For any sets A and B, show that P ( A ∩ B ) = P ( A ) ∩ P ( B ).

 For any sets A and B, show that

P ( A ∩ B ) = P ( A ) ∩ P ( B ).



Answer.

To show that P(A ∩ B) = P(A) ∩ P(B), we need to show that every element in P(A ∩ B) is in P(A) ∩ P(B), and vice versa.

First, let’s show that every element in P(A ∩ B) is in P(A) ∩ P(B). Suppose X is an element of P(A ∩ B), i.e., X is a subset of A ∩ B. This means that every element in X is also in both A and B. Therefore, X is a subset of A, and X is a subset of B. This implies that X is an element of P(A) and also an element of P(B). Hence, X is an element of P(A) ∩ P(B).

Next, let’s show that every element in P(A) ∩ P(B) is in P(A ∩ B). Suppose Y is an element of P(A) ∩ P(B), i.e., Y is a subset of A and also a subset of B. This means that every element in Y is in both A and B. Therefore, Y is a subset of A ∩ B. Hence, Y is an element of P(A ∩ B).

Therefore, we have shown that every element in P(A ∩ B) is in P(A) ∩ P(B), and vice versa. Hence, we can conclude that P(A ∩ B) = P(A) ∩ P(B).

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