Cardinality of Cartesian Product/General Result

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Theorem

Let $\displaystyle \prod_{k \mathop = 1}^n S_k$ be the cartesian product of a (finite) sequence of sets $\left \langle {S_n} \right \rangle$.


Then:

$\left|{\displaystyle \prod_{k \mathop = 1}^n S_k}\right| = \displaystyle \prod_{k \mathop = 1}^n \left|{S_k}\right|$


This can also be written:

$\left|{S_1 \times S_2 \times \ldots \times S_n}\right| = \left|{S_1}\right| \times \left|{S_2}\right| \times \ldots \times \left|{S_n}\right|$


Corollary

Let $S^n$ be a cartesian space.


Then:

$\left|{S^n}\right| = \left|{S}\right|^n$


Proof