How do I prove the following statement about the complement of a cartesian product?

Question

How do I prove that this statement is true?

$$(A\times B)^C=\left(A^C\times B\right)\cup\left(A\times B^C\right)\cup\left(A^C\times B^C\right)$$

Answer 1

Just consider the element method. Suppose $(x,y)\in \left(A\times B\right)^C$. Then either $x\not\in A$, or $y\not\in B$. This leaves you three possibilities: $x\not\in A$ and $y\in B$, $x\in A$ and $y\not\in B$, or $x\not\in A$ and $y\not\in B$ (and one of these must be true). Respectively these three possibilities correspond to the sets $\left(A^C\times B\right)$, $\left(A\times B^C\right)$, and $\left(A^C\times B^C\right)$, so taking the union of these sets gives you the same thing as $\left(A\times B\right)^C$.