I need 8, 11, and 12 answered

With S_1 = {2, 3, 5, 7), S_2 = {2, 4, 5, 6, 9}, and U = {1: 10}, compute S_1 bar union S_2. With S_1 = {2, 3, 5, 7} and S_2 = {2, 4, 5, 8, 9}, compute S_1 times S_2 and S_2 times S_1. For S = {2, 5, 6, 8} and T = {2, 4, 6, 8), compute |S intersection T| + |S union T|? What relation between two sets S and T must hold so that |S union T| = |S| + |T|? Show that for all sets S and T, S - T = S intersection T bar. Prove DeMorgan's laws, Equations (1.2) and (1.3), by showing that if an element x is in the set on one side of the equality, then it must also be in the set on the other side of the equality. Show that if S_1 Subsetequalto S_2, then S_2 bar Subsetequalto S_1 bar. Show that S_1 = S_2 if and only if S_1 union S_2 = S_1 intersection S_2. Use induction on the size of S to show that if S is a finite set, then |2^s| = 2^|s|. Show that if S_1 and S_2 are finite sets with |S_1| = n and |S_2| = m, then |S_1 union S_2| lessthanorequalto n + m. If S_1 and S_2 are finite sets, show that |S_1 times S_2| = |S_1||S_2|. Consider the relation between two sets defined by S_1 identicalto S_2 if and only if |S_1| = |S_2|. Show that this is an equivalence relation. Occasionally, we need to use the union and intersection symbols in a manner analogous to the summation sign sigma. We define union_p elementof {i, j, k, ...} S_p = S_i union S_j union S_k... with an analogous notation for the intersection of several sets. With this notation, the general DeMorgan's laws are written as union_p elementof P S_p bar = intersection_p elementof P S_p bar and intersection_p elementof P S_p bar = union_p elementof P S_p bar. Prove these identities when P is a finite set. Show that S_1 union S_2 = S_1 bar intersection S_2 bar bar