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Functions and relations flashcard set
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Functions and relations flashcard set
Functions and relations flashcard set
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1
Question
What is a function from set A to set B?
Page 2
Answer
An assignment of exactly one element of B to each element of A.
2
Question
How is a function denoted from A to B?
Page 2
Answer
f: A → B
3
Question
What is a relation from set A to set B?
Page 2
Answer
A subset R of the Cartesian product A × B consisting of ordered pairs.
4
Question
What are the elements of a relation?
Page 2
Answer
Ordered pairs (a, b) where a ∈ A and b ∈ B.
5
Question
Give an example of a relation from {a, b, c} to {0, 1, 2, 3}.
Page 2
Answer
R = {(a, 0), (a, 1), (a, 3), (b, 1), (b, 2), (c, 0), (c, 3)}
6
Question
What is a relation from a set A to itself called?
Page 2
Answer
A relation on A.
7
Question
What ordered pairs are in the ≤ relation on {0, 1, 2, 3}?
Page 3
Answer
All (a, b) where a ≤ b, like (0,0), (0,1), (0,2), (0,3), (1,1), etc.
8
Question
How does a relation define a function from A to B?
Page 3
Answer
If it contains exactly one ordered pair (a, b) for every a ∈ A.
9
Question
What is the domain of a function f: A → B?
Page 4
Answer
Set A.
10
Question
What is the codomain of a function f: A → B?
Page 4
Answer
Set B.
11
Question
If f(a) = b, what is b called relative to a?
Page 4
Answer
The image of a.
12
Question
What is a preimage of b under f?
Page 4
Answer
An a such that f(a) = b.
13
Question
What is the range or image of a function f?
Page 4
Answer
The set of all images of elements of the domain.
14
Question
What does it mean for f to map A to B?
Page 4
Answer
f is a function from A to B.
15
Question
What is a one-to-one function?
Page 5
Answer
f(a) = f(b) implies a = b for all a, b in domain.
16
Question
What is another term for one-to-one function?
Page 5
Answer
Injective.
17
Question
How is one-to-one formally defined with quantifiers?
Page 5
Answer
∀a∀b (f(a) = f(b) → a = b), universe is domain.
18
Question
Equivalent logical statement for one-to-one?
Page 5
Answer
∀a∀b (a ≠ b → f(a) ≠ f(b))
19
Question
Is f(x) = x² from integers to integers one-to-one?
Page 6
Answer
No, because f(1) = f(-1) = 1 but 1 ≠ -1.
20
Question
Is f(x) = x + 1 from reals to reals one-to-one?
Page 6
Answer
Yes, because if x + 1 = y + 1 then x = y.
21
Question
What is an increasing function?
Page 7
Answer
∀x∀y (x < y → f(x) ≤ f(y)), universe domain.
22
Question
What is a strictly increasing function?
Page 7
Answer
∀x∀y (x < y → f(x) < f(y))
23
Question
What is a decreasing function?
Page 7
Answer
∀x∀y (x < y → f(x) ≥ f(y))
24
Question
What is a strictly decreasing function?
Page 7
Answer
∀x∀y (x < y → f(x) > f(y))
25
Question
What is an onto function from A to B?
Page 8
Answer
For every b ∈ B, there is a ∈ A with f(a) = b.
26
Question
Another term for onto function?
Page 8
Answer
Surjective.
27
Question
Formal definition of onto with quantifiers?
Page 8
Answer
∀y ∃x (f(x) = y), x in domain, y in codomain.
28
Question
In the example f(a)=3, f(b)=2, f(c)=1, f(d)=3 to {1,2,3}, is it onto?
Page 8
Answer
Yes, because 1,2,3 are all images.
29
Question
If codomain is {1,2,3,4} for that f, is it onto?
Page 9
Answer
No, because 4 has no preimage.
30
Question
Is f(x) = x² from integers to integers onto?
Page 9
Answer
No, because negative numbers like -1 have no integer preimage.