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 $

Parents

Requirements

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