\begin{definition}[Definition:Cartesian Product/Cartesian Space]
Let $S$ be a set.
The '''cartesian $n$th power of $S$''', or '''$S$ to the power of $n$''', is defined as:
:$\ds S^n = \prod_{k \mathop = 1}^n S = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S}$
Thus $S^n = \underbrace {S \times S \times \cdots \times S}_{\text{$n$ times} }$
Alternatively it can be defined recursively:
:$S^n = \begin{cases}
S: & n = 1 \\
S \times S^{n - 1} & n > 1
\end{cases}$
The set $S^n$ called a '''cartesian space'''.
An element $x_j$ of an ordered tuple $\tuple {x_1, x_2, \ldots, x_n}$ of a '''cartesian space''' $S^n$ is known as a basis element of $S^n$.
\end{definition}