Empty Set $\varnothing$

The elements in $M \cap N$ are sets. A set $Z$ is in $M$ if and only if it is a subset of $A.$ It's in $N$ if and only if it's a subset of $B.$ The only set in both $M$ and $N$ is the only set that is a subset of both $A$ and $B,$ namely the empty set. So $M \cap N = \{ \varnothing \}.$

$M$ is the power set of $A$ , which means $M$ contains all the subsets of $A$ .

$N$ is the power set of $B$ .

Since $A \cap B=\varnothing$ , $A$ and $B$ don't have any same subsets but the empty set.

Therefore, $M \cap N=\{\varnothing\}$ .