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Cartesian Product**

Given sets \(X,Y\), their Cartesian product \(X\times Y\) consists of all ordered pairs \((x,y)\) such that \(x\in X\) and \(y\in Y\) (by 'ordered pair' one means that \((x,y)=(x',y')\Leftrightarrow(x=x'\land y=y')\)). This set is constructible within theZermelo-Fraenkel axioms of set theory by identifying the set \(\big\{\{x\},\{x,y\}\big\}\) with the pair \((x,y)\), so that \(X\times Y=\left\{S\in\mathcal{P}(\mathcal{P}(X\cup Y))\middle|\exists x\in X:\exists y\in Y:S=\big\{\{x\},\{x,y\}\big\}\right\}\).