MATH117 Homework 9: due Friday, 4 November
Remember to justify your answers!
1. Required problems
(1) Suppose A = {1, 2} and B = {2, 3}. Find each of the following
(where P(X) is the power set of X):
(a) P(A ∩ B)
(b) P(A)
(c) P(A ∪ B)
(d) P(A × B)
(2) One of the following is true and the other is false. Prove the true statement and find a
counterexample for the false statement.
(a) For all sets A, B, and C, if B ∩ C ⊆ A then (A − B) ∩ (A − C) = ∅.
(b) For all sets A, B, and C,
(A − B) − C = A − (B ∪ C).
(3) Define a function S : Z+ → Z+ as follows: For each positive integer n, S(n) = the sum of
the positive divisors of n. Find the following:
(a) S(17)
(b) S(5)
(c) S(18)
(d) S(21)
(4) Find all function from X = {a, b, c} to Y = {u, v}.
(5) Let F : R → R and G : R → R be functions from R to itself. Define new functions:
F − G : R → R and G − F : R → R as follows: For all x ∈ R,
(F − G)(x) = F (x) − G(x)
(G − F )(x) = G(x) − F (x)
Does F − G = G − F ? Explain.
(6) Define F : Z → Z by the rule F (n) = 2 − 3n, for all integers n.
(a) Is F one-to-one? (Proof or counterexample.)
(b) IS F onto? (Proof or counterexample.)
CONTINUED ON BACK →
(7) Define G : R → R by the rule G(x) = 2 − 3x, for all real numbers x.
(a) Is G one-to-one? (Proof or counterexample.)
(b) IS G onto? (Proof or counterexample.)
(8) Define H : R × R → R × R as follows:
H(x, y) = (x + 1, 2 − y) for all (x, y) ∈ R × R.
Is H a one-to-one correspondence? (Proof or counterexample.)
(9) If F : R → R and G : R → R are functions defined by F (x) = x2 and G(x) =
F ◦ G and G ◦ F . Are F ◦ G and G ◦ F equal?
2. Bonus Problem
Please hand in on a sheet separate from your homework.
(1) Define G : Z+ × Z+ → Z+ and F : Z+ × Z+ → Z+ as follows: For all (m, n) ∈ Z+ ,
G(m, n) = 3m 6n
F (m, n) = 3m 5n
(a) Is G one-to-one? Proof or counterexample.
(b) Is F one-to-one? Proof or counterexample.
√
x, find