---
tag: 08EJHHMB5R
kind: definition
name: Equinumerous sets
msc: ["03E10"]
defines: 0B0KR7FH2Q
uses: [0SNQEF7Z53]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinality
    locator: Equinumerosity
---

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The sets $A$ and $B$ are equinumerous, written $A \approx B$, when there is a [[bijective function|0SNQEF7Z53]] $f \colon A \to B$.

## Remarks

Equinumerosity is reflexive, symmetric and transitive. The identity, the inverse and the composition of bijections give the three properties.
The set $\mathbb{N}$ is equinumerous with the set of even natural numbers, by $n \mapsto 2n$. So an infinite set can be equinumerous with a proper subset.
Other names are "equipotent" and "have the same [[cardinality|0JTGRV4WT4]]".

## Explanations

### L1 human

You do not need to count to compare sizes.
If each chair in a room has one person and each person has one chair, there are as many people as chairs.
Equinumerosity applies the same idea to infinite sets.