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!

  • 2
    $\begingroup$ What are you supposed to prove? $\endgroup$ – Patrick Stevens Dec 4 '16 at 11:49
  • $\begingroup$ Please edit your question. $\endgroup$ – Hans Hüttel Dec 4 '16 at 11:54
  • $\begingroup$ I edited it and i added what we have to prove. $\endgroup$ – Yalom Dec 4 '16 at 11:59
  • $\begingroup$ Your edit is not particularly clear; you speak of an "it" and it is not immediately clear what the "it" refers to. $\endgroup$ – Hans Hüttel Dec 4 '16 at 12:03

Of course we also have that $A \times C = B \times C$ if $A=B$. But in the problem we are assuming that $A\neq B$.

Cf. the usual property of $0$ wrt. multiplication for real numbers: That if $x \neq y$, then $x \cdot k = y \cdot k$ if and only if $k =0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.