\begin{definition}[Definition:Cartesian Product/Finite]
Let $\sequence {S_n}$ be a sequence of sets. 
The '''cartesian product''' of $\sequence {S_n}$ is defined as:
:$\ds \prod_{k \mathop = 1}^n S_k = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S_k}$
It is also denoted $S_1 \times S_2 \times \cdots \times S_n$.
Thus $S_1 \times S_2 \times \cdots \times S_n$ is the set of all ordered $n$-tuples $\tuple {x_1, x_2, \ldots, x_n}$ with $x_k \in S_k$.
In particular:
:$\ds \prod_{k \mathop = 1}^2 S_k = S_1 \times S_2$
\end{definition}