# Definition:Injection/Definition 4

## Definition

Let $f$ be a mapping.

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

- $\forall y \in \Img f: \card {\map {f^{-1} } y} = \card {\set {f^{-1} \sqbrk {\set y} } } = 1$

where:

- $\Img f$ denotes the image set of $f$
- $\card {\, \cdot \,}$ denotes the cardinality of a set
- $\map {f^{-1} } y$ is the preimage of $y$
- $f^{-1} \sqbrk {\set y}$ is the preimage of the subset $\set y \subseteq \Img f$.

That is, if and only if the preimage of $y$ is a singleton for all $y$ in the image set of $f$.

## 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**.

## Sources

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 12$ - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Functions - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.5$

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