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 $$\left|{S}\right|$$.

If $$S$$ is finite, then:


 * $$\left|{S}\right| \ \stackrel {\mathbf {def}} {=\!=} \ S \sim \N_n$$

where $$\N_n$$ is the subset of natural numbers $$\left\{{0, 1, 2, \ldots, n-1}\right\}$$.

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:
 * $$\exists n \in \N: \left|{S}\right| = n \iff S$$ is finite.

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

Notational Variants
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.

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.

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

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