Does equality of cartesian products imply equality for their components?

Question

Prove or disprove: If $A \mathsf x A=B\mathsf xB$ then $A=B$

So I started letting $a \in A$ so then $(a,a) \in A \mathsf x A$ and since $A \mathsf x A=B\mathsf xB$ we know $(a,a) \in B \mathsf x B$ but from here can I make the assumption that $A=B$?

Answer 1

You can't quite make the assumption yet, however since $(a,a) \in B\times B$ we know that $a \in B$ (by definition of $B \times B$) and so $A \subseteq B$. Similarly you can show that $B \subseteq A$ and so $A = B$.

Answer 2

If $a \in A$ then $(a,a) \in A\times A = B\times B$ so $a \in B$ and $A \subset B$.

If $b \in B$ then $(b,b) \in A\times B = A\times A$ so $b \in B$ and $B \subset A$.

So $A = B$.

....

It's even stronger $C\times D = A\times B$ and none of $A,B,C,D$ are empty, then $C = A$ and $B = D$