# 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 defined as

Some authors, working to a particular mathematical agenda, do not discuss the **cardinality of an infinite set**, and instead limit their definition of this concept to the **finite case**.

Some others gloss over the definition for the **cardinality of a finite set**, perhaps on the understanding that the definition is trivial, and instead raise the concept only in the **infinite case**.

## Also known as

Some authors prefer the term **order** instead of **cardinality**, particularly in the context of finite sets.

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 sources cut through all the complicated language and call it the **size**.

Some sources use $\map {\#} S$ (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 $\map C S$, but this is easy to confuse with other uses of the same or similar notation.

A clear but relatively verbose variant is $\Card \paren S$ or $\operatorname{card} \paren S$.

1968: A.N. Kolmogorov and S.V. Fominâ€Ž: *Introductory Real Analysis* use $\map m A$ for the **power** of the set $A$.

Further notations are $\map n A$ and $\overline A$.

## Also see

- 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.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 13$: Arithmetic - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2.5$: The power of a set - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $8$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(e)}$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.6$: Cardinality - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 6$: Finite Sets - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.5$: Definition $\text{A}.25$