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4. Determine whether each of the functions below is onto, and/or one-to-one for f : Z → Z, prove your
answers.
(a) f(x) = 5x − 3
(b) f(x) = 2x
3
(c) f(x) = (2x − 2)2
(d) f(x) =
√
x
6
5. Determine whether each of the functions below is onto, and/or one-to-one for f : R → R, prove your
answers.
(a) f(x) = 5x − 3
(b) f(x) = 2x
3
(c) f(x) = (2x − 2)2
(d) f(x) =
√
x
6
6. Define the following functions (assume that the domains/codomains are defined such that
each composition is valid): f(x) = 2x, g(x) =
x
(1+x)
, h(x) = √
x. Find
(a) f ◦ g ◦ h
(b) h ◦ g ◦ f(c) f ◦ f
(d) g ◦ g
7. Find inverses of the following functions (assume that the domains/codomains are defined such that each
function is a bijection).
(a) f(x) = 5x − 3
(b) f(x) = 2x
3
(c) f(x) = (2x − 2)2
(d) f(x) =
√
x
8.
6
Let f(x) and g(x) be two linear functions (a function is linear if f(x) = ax + b for a, b ∈ R).
a. Prove or disprove: f ◦ g = g ◦ f.
b. Prove or disprove: f ◦ g and g ◦ f are linear functions.
9. Let R be a relation on a set A = {a1, . . . , an} of size n. Let MR be the 0-1 matrix representing R (i.e., the
entry mij = 1 if (ai
, aj ) ∈ R and zero otherwise).
(a) How many unique relations are there on A (in terms of n)?
(b) The complement relation is defined as
R = {(a, b) | (a, b) 6∈ R}
Say that the number of nonzero entries in MR (that is, the number of 1s) is k. How many nonzero
entries are there in MR
? Briefly justify your answer.
(c) How many reflexive relations are there on a set of size n? Briefly justify your answer.
(d) How many symmetric relations are there on a set of size n? Briefly justify your answer.
10. Define the following relation on the set of integers.
R1 = {(a, b) | a > b, a, b ∈ Z}
(a) Give an element that is in R1
(b) Give an element that is not in R1.
(c) Give an element that is in R1 ◦ R1
(d) Give an element that is not in R1 ◦ R1.
(e) Give an element that is in R3
1
(f) Give an element that is not in R3
1
(g) Give a general (set builder) definition for Rn
1
in terms of a, b and n.
11. Prove the following. A relation R is asymmetric if and only if R is irreflexive and antisymmet-ric. Note: a
relation R on the set A is irreflexive if for every a ∈ A, (a, a) 6∈ R. That is, R is irreflexive if no element in A
is related to itself.
Hint: write the definition of what it means to be asymmetric, then “add” the contradiction:

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