Definition:Cartesian Product/Countable

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Definition

Let $\sequence {S_n}_{n \mathop \in \N}$ be an infinite sequence of sets.

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

$\displaystyle \prod_{k \mathop = 1}^\infty S_k = \set {\tuple {x_1, x_2, \ldots, x_n, \ldots}: \forall k \in \N: x_k \in S_k}$


It defines the concept:

$S_1 \times S_2 \times \cdots \times S_n \times \cdots$

Thus $\displaystyle \prod_{k \mathop = 1}^\infty S_k$ is the set of all infinite sequences $\tuple {x_1, x_2, \ldots, x_n, \ldots}$ with $x_k \in S_k$.


Also see

  • Results about Cartesian products can be found here.


Source of Name

This entry was named for René Descartes.


Sources