Critique my proof of: Suppose $A$ and $B$ are sets. Then $A \times B = B \times A \iff A = \emptyset, B = \emptyset,$ or $A = B$

Solution 1:

For the left-to-right direction, you cannot start by letting $P$ be an arbitrary element of $A \times B$, because you are not proving a statement of the form "for all $P$ in $A \times B,\ \ldots$." Your strategy was to prove $A \subseteq B$ and $B \subseteq A$, so that means your proof should have looked like this: "Let $x$ be an arbitrary element of $A$. (Proof of $x \in B$ goes here.) Therefore $A \subseteq B$. Now let $y$ be an arbitrary element of $B$. (Proof of $y \in A$ goes here.) Therefore $B \subseteq A$."

You can tell that there is something wrong with your proof because your proof never used the assumption that $A \ne \varnothing$ and $B \ne \varnothing$, but those assumptions are necessary. If you fill in the proof outline above, you will find that you need to use those assumptions to complete the proof.