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 $\sequence {S_n}$.


Then:

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


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:

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


Proof