Part 2:Proof that(A)(B)(AB).

Consider the sentences in the following scrambled list.

• So every element ofXis in bothAandB, and thusXAB.
• ThereforeX(AB), and hence(A)(B)(AB).
• HenceXAandXB.
• ThenX(A) andX(B).

We prove part 2 by selecting appropriate sentences from the list and putting them in the correct order. (1) Let X E P(A) n P(B). (2) So every element of X is in both A and B, and thus X S An B. O Therefore X E .A(A n B), and hence A(A) n A(B) C A(An B). O Hence X S A and X C B. OThen X E AA) and X E A(B). (3) OSo every element of X is in both A and B, and thus X S An B. O Therefore X E .A(A n B), and hence A(A) n A(B) = A(An B). O Hence X S A and X C B. O Then X E PA) and X E P(B). (4) So every element of X is in both A and B, and thus X S An B. O Therefore X E .A(A n B), and hence MA) n H(B) S A(An B). O Hence X _ A and X C B. Then X E AA) and X E P(B). (5) OSo every element of X is in both A and B, and thus X S An B. OTherefore X E .A(A n B), and hence A(A) n A(B) S A(An B). O Hence X S A and X C B. OThen X E AA) and X E P(B)