Definition:Cardinality

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

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

If $S \sim \N_n$ where $\sim$ denotes set equivalence and $\N_n$ is the set of all natural numbers less than $n$, then we define:


 * $\left\vert{S}\right\vert = n$

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

By 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.

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

The cardinality of an infinite set is often denoted by an aleph number ($\aleph_0, \aleph_1, \ldots$) or a beth number ($\beth_0, \beth_1, \ldots$).

Cardinality of Natural Numbers
When the natural numbers are defined as elements of a Minimal Infinite Successor Set, the cardinality function can be viewed as the identity mapping on $\N$.

That is:
 * $\forall n \in N: \left|{n}\right| := n$

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

Other authors say that two sets that are equivalent have the same power. Compare equipotent as mentioned in the definition of set equivalence.

Georg Cantor used the term power and equated it with the term cardinal number, using the notation $\overline {\overline M}$ for the cardinality of $M$.

Some just cut through all the complicated language and call it the size.

Some sources use $\# \left({S}\right)$ (or a variant) to denote set cardinality. This notation has its advantages in certain contexts, and is used on occasion on this website.

Others use $C \left({S}\right)$, but this is easy to confuse with other uses of the same or similar notation.

A clear but relatively verbose variant is $\operatorname{Card} \left({S}\right)$.

use $m \left({A}\right)$ for the power of the set $A$.

Also see

 * Definition:Cardinal


 * Cardinality of Finite Set is Well-Defined