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

The cardinality of a set $S$ is written $\card S$.

Cardinality of Finite Set

Let $S$ be a finite set.

The cardinality $\card S$ of $S$ is the number of elements in $S$.

That is, if:

$S \sim \N_{< n}$


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

then we define:

$\card S = n$



\(\displaystyle S_1\) \(=\) \(\displaystyle \set {-1, 0, 1}\) $\quad$ $\quad$
\(\displaystyle S_2\) \(=\) \(\displaystyle \set {x \in \Z: 0 < x < 6}\) $\quad$ $\quad$
\(\displaystyle S_3\) \(=\) \(\displaystyle \set {x^2 - x: x \in S_1}\) $\quad$ $\quad$
\(\displaystyle S_4\) \(=\) \(\displaystyle \set {X \in \powerset {S_2}: \card X = 3}\) $\quad$ $\quad$
\(\displaystyle S_5\) \(=\) \(\displaystyle \powerset \O\) $\quad$ $\quad$

Cardinality of $S_1 = \set {-1, 0, 1}$

The cardinality of $S_1$ is given by:

$\card {S_1} = 3$

Cardinality of $S_2 = \set {x \in \Z: 0 < x < 6}$

The cardinality of $S_2$ is given by:

$\card {S_2} = 5$

Cardinality of $S_3 = \set {x^2 - x: x \in S_1}$

The cardinality of $S_3$ is given by:

$\card {S_3} = 2$

Cardinality of $S_4 = \set {X \in \powerset {S_2}: \card X = 3}$

The cardinality of $S_4$ is given by:

$\card {S_4} = 10$

Cardinality of $S_5 = \powerset \O$

The cardinality of $S_5$ is given by:

$\card {S_5} = 1$

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, particularly in the context of finite sets.

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 $\operatorname{Card} \left({S}\right)$ or $\operatorname{card} \left({S}\right)$.

1968: A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis use $m \left({A}\right)$ for the power of the set $A$.

Also see