A homework question asks,
Consider any two sets $A$ and $B$. What should the relation between $A$ and $B$ be, so that $( ∩ ) × = × ( ∩ )$. Prove your answer
I am not looking for an answer to the problem, but I’m not understanding what it is asking for. I think it’s asking “is $A$ a subset of $B$, vice versa, etc” but I’m not sure.
The question indeed asks you to find some inclusion relation between $A$ and $B$ such that the relation holds. Or some other condition that you should find.
You can start by observing that if $B=\emptyset$ or $A\cap B=\emptyset$, then the statement holds. Thus you can go on with the assumption that neither set is empty.
What can you conclude from $X\times Y=Y\times X$ if $X$ and $Y$ are not empty?