Proof of Cartesian product

Question

Let A, B, C be sets with A \ne ∅. Prove that if A × B = A × C, then B = C. Is the statement still true if A = ∅? Prove your answer.

Answer 1

Fix $a\in A\ne\emptyset$ so$$x\in B\iff(a,\,x)\in A\times B\iff(a,\,x)\in A\times C\iff x\in C.$$Any $B\ne C$ gives a counterexample for $A=\emptyset$, since then $A\times B=A\times C=\emptyset$.

Answer 2

No.

Counterexample:

$B = \{ 0 \}$, $C = \{ 1 \}$

With $A = \emptyset$, we then have that $A \times B = A \times C = \emptyset$, but $B \neq C$