# Injection

$\def\Id {\mathop{\rm Id}}$

A function (or mapping) is called injective if distinct arguments have distinct images.

In other words, a function $f : A \to B$ from a set $A$ to a set $B$ is

an injective function or an injection or one-to-one function

if and only if

$a_1 \ne a_2$ implies $f(a_1) \ne f(a_2)$, or equivalently $f(a_1) = f(a_2)$ implies $a_1 = a_2$

for all $a_1, a_2 \in A$.

#### Equivalent conditions

A function $f$ is injective if and only if $f^{-1}(f(S)) = S$ for all subsets $S$ of the domain $A$.

A function $f$ is injective if and only if, for every pair of functions $g,h$ with values in $A$, the condition $f \circ g = f \circ h$ implies $g=h$. (In category theory, this property is used to define monomorphisms.)

A function $f$ is injective if and only if there is a left-inverse function $g$ with $g \circ f = \Id_A$.

#### Related notions

A special case is the inclusion function defined on a subset $A \subset B$ by $f(a)=a$.

A function that is both injective and surjective is called bijective (or, if domain and range coincide, in some contexts, a permutation).

An injective homomorphism is called monomorphism.

Injective mappings that are compatible with the underlying structure are often called embeddings.

How to Cite This Entry:
Injection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injection&oldid=30986
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article