Q:

Power set of cartesian product where is {∅}

X = {∅} × {{~}, *} = {(∅, {~}), (∅, *)}
P(X) = {∅, {(∅, {~})}, {(∅, *)}, {(∅, {~}), (∅, *)}}
is that correct?

A:

The set $X$ consists of $n=2$ elements. Call them $a$ and $b$. The power set ${\cal P}(X)$ then consists of $2^n=4$ elements, namely
$${\cal P}(X)=\bigl\{\emptyset, \{a\},\{b\},\{a,b\}\bigr\}\ .$$
That's exactly what you have written.

