$A\neq \varnothing $, $B\neq \varnothing$, $A\neq B.$ Prove $A\times B \neq B\times A$

Question

$$A\neq \varnothing ,B\neq \varnothing,A\neq B. \\\text{Prove }A\times B \neq B\times A$$

I'm pretty sure this has to do with inverse for relations. But I'm not sure how to begin proofing something like this. Help is appreciated!

Answer 1

Without loss of generality, assume that there is $x\in A$, $x\notin B$. $B$ is not empty, so there is $y\in B$.

Then $(x,y)\in A\times B$ and $(x,y)\notin B\times A$.

Answer 2

HINT: Since $A\neq B$, there is some $a\in A$ such that $a\notin B$, or there is some $b\in B$ such that $b\notin A$. Assume that the first case holds. Since $B$ is not empty, there is some $b\in B$. Now you have two elements from $A$ and $B$, use them to produce a witness for the inequality you are trying to prove.