---
tag: 0PVG1CRAJN
kind: definition
name: Injective function
msc: ["03E20"]
defines: 09NWETVTNY
uses: [0XHDTE5YMZ]
sources:
  - url: https://en.wikipedia.org/wiki/Injective_function
    locator: lead section
---

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is injective when, for all $a, a' \in A$, $f(a) = f(a')$ implies $a = a'$.

## Remarks

Other names are one-to-one function and injection.
The function $x \mapsto 2x$ on $\mathbb{R}$ is injective, and $x \mapsto x^2$ on $\mathbb{R}$ is not, because $f(1) = f(-1)$.
An injective function with a nonempty domain has a [[left inverse|02486SD328]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].

## Explanations

### L1 human

An injective function never sends two inputs to the same place.
To prove injectivity, assume $f(a) = f(a')$ and deduce $a = a'$.