advertisement

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.