Definition:Cartesian Product/Cartesian Space

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Definition

Let $S$ be a set.

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

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


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

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 $S^n$.


Two Dimensions

$n = 2$ is frequently taken as a special case:


The cartesian $2$nd power of $S$ is:

$\displaystyle S^2 = S \times S = \set {\tuple {x_1, x_2}: x_1, x_2 \in S}$


Thus $S^2 = S \times S$


The set $S^2$ called a cartesian space of $2$ dimensions.


Family of Sets

Let $I$ be an indexing set.

Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be an family of sets indexed by $I$.

Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$.


Let $S$ be a set such that:

$\forall i \in I: S_i = S$

Then $\displaystyle \prod_{i \mathop \in I} S_i$ can be denoted $S^I$ and is the Cartesian $I$-space of $S$.


Real Cartesian Space

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


Let $n \in \N_{>0}$.

Then $\R^n$ is the cartesian product defined as follows:

$\displaystyle \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$


Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:

$\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$


Source of Name

This entry was named for René Descartes.


Sources