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$ \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:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article