Cardinality of Cartesian Product/General Result

Theorem

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

Then:

$\ds \card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k}$

This can also be written:

$\card {S_1 \times S_2 \times \ldots \times S_n} = \card {S_1} \times \card {S_2} \times \ldots \times \card {S_n}$

Corollary

Let $S^n$ be a cartesian space.

Then:

$\card {S^n} = \card S^n$