Injection

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

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 $.

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

An injective homomorphism is called monomorphism.

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

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

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.