682 lines
23 KiB
Markdown
682 lines
23 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, $\mathop{\text{dom}}A$, 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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## 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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## 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
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Let $F$ map $A$ onto $B$. Does $F$ map $A$ into $B$?
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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: 1718427443415-->
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END%%
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%%ANKI
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Cloze
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Let $F$ be a function. Then $F$ maps {$\mathop{\text{dom} }F$} onto {$\mathop{\text{ran} }F$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443419-->
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END%%
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%%ANKI
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Basic
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Does the following depict a surjection?
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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: 1718465870552-->
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END%%
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%%ANKI
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Basic
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Does the following depict an onto function?
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![[function-injective.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: 1718465870558-->
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END%%
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%%ANKI
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Basic
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Does the following depict an onto function?
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![[function-surjective.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: 1718465870565-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the following a surjection?
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![[function-general.png]]
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Back: No element of $X$ maps to $a$ or $b$.
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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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<!--ID: 1718465870573-->
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END%%
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## Bijections
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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.
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%%ANKI
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Basic
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What does it mean for a function to be bijective?
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Back: It is both injective and surjective.
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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: 1718464728903-->
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END%%
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%%ANKI
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Basic
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Each element of a bijection's codomain is mapped to by how many elements of the domain?
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Back: Exactly 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: 1718464728907-->
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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:bijective} is to {1:one-to-one correspondence}.
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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: 1718464728899-->
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END%%
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%%ANKI
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Cloze
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{1:Surjective} is to {2:onto} as {2:bijective} is to {1:one-to-one correspondence}.
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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: 1718465870579-->
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END%%
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%%ANKI
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Basic
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Does the following depict a bijection?
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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: 1718465870585-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the following a one-to-one correspondence?
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![[function-injective.png]]
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Back: The function does not map onto $Y$.
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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: 1718465870592-->
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END%%
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%%ANKI
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Basic
|
|
*Why* isn't the following a one-to-one correspondence?
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|
![[function-surjective.png]]
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Back: The function is not one-to-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: 1718465870599-->
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END%%
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%%ANKI
|
|
Basic
|
|
What distinguishes a one-to-one function from a one-to-one correspondence?
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Back: The former is not necessarily surjective.
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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: 1718465870605-->
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END%%
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## Inverses
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|
|
Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}.$$
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%%ANKI
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Basic
|
|
What kind of mathematical object does the inverse operation apply to?
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|
Back: Sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719016770704-->
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END%%
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%%ANKI
|
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Basic
|
|
What is the "arity" of the inverse operation in set theory?
|
|
Back: $1$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719017251246-->
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END%%
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%%ANKI
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|
Basic
|
|
Let $F$ be a set. How is the inverse of $F$ denoted?
|
|
Back: $F^{-1}$
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719016770741-->
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END%%
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%%ANKI
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|
Basic
|
|
What kind of mathematical object does the inverse operation emit?
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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: 1719016770749-->
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END%%
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%%ANKI
|
|
Basic
|
|
How is the inverse of set $F$ defined in set-builder notation?
|
|
Back: $F^{-1} = \{\langle u, v \rangle \mid vFu\}$\
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1719016770752-->
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END%%
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%%ANKI
|
|
Basic
|
|
Consider set $A$. Is $A^{-1}$ a relation?
|
|
Back: Yes.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1719016770755-->
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END%%
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%%ANKI
|
|
Basic
|
|
Consider set $A$. Is $A^{-1}$ a function?
|
|
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: 1719016770759-->
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END%%
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%%ANKI
|
|
Basic
|
|
Consider relation $R$. Is $R^{-1}$ a relation?
|
|
Back: Yes.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1719016770763-->
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END%%
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|
%%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-->
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END%%
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%%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-->
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|
END%%
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|
%%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-->
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|
END%%
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|
%%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-->
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|
END%%
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|
%%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%%
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|
%%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%%
|
|
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|
%%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%%
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|
|
|
%%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%%
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|
%%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%%
|
|
|
|
## 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%%
|
|
|
|
## 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). |