Reading Quiz Section 7.2
1. What does it mean for a relation R ⊆ A × B to be a function? Select all that apply.
(a) dom(R) = A
(b) range(R) = B
(c) For any a ∈ A, if (a, b
1
), (a, b
2
) ∈ R, then b
1
= b
2
(d) For any b ∈ range(R), if (a
1
, b), (a
2
, b) ∈ R, then a
1
= a
2
2. Let f : A → B be a function. If f
−1
: B → A is a function, this means in particular that
dom( f
−1
) = B. This is equivalent to what property of f ?
(a) Injectivity
(b) Surjectivity
(c) dom( f ) = A
(d) That f is a symmetric relation.
3. True or False: a relation R has a domain and range if and only if it is a function.
Practice Problems Section 7.2
1. Let f : A → A be a function. Viewed as a relation, if f is symmetric, what can be said about f ?
2. (a) Express the function f : R → R : x 7→ x
2
as a relation.
(b) What is the inverse relation f
−1
?
(c) Use Definition 7.6 to prove that the relation f
−1
is not a function.
(d) Prove directly from Definition 4.18 that f is not injective and not surjective. Compare your
arguments with your answer to part (c).
