71 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Functions | Obsidian::STEM | set::function |
|
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%%
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).
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).
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).
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).
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).
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).
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).
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).
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).
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 formal-system::predicate
END%%
%%ANKI
Basic
Does the following depict an injection?
!
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?
!
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?
!
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?
!
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 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.
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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 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
Fis single-rooted.
- The image of a difference includes the difference of the images:
F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]- Equality holds if
Fis 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 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 \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. Term "\mathop{\text{coim}}F" is an abbreviation for what?
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 / \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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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.
- “Fiber (Mathematics),” in Wikipedia, April 10, 2024, 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.
- “Operation (Mathematics).” In Wikipedia, October 10, 2024. https://en.wikipedia.org/w/index.php?title=Operation_(mathematics).