\begin{definition}[Definition:Cartesian Product]
Let $S$ and $T$ be sets or classes.
The '''cartesian product''' $S \times T$ of $S$ and $T$ is the set (or class) of ordered pairs $\tuple {x, y}$ with $x \in S$ and $y \in T$:
:$S \times T = \set {\tuple {x, y}: x \in S \land y \in T}$
Another way of defining it is by:
:$\tuple {x, y} \in S \times T \iff x \in S, y \in T$
More specifically:
:$\forall p: \paren {p \in S \times T \iff \exists x: \exists y: x \in S \land y \in T \land p = \tuple {x, y} }$
$S \times T$ can be voiced '''$S$ cross $T$'''.
\end{definition}