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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!

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  • 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
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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 $k =0$.

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