> [!NOTE] Definition/Theorem (Cartesian product) > Let $X$ and $Y$ be [[Sets|sets]]. Then their Cartesian product is defined by the following set of [[Ordered pair|ordered pairs]] $A\times B=\{x\in\mathcal{P}(\mathcal{P}(A\cup B))\mid\exists a\in A\left.\exists b\in B:x=(a,b)\right\}.$ The [[Zermelo Frankel set theory (ZFC)|axiom of choice]] asserts the cartesian product of non-empty sets is non-empty.