-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the "arity" of the inverse operation in set theory?
|
|
Back: $1$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017251246-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ be a set. How is the inverse of $F$ denoted?
|
|
Back: $F^{-1}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770741-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the inverse operation emit?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770749-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the inverse of set $F$ defined in set-builder notation?
|
|
Back: $F^{-1} = \{\langle u, v \rangle \mid vFu\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770752-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider set $A$. Is $A^{-1}$ a relation?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770755-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider set $A$. Is $A^{-1}$ a function?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770759-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider relation $R$. Is $R^{-1}$ a relation?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770763-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider relation $R$. Is $R^{-1}$ a function?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770767-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$. Is $F^{-1}$ a relation?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770772-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$. Is $F^{-1}$ a function?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770778-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ a function?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770782-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ one-to-one?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770787-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ onto $A$?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770792-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$ be a surjection. Is $F^{-1}$ a function?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770796-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F \colon A \rightarrow B$ be a surjection. Is $F^{-1}$ a relation?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770800-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$. What is the domain of $F^{-1}$?
|
|
Back: $\mathop{\text{ran}}F$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770805-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$. What is the range of $F^{-1}$?
|
|
Back: $A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016770812-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F$. How does $(F^{-1})^{-1}$ relate to $F$?
|
|
Back: They are equal.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016946539-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider relation $R$. How does $(R^{-1})^{-1}$ relate to $R$?
|
|
Back: They are equal.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016946547-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider set $A$. How does $(A^{-1})^{-1}$ relate to $A$?
|
|
Back: $(A^{-1})^{-1}$ is a subset of $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719016946554-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
When does $A \neq (A^{-1})^{-1}$?
|
|
Back: If there exists an $x \in A$ such that $x$ is not an ordered pair.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017560113-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is set $\{\langle u, v \rangle \mid vAu\}$ more simply denoted?
|
|
Back: $A^{-1}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644290-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does $\varnothing^{-1}$ evalute to?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644293-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given set $F$, what does $\mathop{\text{dom}}F^{-1}$ evaluate to?
|
|
Back: $\mathop{\text{ran}}F$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719398756549-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given set $F$, what does $\mathop{\text{ran}}F^{-1}$ evaluate to?
|
|
Back: $\mathop{\text{dom}}F$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719398756554-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For any set $F$, {1:$F$} is {2:single-valued} iff {2:$F^{-1}$} is {1:single-rooted}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719398756558-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon \varnothing \rightarrow B$. What is $F^{-1}$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719681913530-->
|
|
END%%
|
|
|
|
## Compositions
|
|
|
|
Let $F$ and $G$ be arbitrary sets. The **composition** of $F$ and $G$ is the set $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the composition operation apply to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017251256-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the composition operation emit?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017251259-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ and $G$ be arbitrary sets. How is the composition of $G$ and $F$ denoted?
|
|
Back: $G \circ F$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017251252-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ and $G$ be arbitrary sets. How is the composition of $F$ and $G$ denoted?
|
|
Back: $F \circ G$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017251262-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the "arity" of the composition operation in set theory?
|
|
Back: $2$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017251265-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
{$(F \circ G)(x)$} is alternatively written as {$F(G(x))$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017560120-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the composition of sets $F$ and $G$ defined in set-builder notation?
|
|
Back: $F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719017560123-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is set $\{\langle u, v \rangle \mid \exists t, uBt \land tAv \}$ more simply denoted?
|
|
Back: $A \circ B$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644294-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ be an arbitrary set. What is $F \circ \varnothing$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644295-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ be an arbitrary set. What is $\varnothing \circ F$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644296-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Let $F$ be a {function}. If $t \in$ {$\mathop{\text{ran} }F$}, then $F(F^{-1}(t)) = t$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719398756562-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Let $F$ be an {injection}. If $t \in$ {$\mathop{\text{dom} }F$}, then $F^{-1}(F(t)) = t$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719398756565-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If $F$ is a relation and $G$ is a function, is $F \circ G$ a function?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719406791406-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If $F$ is a function and $G$ is a relation, is $F \circ G$ a function?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719406791410-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If $F$ is a function and $G$ is a function, is $F \circ G$ a function?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719406791413-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ and $G$ be functions. How is $\mathop{\text{dom}}(F \circ G)$ defined using set-builder notation?
|
|
Back: $\{x \in \mathop{\text{dom}}G \mid G(x) \in \mathop{\text{dom}}F\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719406791415-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For any sets $F$ and $G$, {$(F \circ G)^{-1}$} $=$ {$G^{-1} \circ F^{-1}$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719666552283-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How might you explain $(F \circ G)^{-1} = G^{-1} \circ F^{-1}$ in plain English?
|
|
Back: The opposite of applying $G$ then $F$ is to undo $F$ then $G$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719666552291-->
|
|
END%%
|
|
|
|
## Restrictions
|
|
|
|
Let $F$ and $A$ be arbitrary sets. The **restriction of $F$ to $A$** is the set $$F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the restriction operation apply to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644297-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
$F \restriction A$ is the restriction of $F$ {to} $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644298-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the restriction operation emit?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644299-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the "arity" of the restriction operation in set theory?
|
|
Back: $2$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644300-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the restriction of $F$ to $A$ denoted?
|
|
Back: $F \restriction A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644301-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the restriction of $F$ to $A$ defined?
|
|
Back: $F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644302-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction A)$?
|
|
Back: They are equal.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644303-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction A)$?
|
|
Back: They are equal.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644304-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction C)$?
|
|
Back: $\mathop{\text{dom}}(F \restriction C) \subseteq \mathop{\text{dom}}F$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644305-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $F \restriction A$ pronounced?
|
|
Back: The restriction of $F$ to $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644306-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction C)$?
|
|
Back: $\mathop{\text{ran}}(F \restriction C) \subseteq \mathop{\text{ran}}F$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644307-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is set $\{\langle u, v \rangle \mid uAv \land u \in B\}$ more simply denoted?
|
|
Back: $A \restriction B$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644308-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ be an arbitrary set. What is $F \restriction \varnothing$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644309-->
|
|
END%%
|
|
|
|
## Images
|
|
|
|
Let $F$ and $A$ be sets. Then the **image of $F$ under $A$** is $$F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the image operation apply to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644310-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object does the image operation emit?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644311-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
$F[\![A]\!]$ is the image of $F$ {under} $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644312-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the "arity" of the image operation in set theory?
|
|
Back: $2$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644313-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the image of $F$ under $A$ denoted?
|
|
Back: $F[\![A]\!]$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644314-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the image of $F$ under $A$ defined?
|
|
Back: $F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644315-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the image of $F$ under $A$ defined in terms of restrictions?
|
|
Back: $F[\![A]\!] = \mathop{\text{ran}}(F \restriction A)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644316-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is set $\{v \mid \exists u \in B, uAv\}$ more simply denoted?
|
|
Back: $A[\![B]\!]$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644317-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Enderton says "multiple-valued functions" are actually what?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644318-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Enderton says "multiple-valued functions" are actually what?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Enderton says "$F^{-1}(9) = \pm 3$" is preferably written in what way?
|
|
Back: $F^{-1}[\![\{9\}]\!] = \{-3, 3\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644319-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $F$ be an arbitrary set. What is $F[\![\varnothing]\!]$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1719103644321-->
|
|
END%%
|
|
|
|
## Bibliography
|
|
|
|
* “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
|
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |