---
tag: 0SNQEF7Z53
kind: definition
name: Bijective function
msc: ["03E20"]
defines: 0GG9TVW90M
uses: [0PVG1CRAJN, 0XKH881BZ7]
sources:
  - url: https://en.wikipedia.org/wiki/Bijection
    locator: lead section
---

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is bijective when it is [[injective|0PVG1CRAJN]] and [[surjective|0XKH881BZ7]].
Equivalently, for each $b \in B$, there is exactly one $a \in A$ with $f(a) = b$.

## Remarks

Other names are bijection and one-to-one correspondence.
The function $x \mapsto 2x + 1$ on $\mathbb{R}$ is bijective.
A function is bijective if and only if it has an [[inverse function|0W766JAPSN]]. See [[bijective iff it has a two-sided inverse|01VG2A0BHW]].
Two finite sets have the same number of elements exactly when there is a bijection between them.

## Explanations

### L1 human

A bijection pairs the elements of $A$ and the elements of $B$ perfectly.
Each element of $A$ has one partner in $B$, and each element of $B$ has one partner in $A$.