Cartesian product proof with counterexample

Question

I was asked to disprove the following statement by counterexample: Let A, B and C be sets. If A x C = B x C then A = B

I was under the impression that:

(x1, y1) = (x2, y2) if and only if x1 = x2 and y1 = y2.

So with this definition I can't really think of any counterexamples to disprove this. Can you please hint at or provide me with a counterexample?

Please notice that your question does not address the field of proof theory which is about formal proofs and proof techniques. — rindPHI, Sep 24 '14 at 07:18

score 0 · Answer 1 · answered Sep 24 '14 at 07:17

As Eoin already said in the above comment, consider the case that $C$ is empty. So we get the following easy counterexample:

Let $A:=\{1\},B:=\{2\},C:=\emptyset$. Then $$A\times C=\{(a,c):a\in A\land c\in C\}=\emptyset =\{(b,c):b\in B\land c\in C\}=B\times C,$$ just because there is no $c\in C$. But obviously $A\neq B$! So the implication is not valid.

Cartesian product proof with counterexample

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