Cartesian productCartesianProduct2001-10-19 00:41:332006-10-12 08:10:51Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}For any sets $A$ and $B$, the {\em Cartesian product} $A \times B$ is the set consisting of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$.
The Cartesian product satisfies the following properties, for all sets $A$, $B$, $C$, and $D$:
\begin{itemize}
\item $A\times \emptyset = \emptyset$
\item $(A \times B) \cap (C \times D) = (A\cap C) \times (B\cap D)$
\item $(A \times B)^\complement = (A^\complement \times B^\complement)
\cup (A^\complement \times B)
\cup (A \times B^\complement)$
\end{itemize}
Here $\emptyset$ denotes the empty set, $\cap$ denotes intersection, $\cup$ denotes union, and ${}^\complement$ denotes complement with respect to some universal set $U$ containing $A$ and $B$.