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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 $B$.
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