I need 8, 11, and 12 answered With S_1 ={2, 3, 5, 7}, S_2 = {2, 4, 5, 8,9}, and

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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

Explanation / Answer

8. S1 = S2 if and only if S1 Union S2 = S2 Intersection S2

    As we know S1 union S2 means that the element belonging to this union will be either a member of S1 or S2 and there won't be any repeatition. So if both the sets are equal then this will consists of all the elemnents of S1 or S2.

Looking at the intesection of S1 and S2 , we know that this consists of elements common to both S1 and S2. But if it is equal to union of S1 and S2 , then it means that all the elemnts are common and this is only possible if both the sets are same.

11. S1 x S2 means set of all pairs of (x,y) such that x belongs to S1 and y belongs to S2. So the cradinality of this set will be as follows:
   every elemnet of S1 will pair with all the elements of S2.If the number of elements in S1 is m and number of elements is n for S2, then total number of elements in S1 X S2 will be m*n which is nothing but |S1| * |S2|.Hence proved.

12. The relation |s1| = |s2| is an equivalene relation because this is symmetric because |s1| = |s2| is same as |s2| = |s1|. It is reflexive because |s1| = |s1| and it is transitive . If |s1| = |s2| and |s2| = |s3| then |s1| = |s3|.

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