Comparation of two sets that use cartesian product If $A$, $B$ and $C$ are sets and $C = \\{ S \times T : (S,T) \in \mathcal P(A) \times \mathcal P(B)\\}$, are $P(A \times B)$ and $C$ are always equal? Is one always a subset of the other or there is no "⊆" relation between them? Thanks!

Well, $C$ is the set of Cartesian products $S\times T$ where $S\subseteq A$ and $T\subseteq B.$ Take any $S\times T\in C.$ If $(x,y)\in S\times T,$ then $x\in S\subseteq A$ and $y\in T\subseteq B,$ so $x\in A$ and $y\in B,$ so $(x,y)\in A\times B$. Hence, $S\times T\subseteq A\times B,$ and so $S\times T\in\mathcal P(A\times B).$ This holds for all $S\times T\in C,$ so that $C\subseteq\mathcal P(A\times B).$

The reverse inclusion need not hold in general, and will _fail_ to hold precisely when $A,B$ both have more than one element. In particular, suppose $a_1,a_2\in A$ are distinct and $b_1,b_2\in B$ are distinct. Now consider the set $\bigl\\{(a_1,b_1),(a_2,b_2)\bigr\\}.$ Does this set lie in $C$? In $\mathcal P(A\times B)$?