# Definition:Injection/Definition 3

## Definition

Let $f$ be a mapping.

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

- $f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping

where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.

## 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

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Functions - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.3$: Definition $1.9 \ \text{(a)}$