(a) Let A = {1,2,3}, B = {1,3}, 0 = {a,b,c} and D = {b,c}. Write down the elements of the following sets, you do not need to provide any justification: (i) (A\\B) x 0: (ii) B X (C\\D); (iii) (A x C)\\(B x D). [3 marks] (b) Prove, using a suitable counter example, that the following statements are false. (i) For all :5, y E R. lx+y| Z W + '9'- 1 (ii) For allzcEZ, '3 6741. IL' (iii) LetA = {n E Z | n = 10a+b, a 6 {1,2,3}, (3 6 {2,3, 6}} For all n = (10a+b) E A, ifb=6,thena=1 or2. [3 marks] (c) Determine whether the following statements are true or false. Give brief reasons for your answers. (No credit is available for writing true or false without justification.) (i) For all sets A and B in a universal set 9, if B Q Ac, then A F] B 72 E). (ii) For all sets A, B and C in a universal set (2, A\\B and C\\B are disjoint. (iii) For all finite sets A and B in a universal set S], I'P((A x B) U (A x B))| = |P((A x B) U (B x A))| if and only if A = B. [6 marks] (d) Define f : N > 'P(N) by f(n) = {n} for n E N. Determine, with proof, if f is: (i) injective; (ii) surjective. [4 marks] (e) (i) Let A and B be sets in a universal set 9. Prove that if X 6 P(B\\A) is nonempty, then X E 73(B)\\'P(A). Determine, with justification, if the statement is true when X is the empty set. [4 marks] (ii) Let A, B and C be sets in a universal set 9. Prove that, if A g (B 0\"), then C Q AC. Determine, with justification, if the converse statement is true. [5 marks]