Definition:Cartesian Product/Cartesian Space

Definition
Let $S$ be a set.

Then the cartesian $n$th power of $S$, or $S$ to the power of $n$, is defined as:


 * $\displaystyle S^n = \prod_{k \mathop = 1}^n S = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S}\right\}$

Thus $S^n = \underbrace{S \times S \times \cdots \times S}_{n \text{ times} }$

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 $\left({x_1, x_2, \ldots, x_n}\right)$ of a cartesian space $S^n$ is known as a basis element of $S^n$.

Real Cartesian Space
When $S$ is the set of real numbers $\R$, the cartesian product takes on a special significance.