Which of the following is correct, if P(A) is power set of the set A?

A
$P (A) \cup P (B) = P (A \cup B)$
B
$P (A) \cap P (B) = P (A \cap B)$
C
$P (A) - P (B) = P (A - B)$
D
None of these

Solution:

$A \cap B \subseteq A and A \cap B \subset B$

$\begin{aligned} \Rightarrow P (A \cap B) \subseteq P (A) and P (A \cap B) \subseteq P (B) \end{aligned}$

$\Rightarrow P (A \cap B) \subseteq P (A) \cap P (B)$ ..(i)

Now, $C \in P (A) \cap P (B)$

$\begin{array}{l} \Rightarrow C \in P (A) and C \in P (B) \Rightarrow C \subset A and C \subseteq B \\ \Rightarrow C \subseteq A \cap B \Rightarrow C \in P (A \cap B) \end{array}$

$\Rightarrow P (A) \cap P (B) \subseteq P (A \cap B)$ …(ii)

From (i) ,from (ii),

$P (A \cap B) = P (A) \cap P (B)$

Which of the following is correct, if $P (A)$ is power set of the set A?

Solution:

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