--- 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, $\mathop{\text{dom}}F = A$, 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%% %%ANKI Basic Is $\varnothing$ a function? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$, $G$ be functions such that $F \subseteq G$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}G$? Back: $\mathop{\text{dom}}F \subseteq \mathop{\text{dom}}G$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$, $G$ be functions such that $F \subseteq G$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}G$? Back: $\mathop{\text{ran}}F \subseteq \mathop{\text{ran}}G$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$, $G$ be functions. Is $F \cap G$ a function? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$, $G$ be functions. When is $F \cap G$ a function? Back: Always. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$, $G$ be functions. Is $F \cup G$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$, $G$ be functions. When is $F \cup G$ a function? Back: Iff $f(x) = g(x)$ for every $x \in \mathop{\text{dom}}F \cap \mathop{\text{dom}}G$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% An **operation** on some set (say) $S$ is a function with "signature" $S \times \cdots \times S \rightarrow S$. More precisely, an $n$-ary operation on $S$ is a function $S^n \rightarrow S$ where $n \geq 0$. %%ANKI Basic Let $A$ and $B$ be disjoint sets. Is $f \colon A \rightarrow B$ a function, operation, both, or neither? Back: Function. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic Let $A \subseteq B$. Is $f \colon A \rightarrow B$ a function, operation, or both? Back: Both. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic Let $A$ and $B$ be disjoint sets. $f \colon A \rightarrow B$ is an operation on what set? Back: N/A. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic Let $A \subseteq B$. $f \colon A \rightarrow B$ is an operation on what set? Back: $B$. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic What is the arity of operation $f \colon A \rightarrow A$? Back: $1$ Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic What is the arity of operation $f \colon A \times A \rightarrow A$? Back: $2$ Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic What is the arity of operation $f \colon A \times \cdots \times A \rightarrow A$? Back: The number of terms in $A \times \cdots \times A$. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic Why is it incomplete to state function $f$ is an operation? Back: We have to ask what set $f$ is an operation on. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% %%ANKI Basic Which of operations or functions is the more general concept? Back: Functions. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). 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 formal-system::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%% ### Left Inverses Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a **left inverse**) such that $G \circ F = I_A$ if and only if $F$ is one-to-one. %%ANKI Basic What is the most specific mathematical object that describes a left inverse? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is a left inverse of $F \colon A \rightarrow B$ defined? Back: As a function $G \colon B \rightarrow A$ such that $G \circ F = I_A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is a left inverse of set $A$ defined? Back: N/A. Left inverses are only applicable to functions. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its domain? Back: $B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does $I_A$ usually denote? Back: The identity function on set $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the identity function on set $B$ denoted? Back: $I_B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its codomain? Back: $A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more compactly write $G \circ F$? Back: $I_A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more compactly write $F \circ G$? Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be a left inverse of function $G$. How do they interestingly compose? Back: As $F \circ G$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:left} inverse iff $F$ is {3:one-to-one}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does proving "left inverses iff injective" rely on AoC? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What are the hypotheses of "left inverses iff injective"? Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. *Why* does "left inverses iff injective" assume $A \neq \varnothing$? Back: Because a mapping from nonempty $B$ to $\varnothing$ cannot be a function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "left inverses iff injective" require AoC? Back: It doesn't. 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 formal-system::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 $a$ or $b$. Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163). END%% ### Right Inverses Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a right inverse) such that $F \circ G = I_B$ if and only if $F$ maps $A$ onto $B$. %%ANKI Basic What is the most specific mathematical object that describes a right inverse? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is a right inverse of $F \colon A \rightarrow B$ defined? Back: As a function $G \colon B \rightarrow A$ such that $F \circ G = I_B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is a right inverse of set $A$ defined? Back: N/A. Right inverses are only applicable to functions. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {1:Left} inverses are to {2:injections} whereas {2:right} inverses are to {1:surjections}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its domain? Back: $B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its codomain? Back: $A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more compactly write $G \circ F$? Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more compactly write $F \circ G$? Back: $I_B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be a right inverse of function $G$. How do they interestingly compose? Back: As $G \circ F$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:right} inverse iff $F$ is {3:onto $B$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does proving "right inverses iff surjective" rely on AoC? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What are the hypotheses of "right inverses iff surjective"? Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. *Why* does "right inverses iff surjective" assume $A \neq \varnothing$? Back: Because a mapping from nonempty $B$ to $\varnothing$ cannot be a function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "right inverses iff surjective" require AoC? Back: There is no other mechanism for choosing an $x \in A$ for *each* $y \in B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). 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%% ## Inverses Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}$$ %%ANKI Basic What is the most specific mathematical object that describes an inverse? Back: A relation. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object does the inverse operation apply to? Back: Sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the "arity" of the inverse operation in set theory? Back: $1$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be a set. How is the inverse of $F$ denoted? Back: $F^{-1}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object does the inverse operation emit? Back: Relations. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the inverse of set $F$ defined in set-builder notation? Back: $F^{-1} = \{\langle u, v \rangle \mid vFu\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider set $A$. Is $A^{-1}$ a relation? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider set $A$. Is $A^{-1}$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider relation $R$. Is $R^{-1}$ a relation? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider relation $R$. Is $R^{-1}$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. Is $F^{-1}$ a relation? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. Is $F^{-1}$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ a function? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ one-to-one? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ onto $A$? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ be a surjection. Is $F^{-1}$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ be a surjection. Is $F^{-1}$ a relation? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. What is the domain of $F^{-1}$? Back: $\mathop{\text{ran}}F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). 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). 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). 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). 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). 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). END%% %%ANKI Basic How is set $\{\langle u, v \rangle \mid vAu\}$ more simply denoted? Back: $A^{-1}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does $\varnothing^{-1}$ evalute to? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given set $F$, what does $\mathop{\text{dom}}F^{-1}$ evaluate to? Back: $\mathop{\text{ran}}F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given set $F$, what does $\mathop{\text{ran}}F^{-1}$ evaluate to? Back: $\mathop{\text{dom}}F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For any set $F$, {1:$F$} is {2:single-valued} iff {2:$F^{-1}$} is {1:single-rooted}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon \varnothing \rightarrow B$. What is $F^{-1}$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). 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). 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). 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). 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). 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). 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). 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). END%% %%ANKI Basic How is set $\{\langle u, v \rangle \mid \exists t, uBt \land tAv \}$ more simply denoted? Back: $A \circ B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be an arbitrary set. What is $F \circ \varnothing$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be an arbitrary set. What is $\varnothing \circ F$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Let $F$ be an {injection}. If $t \in$ {$\mathop{\text{ran} }F$}, then $F(F^{-1}(t)) = t$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Let $F$ be an {injection}. If $t \in$ {$\mathop{\text{dom} }F$}, then $F^{-1}(F(t)) = t$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $A$ is single-valued and $B$ is single-valued, is $A \circ B$ single-valued? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $A$ is single-valued and $B$ is single-rooted, is $A \circ B$ single-valued? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $A$ is single-rooted and $B$ is single-rooted, is $A \circ B$ single-rooted? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is a relation and $G$ is a function, is $F \circ G$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is a function and $G$ is a relation, is $F \circ G$ a function? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is a function and $G$ is a function, is $F \circ G$ a function? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is an injection and $G$ is an injection, is $F \circ G$ an injection? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is an injection and $G$ is a surjection, is $F \circ G$ a bijection? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is an injection and $G$ is a bijection, is $F \circ G$ a bijection? Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $F$ is a bijection and $G$ is a bijection, is $F \circ G$ a bijection? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ and $G$ be functions. How is $\mathop{\text{dom}}(F \circ G)$ defined using set-builder notation? Back: $\{x \in \mathop{\text{dom}}G \mid G(x) \in \mathop{\text{dom}}F\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For any sets $F$ and $G$, {$(F \circ G)^{-1}$} $=$ {$G^{-1} \circ F^{-1}$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How might you explain $(F \circ G)^{-1} = G^{-1} \circ F^{-1}$ in plain English? Back: The opposite of applying $G$ then $F$ is to undo $F$ then $G$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is composition commutative? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is composition associative? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For sets $A$, $B$, and $C$, {$(A \circ B)[\![C]\!]$} $=$ {$A[\![B[\![C]\!]]\!]$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Restrictions Let $F$ and $A$ be arbitrary sets. The **restriction of $F$ to $A$** is the set $$F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$$ %%ANKI Basic What kind of mathematical object does the restriction operation apply to? Back: Sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze $F \restriction A$ is the restriction of $F$ {to} $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object does the restriction operation emit? Back: Relations. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the "arity" of the restriction operation in set theory? Back: $2$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the restriction of $F$ to $A$ denoted? Back: $F \restriction A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the restriction of $F$ to $A$ defined? Back: $F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction A)$? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction A)$? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction C)$? Back: $\mathop{\text{dom}}(F \restriction C) \subseteq \mathop{\text{dom}}F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $F \restriction A$ pronounced? Back: The restriction of $F$ to $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction C)$? Back: $\mathop{\text{ran}}(F \restriction C) \subseteq \mathop{\text{ran}}F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{\langle u, v \rangle \mid uAv \land u \in B\}$ more simply denoted? Back: $A \restriction B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be an arbitrary set. What is $F \restriction \varnothing$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Let $Q$, $A$, and $B$ be sets. Then {$Q \restriction (A \cup B)$} $=$ {$(Q \restriction A) \cup (Q \restriction B)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Let $Q$, $A$, and $B$ be sets. Then {$Q \restriction (A \cap B)$} $=$ {$(Q \restriction A) \cap (Q \restriction B)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Let $Q$, $A$, and $B$ be sets. Then {$Q \restriction (A - B)$} $=$ {$(Q \restriction A) - (Q \restriction B)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider sets $A$ and $B$. How is $B \restriction A$ rewritten as a composition? Back: $B \circ I_A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider sets $A$ and $B$. How is $A \circ I_B$ rewritten as a restriction? Back: $A \restriction B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider sets $A$ and $B$. How is $A \cap B$ rewritten as a function under some image? Back: $I_A[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider sets $A$ and $B$. How is $I_B[\![A]\!]$ rewritten as a simpler set operation? Back: $B \cap A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Images Let $F$ and $A$ be sets. Then the **image of $F$ under $A$** is $$F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$$ %%ANKI Basic What kind of mathematical object does the image operation apply to? Back: Sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object does the image operation emit? Back: Sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze $F[\![A]\!]$ is the image of $F$ {under} $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the "arity" of the image operation in set theory? Back: $2$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the image of $F$ under $A$ denoted? Back: $F[\![A]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the image of $F$ under $A$ defined? Back: $F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the image of $F$ under $A$ defined in terms of restrictions? Back: $F[\![A]\!] = \mathop{\text{ran}}(F \restriction A)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{v \mid \exists u \in B, uAv\}$ more simply denoted? Back: $A[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Enderton says "multiple-valued functions" are actually what? Back: Relations. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Enderton says "multiple-valued functions" are actually what? Back: Relations. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Enderton says "$F^{-1}(9) = \pm 3$" is preferably written in what way? Back: $F^{-1}[\![\{9\}]\!] = \{-3, 3\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F$ be an arbitrary set. What is $F[\![\varnothing]\!]$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% The following holds for any sets $F$, $A$, $B$, and $\mathscr{A}$: * The image of unions is the union of the images: * $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ * The image of intersections is a subset of the intersection of images: * $F[\![\bigcap \mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$ * Equality holds if $F$ is single-rooted. * The image of a difference includes the difference of the images: * $F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$ * Equality holds if $F$ is single-rooted. %%ANKI Basic How does the image of unions relate to the union of the images? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does the union of images relate to the images of the unions? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does $F[\![A \cup B]\!]$ relate to $F[\![A]\!] \cup F[\![B]\!]$? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of identity $F[\![A \cup B]\!] = F[\![A]\!] \cup F[\![B]\!]$? Back: $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$? Back: $F[\![A \cup B]\!] = F[\![A]\!] \cup F[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* is the following identity intuitively true? $$F[\![A \cup B]\!] = F[\![A]\!] \cup F[\![B]\!]$$ Back: $F(x)$ is in the range of $F$ regardless of whether $x \in A$ or $x \in B$ (or both). Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does the image of intersections relate to the intersection of the images? Back: The former is a subset of the latter. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does the intersection of images relate to the image of the intersections? Back: The latter is a subset of the former. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition on set $F$ makes the following true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$ Back: N/A. This is always true. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition on set $F$ makes the following true? $$F[\![A \cap B]\!] = F[\![A]\!] \cap F[\![B]\!]$$ Back: $F$ is single-rooted. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition on set $F$ makes the following true? $$F[\![A]\!] \cap F[\![B]\!] \subseteq F[\![A \cap B]\!]$$ Back: $F$ is single-rooted. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of the following identity? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$ Back: $F[\![\bigcap\mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of the following identity? $$F[\![\bigcap\mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$$ Back: $F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What $\varnothing$-based example is used to show the following is intuitively true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$ Back: $A$ and $B$ might be disjoint even if $F[\![A]\!]$ and $F[\![B]\!]$ are not. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following is analagous to what logical expression of commuting quantifiers? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$ Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given single-rooted $R$, the following is analagous to what logical expression of commuting quantifiers? $$R[\![A \cap B]\!] = R[\![A]\!] \cap R[\![B]\!]$$ Back: $\exists x, \forall y, P(x, y) \Leftrightarrow \forall y, \exists x, P(x, y)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does the image of differences relate to the difference of the images? Back: The latter is a subset of the former. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does the difference of images relate to the image of the differences? Back: The former is a subset of the latter. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What $\varnothing$-based example is used to show the following is intuitively true? $$F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$$ Back: $F[\![A]\!]$ and $F[\![B]\!]$ might be the same sets even if $A \neq B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition on set $F$ makes the following true? $$F[\![A - B]\!] \subseteq F[\![A]\!] - F[\![B]\!]$$ Back: $F$ is single-rooted. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition on set $F$ makes the following true? $$F[\![A - B]\!] = F[\![A]\!] - F[\![B]\!]$$ Back: $F$ is single-rooted. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition on set $F$ makes the following true? $$F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$$ Back: N/A. This is always true. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $A \subseteq B$. How does $F[\![A]\!]$ relate to $F[\![B]\!]$? Back: $F[\![A]\!] \subseteq F[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Closures If $S$ is a function and $A$ is a subset of $\mathop{\text{dom}}S$, then $A$ is said to be **closed** under $S$ if and only if whenever $x \in A$, then $S(x) \in A$. This is equivalently expressed as $S[\![A]\!] \subseteq A$. %%ANKI Basic Let $A$ be closed under $S$. Then $A$ is a subset of what other set? Back: $\mathop{\text{dom}}S$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $A$ be closed under $S$. What kind of mathematical object is $A$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $A$ be closed under $S$. What kind of mathematical object is $S$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic In FOL, what does it mean for set $A$ to be closed under function $S$? Back: $\forall x \in A, S(x) \in A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What concept is being expressed in "$\forall x \in A, S(x) \in A$"? Back: Set $A$ is closed under $S$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How can we more compactly express "$\forall x \in A, S(x) \in A$"? Back: $S[\![A]\!] \subseteq A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze If $S[\![A]\!] \subseteq A$, then {1:$A$} is closed {2:under} {1:$S$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $A$ is closed under function $S$. What imagery does the term "closed" invoke? Back: Applying a member of $A$ to $S$ always yields an element in $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% Let $f$ be a function from $B$ into $B$ and assume $A \subseteq B$. There are two possible methods for constructing the **closure** $C$ of $A$ under $f$. The top-down approach defines $C^*$ to be the intersection of all closed supersets of $A$: $$C^* = \bigcap\, \{X \mid A \subseteq X \subseteq B \land f[\![X]\!] \subseteq X \}$$ The bottom-up approach defines $C_*$ to be $$C_* = \bigcup_{i \in \omega} h(i)$$ where $h \colon \omega \rightarrow \mathscr{P}(B)$ is recursively defined as: $$\begin{align*} h(0) & = A, \\ h(n^+) &= h(n) \cup f[\![h(n)]\!]. \end{align*}$$ Note that the [[natural-numbers#Recursion Theorem|recursion theorem]] proves $h$ is indeed a function. %%ANKI Basic Let $f \colon B \rightarrow B$ and $A \subseteq B$. How is the top-down closure $C^*$ of $A$ under $f$ defined? Back: $\bigcap\, \{ X \mid A \subseteq X \subseteq B \land f[\![X]\!] \subseteq X \}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the smallest set the closure $C^*$ of $A$ under $f$ can be? Back: $A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the largest set the closure $C^*$ of $A$ under $f$ can be? Back: $B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \colon B \rightarrow B$ and $A \subseteq B$. How is the bottom-up closure $C_*$ of $A$ under $f$ defined assuming appropriate $h \colon \omega \rightarrow \mathscr{P}(B)$? Back: $\bigcup \mathop{\text{ran}} h$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the smallest set the closure $C_*$ of $A$ under $f$ can be? Back: $A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the largest set the closure $C_*$ of $A$ under $f$ can be? Back: $B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. What kind of mathematical entity is $A$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. What kind of mathematical entity is $f$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. What kind of mathematical entity is $C$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. What two ways can $C$ be defined? Back: Bottom-up or top-down. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. How is the top-down closure denoted? Back: As $C^*$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. How is the bottom-up closure denoted? Back: As $C_*$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C$ be the closure of $A$ under $f$. What is the "signature" of $f$? Back: $f \colon B \rightarrow B$ for some $B \supseteq A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What is $h$'s domain? Back: $\omega$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What is $h$'s codomain? Back: Assume $A \subseteq B$ and $f \colon B \rightarrow B$. Then $h$'s codomain is $\mathscr{P}(B)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What does $h(0)$ evaluate to? Back: $A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What does $h(n^+)$ evaluate to? Back: $h(n) \cup f[\![h(n)]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What theorem proves $h$'s existence? Back: The recursion theorem. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze The top-down closure $C^*$ of $A$ under $f$ is the {intersection} of all {closed supersets} of $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Kernels Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$ Relation $\sim$ is called the **(equivalence) kernel** of $f$. The [[relations#Partitions|partition]] induced by $\sim$ on $A$ is called the **coimage** of $f$ (denoted $\mathop{\text{coim}}f$). The **fiber** of an element $y$ under $F$ is $F^{-1}[\![\{y\}]\!]$, i.e. the preimage of singleton set $\{y\}$. Therefore the equivalence classes of $\sim$ are also known as the fibers of $f$. %%ANKI Basic What kind of mathematical object is the kernel of $F \colon A \rightarrow B$? Back: An equivalence relation. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the kernel of $F \colon A \rightarrow B$ defined? Back: As equivalence relation $\sim$ such that $x \sim y \Leftrightarrow F(x) = F(y)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. What name does the following relation $\sim$ go by? $$x \sim y \Leftrightarrow F(x) = F(y)$$ Back: The kernel of $F$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. The partition induced by the kernel of $F$ is a partition of what set? Back: $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. Term "$\mathop{\text{coim}}F$" is an abbreviation for what? Back: The **coim**age of $F$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the coimage of function $F \colon A \rightarrow B$ defined? Back: As $A / \mathop{\text{ker}}(F)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. What term refers to a member of $\mathop{\text{coim}}F$? Back: A fiber. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. How is the fiber of $y$ under $F$ defined? Back: As set $F^{-1}[\![\{y\}]\!]$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. The fibers of $F$ make up what set? Back: $\mathop{\text{coim}}F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$. How is $\mathop{\text{coim}}F$ denoted as a quotient set? Back: As $A / \mathop{\text{ker}}(F)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $F \colon A \rightarrow B$ and $\sim$ be the kernel of $F$. How does $F$ factor into $\hat{F} \colon A / {\sim} \rightarrow B$? Back: $F = \hat{F} \circ \phi$ where $\phi$ is the natural map. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name does $\phi$ go by? ![[function-kernel.png]] Back: The natural map (with respect to $\sim$). Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. How is $\phi$ defined? ![[function-kernel.png]] Back: $\phi(x) = [x]_{\sim}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name does $A /{\sim}$ go by? ![[function-kernel.png]] Back: $\mathop{\text{coim}} F$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name do the members of $A / {\sim}$ go by? ![[function-kernel.png]] Back: The fibers of $F$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What composition is $F$ equal to? ![[function-kernel.png]] Back: $F = \hat{F} \circ \phi$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ injective? ![[function-kernel.png]] Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ surjective? ![[function-kernel.png]] Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ bijective? ![[function-kernel.png]] Back: Not necessarily. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). 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). * “Fiber (Mathematics),” in _Wikipedia_, April 10, 2024, [https://en.wikipedia.org/w/index.php?title=Fiber_(mathematics)&oldid=1218193490](https://en.wikipedia.org/w/index.php?title=Fiber_(mathematics)&oldid=1218193490). * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). * “Kernel (Set Theory),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Kernel_(set_theory)&oldid=1225189560](https://en.wikipedia.org/w/index.php?title=Kernel_(set_theory)&oldid=1225189560). * “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938).