# Definition:Injection/Definition 5

## Definition

Let $f: S \to T$ be a mapping where $S \ne \O$.

Then $f$ is an injection if and only if:

$\exists g: T \to S: g \circ f = I_S$

where $g$ is a mapping.

That is, if and only if $f$ has a left inverse.

## Also known as

Authors who prefer to limit the jargon of mathematics tend to use the term:

one-one (or 1-1) or one-to-one for injective
one-one mapping or one-to-one mapping for injection.

However, because of the possible confusion with the term one-to-one correspondence, it is standard on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the technical term injection to be used instead.

E.M. Patterson's idiosyncratic Topology, 2nd ed. of $1959$ refers to such a mapping as biuniform.

This is confusing, because a casual reader may conflate this with the definition of a bijection, which in that text is not explicitly defined at all.

An injective mapping is sometimes written:

$f: S \rightarrowtail T$ or $f: S \hookrightarrow T$

In the context of class theory, an injection is often seen referred to as a class injection.

## Also see

• Results about injections can be found here.