1) a) Define the sets A = {2n + 1 | n Z}, B = {3m + 2 | m Z}, C = {12k + 5 | k Z}. Prove that C A B. Hint: Prove that C A B, and then find an element of A B that is not in C to show that the sets are not equal. Remember to justify that your element is in A B but not in C.

b)Suppose A and B are subsets of a set X. (i) Show that A B B^C . (ii) Show that A B = B^C if and only if X = A B.

c) Suppose A, B, and C are sets, with A . Prove that if A B A C, then B C.

d) Define a sequence (x_n)_n=1 by x₁ = 2, x_n+1 = 3x_n + 2, n 1. Determine an explicit formula for x_n

e) Use induction to prove that ⁿ_k=1 k(k + 1) = (n(n + 1)(n + 2))/3 for all n N.