13.1.5 Let A and B he sets. Prove that If A is a nite set, then A H B is a finite set. If A U B is a nite set, then A and B are nite sets. If A H B is an infinite set, then A is an innite set. (a [h (c (d ) ) l ) If A is an infinite set or B is an infinite set, then A U B is an innite set. 9.1.7 Prove the following proposition: (b) If A,B, C, and D are sets with A H B and C m D, and if A and C are disjoint and B and D are disjoint, then A U C m B U D. 9.2.2. Prove that each of the following sets is countably innite. (a) The set Fl\" of all natural numbers that are multiples of 5 (11) The set F of all integers that are multiples of 5 (c) {iik A: EN} ((1) {n E Z | n I: 11]} (1k) Optional Challenge: Replace one of the problems above with 9.2.9. 9.2.9. Dene f : N x N > N as follows: For each (m,n) E N x N, m, n) = 2m_1(2n 1). 1. Prove that f is an injection. (See the textbook for hints.) 2. Prove that f is a s111'jeotion. (See the textbook for hints.) 3. Prove that N X N M N and hence that card(N X N) = N0