# 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:

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**.

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections