MATH117 Homework 10: due Friday, 11 November

MATH117 Homework 10: due Friday, 11 November
Remember to justify your answers!
1. Required problems
(1) Prove that if f : X → Y is a one-to-one correspondence with inverse function f −1 : Y → X,
then f ◦ f −1 = IY , where IY is the identity function on Y .
(2) True or False? Given any set X and given any three functions f : X → X, g : X → X,
and h : X → X, if h is one-to-one and f ◦ h = g ◦ h, then f = g. Justify your answer with
either a proof or a counterexample.
(3) Let A be the set of all strings of 0’s, 1’s, and 2’s of length 4. Define a relation R on A as
follows: For all s, t ∈ A, sRt if and only if the sum of the characters in s equals the sum of
the characters in t.
(a) Is 0121 R 2200?
(b) Is 1011 R 2101?
(c) Is 2212 R 2121?
(d) Is 1220 R 2111?
(4) Define relations R and S on R as follows:
R = {(x, y) ∈ R × R|x < y}
S = {(x, y) ∈ R × R|x = y}.
Graph R, S, R ∪ S, and R ∩ S in the cartesian plane. (The Cartesian plane is the usual
xy-plane where you graph points using their x and y coordinates.)
(5) Let C be the circle relation on the set R: For all x, y ∈ R, xCy ⇐⇒ x2 + y 2 = 1.
(a) Is the circle relation reflexive?
(b) Is the circle relation symmetric?
(c) Is the circle relation transitive?
(6) Let X = {a, b, c} and P(X) be the power set of X. A relation N is defined on P(X) as
follows: For all A, B ∈ P(X), ANB ⇐⇒ the number of elements in A is NOT equal to
the number of elements in B.
(a) Is N reflexive?
(b) Is N symmetric?
(c) Is N transitive?
(7) If R and S are transitive, is R ∩ S transitive? Why or why not?
(8) Let T = {(0, 2), (1, 0), (2, 3), (3, 1)}. Find T t , the transitive closure of T .
2. Bonus Problem
Please hand in on a sheet separate from your homework.
(1) If f : X → Y and g : Y → Z are functions and g ◦ f is onto, must f be onto? Prove or give
a counterexample.