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:

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


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 $S^n$.


Two Dimensions

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


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

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


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


Three Dimensions

$n = 3$ is another special case:


The cartesian $3$rd power of $S$ is:

$S^3 = S \times S \times S = \set {\tuple {x_1, x_2, x_3}: x_1, x_2, x_3 \in S}$


The set $S^3$ called a cartesian space of $3$ dimensions.


Family of Sets

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

Let $\ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.

Let $S$ be a set such that:

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


Definition 1

The Cartesian space of $S$ indexed by $I$ is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S$ for each $i \in I$:

$S_I := \ds \prod_I S = \set {\family {s_i}_{i \mathop \in I}: s_i \in S}$


Definition 2

The Cartesian space of $S$ indexed by $I$ is defined and denoted as:

$\ds S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$


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:

$\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$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

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