23 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}}A, and \mathop{\text{ran}}F \subseteq B.
%%ANKI Basic Which of relations or functions is the more general concept? Back: Relations. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is a function?
Back: A relation F such that for each x \in \mathop{\text{dom}}F, there exists a unique y such that xFy.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For function F and x \in \mathop{\text{dom}}F, what name is given to F(x)?
Back: The value of F at x.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Who introduced the function notation F(x)?
Back: Leonhard Euler.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let F be a function and \langle x, y \rangle \in F. Rewrite the membership as an expression excluding y.
Back: \langle x, F(x) \rangle \in F
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let F be a function and \langle x, y \rangle \in F. Rewrite the membership as an expression excluding x.
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider notation F(x). What assumption is F assumed to satisfy?
Back: It is assumed to be a function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider notation F(x). What assumption is x assumed to satisfy?
Back: It is assumed to be in the domain of F.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze A function is a {single-valued} relation. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is F \colon A \rightarrow B pronounced?
Back: F maps A into B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What three conditions hold iff F maps A into B?
Back: F is a function, \mathop{\text{dom}}F = A, and \mathop{\text{ran}}F \subseteq B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider function F \colon A \rightarrow B. What term is used to refer to A?
Back: The domain.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider function F \colon A \rightarrow B. What term is used to refer to B?
Back: The codomain.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic How does the range of a function compare to its codomain? Back: The range is a subset of the codomain. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Injections
A function is injective or one-to-one if each element of the codomain is mapped to by at most one element of the domain.
%%ANKI Basic What does it mean for a function to be injective? Back: Each element of the codomain is mapped to by at most one element of the domain. Reference: “Bijection, Injection and Surjection,” in Wikipedia, May 2, 2024, https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection.
END%%
%%ANKI Basic What does it mean for a function to be one-to-one? Back: Each element of the codomain is mapped to by at most one element of the domain. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Each element of an injection's codomain is mapped to by how many elements of the domain? Back: At most one. Reference: “Bijection, Injection and Surjection,” in Wikipedia, May 2, 2024, https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection.
END%%
%%ANKI
Basic
Suppose Function.Injective f for f \colon A \rightarrow B. What predicate logical formula describes f?
Back: \forall a_1, a_2 \in A, (f(a_1) = f(a_2) \Rightarrow a_1 = a_2)
Reference: “Bijection, Injection and Surjection,” in Wikipedia, May 2, 2024, https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection.
Tags: lean logic::predicate
END%%
%%ANKI
Basic
Does the following depict an injection?
!
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%%
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?
!
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%%
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 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%%
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%%
Bibliography
- “Bijection, Injection and Surjection,” in Wikipedia, May 2, 2024, https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection.
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).