Definition:Cartesian Product/Finite

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Definition

Let $\sequence {S_n}$ be a sequence of sets.

The cartesian product of $\sequence {S_n}$ is defined as:

$\ds \prod_{k \mathop = 1}^n S_k = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S_k}$


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 $\tuple {x_1, x_2, \ldots, x_n}$ with $x_k \in S_k$.


In particular:

$\ds \prod_{k \mathop = 1}^2 S_k = S_1 \times S_2$


Also known as

The concept $\ds \prod_{k \mathop = 1}^n S_k$ is also seen defined as the direct product of $\sequence {S_n}$.


Some sources use the notation $\huge {\boldsymbol \times}$ instead of $\ds \prod$.


Also see

  • Results about Cartesian products can be found here.


Source of Name

This entry was named for René Descartes.


Sources

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