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.

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

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.

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:


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

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:


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

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:


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

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


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

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.