444 lines
17 KiB
Markdown
444 lines
17 KiB
Markdown
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---
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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-->
|
||
|
|
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 $b \in Y$.
|
||
|
|
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: 1718465870573-->
|
||
|
|
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%%
|
||
|
|
|
||
|
|
## 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).
|