Definition:Cartesian Product

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Definition

The cartesian product (or Cartesian product) of two sets $S$ and $T$ is the set of ordered pairs $\left({x, y}\right)$ with $x \in S$ and $y \in T$.

This is denoted:

$S \times T = \left\{{\left({x, y}\right) : x \in S \land y \in T}\right\}$

Some authors call this the direct product of $S$ and $T$.

Some call it the cartesian product set, others just the product set.


Another way of defining it is by:

$\left({x, y}\right) \in S \times T \iff x \in S, y \in T$


It is also known as the cross product of two sets, but this can be confused with other usages of this term.


Factors

In a cartesian product $S \times T$, the sets $S$ and $T$ are called the factors of $S \times T$.


Generalized Definition

Let $\left \langle {S_n} \right \rangle$ be a sequence of sets.

The cartesian product of $\left \langle {S_n} \right \rangle$ is defined as:

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


It is also denoted $S_1 \times S_2 \times \ldots \times S_n$.

Thus $S_1 \times S_2 \times \ldots \times S_n$ is the set of all ordered $n$-tuples $\left({x_1, x_2, \ldots, x_n}\right)$ with $x_k \in S_k$.


Cartesian Space

Let $S$ be a set.

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

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


Thus $S^n = S \times S \times \ldots \left({n}\right) \ldots \times S$

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 a 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.


Let $n \in \N^*$.

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

$\displaystyle \R^n = \R \times \R \times \cdots \left({n}\right) \cdots \times \R = \prod_{k=1}^n \R$


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

$\R^n = \left\{{\left({x_1, x_2, \ldots, x_n}\right): x_1, x_2, \ldots, x_n \in \R}\right\}$

It can be shown that:

  • $\R^2$ is isomorphic to any infinite flat plane in space;
  • $\R^3$ is isomorphic to the whole of space itself.


See Real Vector Space.


Axiomatic Set Theory

The concept of the cartesian product is shown in Kuratowski Formalization of Ordered Pair to be constructible from the Zermelo-Fraenkel axioms.


Notes

The notation for the cartesian power of a set $S^n$ should not be confused with the notation used for the conjugate of a set.

Also beware not to confuse the name of the concept itself with that of the power set $\mathcal P \left({S}\right)$ of $S$.


Also see


  • Results about Cartesian products can be found here.


Source of Name

This entry was named for René Descartes.


Sources

Retrieved from "http://www.proofwiki.org/w/index.php?title=Definition:Cartesian_Product&oldid=76295#Cartesian_Space"
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