Definition:Cardinality/Finite

Definition
Let $S$ be a finite set.

The cardinality $\left\vert{S}\right\vert$ of $S$ is the number of elements in $S$.

That is, if:
 * $S \sim \N_{< n}$

where:
 * $\sim$ denotes set equivalence
 * $\N_{< n}$ is the set of all natural numbers less than $n$

then we define:
 * $\left\vert{S}\right\vert = n$

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

Also known as
Some authors prefer the term order instead of cardinality.

Also see

 * Definition:Cardinal


 * Cardinality of Finite Set is Well-Defined


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