Cartesian Product

Definition 1. The Cartesian product of two sets $X$ and $Y$ is

\[X \times Y = \{(x, y) : x \in X \,\land\, y \in Y\}.\]

This is a set by Separation Schema:

\[X \times Y \subseteq \powerset(\powerset(X \cup Y)).\]

For any sets $A$, $B$, $C$, $D$, etc., we write

\[\begin{align*} A \times B \times C &= (A \times B) \times C, \\ A \times B \times C \times D &= (A \times B \times C) \times D, \\ &\text{etc.} \end{align*}\]

For any set $X$ and any $n \in \N^+$, we write

\[X^n = \underbrace{X \times \cdots \times X}_{\text{$n$ times}}.\]

Proposition 2. For any set $X$,

\[X \times \empt = \empt \times X = \empt.\]

Proposition 3. For any sets $A$, $B$, $C$ and $D$,

\[(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D).\]

Proposition 4 (Distributivities). For any sets $A$, $B$ and $C$:

1. \[A \times (B \cap C) = (A \times B) \cap (A \times C).\]
2. \[A \times (B \cup C) = (A \times B) \cup (A \times C).\]
3. \[A \times (B \setdif C) = (A \times B) \setdif (A \times C).\]