CBSE BoardEnglish MediumSTD 11 ScienceMathsSets2 Marks
Question
For sets A and B, show that: $P(A \cap B) = P(A) \cap P(B)$
✓
Answer
Let $x \in P(A \cap B)$ $\Rightarrow x \subset (A \cap B)$ $\Rightarrow x \subset A$ and $x \subset B$ $\Rightarrow x \in P(A)$ and $x \in P(B)$ $\Rightarrow x \in P(A) \cap P(B)$ $\Rightarrow x \subset P(A) \cap P(B)$ $\therefore P(A \cap B) \subset P(A) \cap P(B)$. . . (i) Let $x \in P(A) \cap P(B)$ $\Rightarrow x \in P(A)$ and $x \in P(B)$ $ \Rightarrow x \subset A$ and $\Rightarrow x \subset B$ $\Rightarrow x \subset A \cap B$ $ \Rightarrow x \subset P(A \cap B)$ $\therefore P(A) \cap P(B) \subset P(A \cap B)$. . . . (ii) From (i) and (ii), we have $P(A \cup B) = P(A) \cap P(B)$
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