Use an element argument to prove the statement. (Assume that all sets are subsets of a universal set U.) Statement: For all sets A, B, and C, (AB) n (CB) = (A nC) - B. Proof: Suppose A, B, and C are any sets. [To show that (AB) n (CB) = (An C) - B, we must show that (A - B) n (CB) C (An C) - B and that (An C) - BC (AB) n (C-B).] Part 1: Proof that (A - B)n (CB) C (An C) - B Consider the sentences in the following scrambled list. By definition of intersection, x EA - B and x EC - B. By definition of set difference, x EA - B and x EC - B. Therefore x E (An C) - B by the definition of set difference. By definition of set difference, x EA and x B and x E C and x € B. By definition of intersection, x EA and x B and x E C and x € B. Thus x E An C by definition of intersection, and, in addition, x # B. To prove Part 1, select sentences from the list and put them in the correct order. 1. Suppose x E (AB) n (CB). 2. Thus x € An C by definition of intersection, and, in addition, x # B. 3. ---Select--- 4. ---Select--- 5. ---Select--- 6. Hence, (AB) n (CB) C (An C) - B by definition of subset. Part 2: Proof that (An C) - BC (AB) n (CB) Consider the sentences in the following scrambled list. So by definition of set difference, x EA - B and x E C - B. By definition of set difference, x EA and x E C. Thus, by definition of intersection, x E A and x E C, and, in addition, x € B. Hence both x EA and x B and also x E C, and x # B. By definition of intersection, x E (A - B) n (CB). By definition of intersection x E An C and x € B. By definition of set difference x E An C and x € B.