Definition:Cartesian Product/Finite

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:


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

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

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

In particular:
 * $\displaystyle \prod_{k \mathop = 1}^2 S_k = S_1 \times S_2$

Infinite Sequence
The same notation can be used to define the cartesian product of an infinite sequence:

Also known as
The concept $\displaystyle \prod_{k \mathop = 1}^n S_k$ is also seen defined as the direct product of $\left \langle {S_n} \right \rangle$.

Also see

 * Generalized Cartesian products of algebraic structures:
 * Definition:External Direct Product/General Definition
 * Definition:Internal Direct Product/General Definition
 * Definition:Group Direct Product/General Definition
 * Definition:Internal Group Direct Product/General Definition