# 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