Question 1. SETS
a) For a set X, let 2 denote the set of subsets of X, also called the power set of X. Let A = {o, {}, {{}}}. Which of the following is true: (i) A ? 2 (ii) 2 ? A and (iv) Neither i nor ii.
b) Let A = {, a, b}, B = {a, {a}}, C = {c}. Determine the following sets: (i) A ? B, A ? B, BA, AB, A ? B ? A (!!), A ? A, B ? C, C ? B, P(B), P(P(C)), (i1i) A, C ? O, C ? P, C ? A, XA, BXC.
c) Which of the following pairs of sets are equal? (i) {1, 2, 3}, {1, 3, 2} (ii) {1, 1, 3}, {1, 3, 3} (iii) {x ? N, x + x = 2}, {12} (iv) {x ? R, x + x - 2}, {12} (v) A ? B ? A ? A for all sets A, B (vi) A ? B ? (BA)B for all sets A, B
d) Prove the following statements for arbitrary subsets A, B, and C of U: (i) A ? B ? C ? A ? B ? A ? C (ii) A ? C ? B ? C = A ? B ? C