I have a question concerning a assignement we got at university:
$A, B, C$ are sets, and $A\neq B$
$A\times C=B\times C$
What I'm supposed to prove, is that it's only true if $C=\emptyset$ .
I have seen the proof (it's proven with contraposition), I understand it BUT I don't understand the interpretation of it.
What we know is, that $A\times C=\emptyset$ (whereas $C=\emptyset$), analog for $B$. This means that it doesn't matter what sets $A$ and $B$ are, since $C$ is an empty set. So why does the proof state, that the equation is only true if $A\neq B$ ?
Thank you in advance!