Definition:Cartesian Product

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 the definition of a 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.

Also see

 * Cartesian products of algebraic structures:
 * External Direct Product
 * Internal Direct Product
 * (External) Group Direct Product
 * Internal Group Direct Product