For sets A and B, show that: P(A B) = P(A) P(B) - Vidyadip

Question

For sets A and B, show that: $P(A \cap B) = P(A) \cap P(B)$

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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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