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

Solution:

$A\cap B\subseteq A\text{ and }A\cap B\subset B$

   $\begin{array}{r}\Rightarrow P(A\cap B)\subseteq P\left(A\right)\text{ and }P(A\cap B)\subseteq P\left(B\right)\\ \end{array}$

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

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

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

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

From (i) ,from  (ii), 

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