Order of External Direct Product

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Theorem

Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.

Then the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$.


Proof

By definition the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$ is the cardinality of the underlying set $S \times T$.

The result follows directly from Cardinality of Cartesian Product.

$\blacksquare$


Sources