prove $A=B \iff A \times A = B \times B$ Tried to prove it. wasn't 100% sure I did everything correctly so I would appreciate a lot your advice. this is what I've done:

This is basically fine.  But you don't even need to prove anything for part (a).  If $A= B$ then you can substitute $B$ in for $A$ in any expression and the two expressions will be equal.  $A$ and $B$ are exactly the same, so all you're doing is calling $A$ by a different name.  Also, I would try to write in complete, grammatical sentences.  

Suppose $a \in A$.  Then $(a,a) \in A \times A$, so ...

