For any sets A and B. show that `P(A nnB) = P(A)nn P(B)dot`

Explanation

To prove that \( P(A \cap B) = P(A) \cap P(B) \), we will show that \( P(A \cap B) \subseteq P(A) \cap P(B) \) and \( P(A) \cap P(B) \subseteq P(A \cap B) \).

### Step 1: Prove \( P(A \cap B) \subseteq P(A) \cap P(B) \)

1. Let \( C \in P(A \cap B) \).
2. By definition of power set, this means \( C \subseteq A \cap B \).
3. Since \( C \subseteq A \cap B \), it follows that \( C \subseteq A \) and \( C \subseteq B \).
4. Therefore, \( C \in P(A) \) and \( C \in P(B) \).
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