Exercise 2.16. Suppose that A and B are countably infinite sets. (A) Show that AUB is countably infinite (B) Show that A x B is countably infinite

7. Let A and B be bounded nonempty subsets of R, and let A + B := {a+b : a € A, b € B}. Prove that sup(A + B) = sup A + sup B and inf(A + B) = inf A +inf B.

1. Let A = {-30, -10, 10, 30, 50} and B = {-20, -10, 0, 10} be subsets of the universal set ε = {10x x € ZZ and -3≤ x <6}. (a) List the elements of An B and (AUB)'. (b) Evaluate: (i) │A\ B|; (ii) |A × B|; [1] (iii) |P(A)|; (iv) P(B) × P(A)|. [2] (c) Are the following statements are true or false? Explain your reasoning. (i) 0 СА; (ii) {0} = P(A); (iii) {10} E AUB; (iv) {10} is a proper subset of An B. (d) List the elements of P(A) P(B). [2] [2] (e) If the set C is also a subset of Ɛ, with |C| = 5, An C = {-10, 30} and BNC {-10, 20}, list the elements of C. = [2]

Provide Cantor’s proof that the set (0, 1) is uncountable.