Definition:Cartesian Product

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 $$T$$.

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.

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:

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

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 Power of a Set
Let $$S$$ be a set.

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

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

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

Alternatively it can be defined recursively:

$$ S^n = \begin{cases} S: & n = 1 \\ S \times S^{n-1} & n > 1 \end{cases} $$

Cartesian Product of Reals
When $$S$$ is the set of real numbers $$\mathbb{R}$$, the cartesian product takes on a special significance.

Let $$n \in \mathbb{N}^*$$.

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

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

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

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

It can be shown that:
 * $$\mathbb{R}^2$$ is isomorphic to any infinite flat plane in space;
 * $$\mathbb{R}^3$$ is isomorphic to the whole of space itself.