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title TARGET DECK FILE TAGS tags
Functions Obsidian::STEM set::function
function
set

Overview

A function F is a single-valued relations. 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.

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.

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. 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.

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.

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.

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.

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.

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.

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. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

END%%

Bibliography