Mathematics 220
Homework 8
1. Find the domain and range of the function f (x) =
must provide detailed proof).
Due Monday/Tuesday March 14/15
√
x−1
x
(assume x is real). (Remember, you
2. Let f : R → R be the function defined by f (x) = x2 + 3x + 4.
(a) Prove that f is not injective.
(b) Find all pairs a, b of real numbers so that f (a) = f (b).
3. Let h : Z → Z be a function defined by h(n) = 3n − 8. Determine whether it is (a) injective
and (b) surjective. Remember, you must prove your answer.
4. Question 9.26. Give an example of a function f : N → N that is
(a)
(b)
(c)
(d)
one-to-one and onto
one-to-one but not onto
onto but not one-to-one
neither one-to-one more onto
5. Prove or disprove: For every set A there is an injective function f : A → P(A).
6. Remember — in all parts of this problem, you have to prove that your function satisfies the
required property.
(a) Give an example of a bijection from (0, 1) to (1, ∞).
(b) Give an example of a surjection from R to Q.
(c) Give an example of a bijection from N to Z.
7. Let f : A → B be a function and if D ⊂ B, recall that the inverse image of D under f is by
definition the set f −1 (D) = {x ∈ A : f (x) ∈ D}. Note that f −1 (D) is a set, and is defined
for any function f , even if f does not have an inverse.
(a) If D1 , D2 ⊆ B, prove that
f −1 (D1 ∩ D2 ) = f −1 (D1 ) ∩ f −1 (D2 )
(b) If D1 , D2 ⊆ B, prove that
f −1 (D1 ∪ D2 ) = f −1 (D1 ) ∪ f −1 (D2 )
(c) If D ⊂ B, prove that
f −1 (B − D) = A − f −1 (D)
8. Let A be a well-ordered set. Let f : P(A) → A be the function that assigns to every for
B ⊆ A the smallest element of B.
(a) Is f is a function?
(b) Is f injective?
(c) Is f surjective?
9. (a) Prove that if f : A → B is an injective function, then f (C1 ∩ C2 ) = f (C1 ) ∩ f (C2 ) for all
C1 , C2 ⊆ A.
(b) Prove that if f : A → B is a surjective function, then f (f −1 (D)) = D for every subset
D ⊆ B.
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