notebook/notes/set/functions.md

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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}}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%%

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%%

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. 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.

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 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.

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. 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.

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%%

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 a {function}. 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 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 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 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%%

Kernels

Let F \colon A \rightarrow B. Define relations#Equivalence Relations \sim as $x \sim y \Leftrightarrow f(x) = f(y)$ Relation \sim is called the (equivalence) kernel of f. The relations#Partitions 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. What does \mathop{\text{coim}}F refer to? Back: The coimage 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 / {\sim} where 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 specific name does a member of \mathop{\text{coim}}F go by? 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 / {\sim} where 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 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 \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