# Definition:Injection/Class Theory

## Definition

### Definition 1

A mapping $f$ is **an injection**, or **injective** if and only if:

- $\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$

That is, an **injection** is a mapping such that the output uniquely determines its input.

### Definition 2

This can otherwise be put:

- $\forall x_1, x_2 \in \Dom f: x_1 \ne x_2 \implies \map f {x_1} \ne \map f {x_2}$

## 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

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 9$ Functions