Cartesian product

Set

 context $X,Y$ … small set definiendum $p\in X \times Y$ range $x\in X$ range $y\in Y$ postulate $\exists x,y.\,p=\langle x,y\rangle$

Discussion

$X \times Y\subset \mathcal{P}(\mathcal{P}(X \cap Y))$

Definitions

In accordance to the defintion of the n-tuple in Ordered pair, we set

$X_1\times X_2\times X_3 \equiv (X_1\times X_2)\times X_3$

and inductively for

$X_1\times X_1\times X_3\times \ \dots\ \times X_{n-1}\times X_n \equiv ((\dots((X_1\times X_2)\times X_3)\times\ \dots\ )\times X_{n-1})\times X_n$