--- title: Functions TARGET DECK: Obsidian::STEM FILE TAGS: set::function tags: - function - set --- ## Overview 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 $\mathop{\text{ran}}F \subseteq B$. %%ANKI Basic Which of relations or functions is the more general concept? Back: Relations. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What *is* a function? Back: A relation $F$ such that for each $x \in \mathop{\text{dom}}F$, there exists a unique $y$ such that $xFy$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For function $F$ and $x \in \mathop{\text{dom}}F$, what name is given to $F(x)$? Back: The value of $F$ at $x$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Who introduced the function notation $F(x)$? Back: Leonhard Euler. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be a function and $\langle x, y \rangle \in F$. Rewrite the membership as an expression excluding $y$. Back: $\langle x, F(x) \rangle \in F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be a function and $\langle x, y \rangle \in F$. Rewrite the membership as an expression excluding $x$. Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider notation $F(x)$. What assumption is $F$ assumed to satisfy? Back: It is assumed to be a function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider notation $F(x)$. What assumption is $x$ assumed to satisfy? Back: It is assumed to be in the domain of $F$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze A function is a {single-valued} relation. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $F \colon A \rightarrow B$ pronounced? Back: $F$ maps $A$ into $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What three conditions hold iff $F$ maps $A$ into $B$? Back: $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F \subseteq B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. What term is used to refer to $A$? Back: The domain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. What term is used to refer to $B$? Back: The codomain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does the range of a function compare to its codomain? Back: The range is a subset of the codomain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Injections A function is **injective** or **one-to-one** if each element of the codomain is mapped to by at most one element of the domain. %%ANKI Basic What does it mean for a function to be injective? Back: Each element of the codomain is mapped to by at most one element of the domain. 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). END%% %%ANKI Basic What does it mean for a function to be one-to-one? Back: Each element of the codomain is mapped to by at most one element of the domain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Each element of an injection's codomain is mapped to by how many elements of the domain? Back: At most 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). END%% %%ANKI Basic Suppose `Function.Injective f` for $f \colon A \rightarrow B$. What predicate logical formula describes $f$? Back: $\forall a_1, a_2 \in A, (f(a_1) = f(a_2) \Rightarrow a_1 = a_2$) 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). Tags: lean logic::predicate END%% %%ANKI Basic Does the following depict an injection? ![[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). END%% %%ANKI Basic Does the following depict a one-to-one function? ![[function-injective.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). END%% %%ANKI Basic Does the following depict a one-to-one function? ![[function-surjective.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). END%% %%ANKI Basic *Why* isn't the following an injection? ![[function-general.png]] Back: Both $1 \mapsto d$ and $2 \mapsto d$. 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). END%% %%ANKI Basic Is a single-valued set a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is a single-valued relation a function? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is a single-rooted set a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is a single-rooted relation a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {One-to-one} is to functions whereas {single-rooted} is to relations. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is a one-to-one function a single-rooted relation? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is a single-rooted relation a one-to-one function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is a single-rooted function a one-to-one function? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Surjections 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 $\mathop{\text{ran}}F = B$. %%ANKI Basic What does it mean for function to be surjective? Back: Each element of the codomain is mapped to by at least one element of the domain. 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). END%% %%ANKI Basic What does it mean for a function to be onto? Back: Each element of the codomain is mapped to by at least one element of the domain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Each element of a surjection's codomain is mapped to by how many elements of the domain? Back: At least 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). END%% %%ANKI Basic Suppose `Function.Surjective f` for $f \colon A \rightarrow B$. What predicate logical formula describes $f$? Back: $\forall b \in B, \exists a \in A, f(a) = b$ 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). Tags: lean logic::predicate END%% %%ANKI Cloze {1:Injective} is to {2:one-to-one} as {2:surjective} is to {1:onto}. 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). END%% %%ANKI Basic What three conditions hold iff $F$ maps $A$ onto $B$? Back: $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F = B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ map $A$ into $B$. Does $F$ map $A$ onto $B$? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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).