For any sets A and B. show that $P\left(A\cap B\right)=P\left(A\right)\cap P\left(B\right)\dot{}$

Solution: let $P\left(A\cap B\right),P\left(A\right),P\left(B\right)$are the power sets of the set $A\cap B,A,B$ respectively
let c is a set where $C\in P\left(A\cap B\right)$
elements of C$\in \left(A\cap B\right)$
so, $C\in A$
$C\in B$
so, \displaystyle{C}\in{P}{\left({A}\right)}&{C}\in{P}{\left({B}\right)}
$C\in \left[P\left(A\right)\cap P\left(B\right)\right]$ = LHS
hence proved