---
tag: 0BBTH3YSZA
kind: definition
name: Finite set
msc: ["03E10"]
defines: 070R982G4B
uses: [08EJHHMB5R]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_set
    locator: lead section
---

## Definition

A [[set|0SJ3RX9TXA]] $A$ is finite when there is a natural number $n$ such that $A$ is [[equinumerous|08EJHHMB5R]] with $\{1, 2, \dots, n\}$.
For $n = 0$, this set is the [[empty set|0ECVK59Y03]].
The number $n$ is unique, and it is the number of elements of $A$.

## Remarks

Subsets, finite unions, finite products and the power set of finite sets are finite.
A finite set is not equinumerous with any proper subset of itself. This is the pigeonhole principle.
A set that is not finite is [[infinite|01MKW4R283]].

## Explanations

### L1 human

A set is finite when you can count its elements and stop.
The set of planets of the Solar System is finite, and so is the set of primes less than $100$.