I need 8, 11, and 12 answered

With S_1 ={2, 3, 5, 7}, S_2 = {2, 4, 5, 8,9}, 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_ S_x = 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 S_p bar = Intersection S_p bar