Definition:Cardinality

Two sets (either finite or infinite) which are equivalent are said to have the same cardinality.

The cardinality of a set $$S$$ is written $$\left|{S}\right|$$.

If $$S$$ is finite, then:

$$\mathbf {Define:} \ \left|{S}\right| \ \stackrel {\mathbf {def}} {=\!=} \ S \sim \mathbb{N}_n$$

That is, if $$S$$ is finite, $$\left|{S}\right|$$ is the number of elements in $$S$$.

By Set Equivalence an Equivalence Relation, to show that $$\left|{S}\right| = n$$, it is sufficient to show that it is equivalent to a set already known to have $$n$$ elements.

Also note that from the definition of finite, $$\exists n \in \mathbb{N}: \left|{S}\right| = n \iff S$$ is finite.