Cardinal number

In mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set ⁠

A

{\displaystyle A}

⁠ is generally denoted by ⁠

|

A

|

{\displaystyle \vert A\vert }

⁠, with a vertical bar on each side, though it may also be denoted by

A

{\displaystyle A}

,

card

⁡

(

A

)

,

{\displaystyle \operatorname {card} (A),}

or

#

A

.

{\displaystyle \#A.}

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. The cardinality of a finite set can be identified with a natural number, which can be found simply by counting its elements. For example, the sets ⁠

{

1

,

2

,

3

}

{\displaystyle \{1,2,3\}}

⁠ and ⁠

{

4

,

5

,

6

}

{\displaystyle \{4,5,6\}}

⁠ both have the same cardinality 3, as evidenced by the bijection ⁠

{

1

↦

4

,

2

↦

5

,

3

↦

6

}

{\displaystyle \{1\mapsto 4,2\mapsto 5,3\mapsto 6\}}

⁠.

The behavior of cardinalities of infinite sets is more complex. For example, there exists a bijection between the set of all natural numbers ⁠

N

{\displaystyle \mathbb {N} }

⁠ and the set of all rational numbers ⁠

Q

{\displaystyle \mathbb {Q} }

⁠, and thus ⁠

|

N

|

=

|

Q

|

{\displaystyle \vert \mathbb {N} \vert =\vert \mathbb {Q} \vert }

⁠ even though ⁠

N

{\displaystyle \mathbb {N} }

⁠ is a proper subset of ⁠

Q

{\displaystyle \mathbb {Q} }

⁠—something that cannot happen with proper subsets of finite sets. However, a fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers ⁠

R

{\displaystyle \mathbb {R} }

⁠ is greater than the cardinality of ⁠

N

{\displaystyle \mathbb {N} }

⁠.

The cardinality of ⁠

N

{\displaystyle \mathbb {N} }

⁠ is usually denoted by ⁠

ℵ

0

{\displaystyle \aleph _{0}}

⁠ (aleph-null), since it is the smallest aleph number.