Question

Use the set identities given in the table to prove the following new identities. Label each step in your proof with the set identity used to establish that step. (a) (An C) (An C) = 0 (b) (BUA) n(BUA) = A (c) AnB = AUB (d) An (AUB) = ĀnB (e) AU (A n B) = ĀUB (f) An ( BB) = 0 (g) AU (B U B) = 0

Answer 1

In this solution, we use set identities to prove various new identities. We provide step-by-step explanations for each identity using relevant set identities as references.

Step-by-step explanation:

(a) To prove (An C) (An C) = 0, we can use the identity: A n A' = 0. So, starting with (An C) (An C), we can apply the identity to get (An C) (An C) = (An C)' = 0.

(b) To prove (BUA) n (BUA) = A, we can use the identity: A U (B n C) = (A U B) n (A U C). So, starting with (BUA) n (BUA), we can apply the identity to get (BUA) n (BUA) = (B n A) U (A n A) = B n A = A.

(c) To prove AnB = AUB, we can use the identity: A n (B U C) = (A n B) U (A n C). So, starting with AnB, we can apply the identity to get AnB = A n (B U A) = (A n B) U (A n A) = (A n B) U A = AUB.

(d) To prove An (AUB) = ĀnB, we can use the identity: A n (B U C) = (A n B) U (A n C). So, starting with An (AUB), we can apply the identity to get An (AUB) = (A n A) U (A n B) = ĀnB.

(e) To prove AU (A n B) = ĀUB, we can use the identity: A U (B n C) = (A U B) n (A U C). So, starting with AU (A n B), we can apply the identity to get AU (A n B) = (A U A) n (A U B) = ĀUB.

(f) To prove An (BB) = 0, we can use the identity: A n A' = 0. So, starting with An (BB), we can apply the identity to get An (BB) = An (B n B') = An (B n B) = An (B) = 0.

(g) To prove AU (B U B) = 0, we can use the identity: A U A' = A'. So, starting with AU (B U B), we can apply the identity to get AU (B U B) = (AU B) U (AU B) = B' U B' = 0.