Definition:Cardinality/Finite

Definition
Let $S$ be a finite set.

The cardinality $\card S$ 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:
 * $\card S = n$

Also note that from the definition of finite:
 * $\exists n \in \N: \card S = n \iff S$ is finite.

Also denoted as
Some sources indicate that $S$ is finite by writing:
 * $\card S < \infty$

Also see

 * Definition:Cardinal
 * Definition:Set Equivalence


 * Cardinality of Finite Set is Well-Defined


 * Set Equivalence behaves like Equivalence Relation: to show that $\card S = n$, it is sufficient to show that it is equivalent to a set already known to have $n$ elements.