# Definition:Injection/Definition 1

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## Definition

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 1 a

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.

An injective mapping is sometimes written:

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

The $\LaTeX$ code for $f: S \rightarrowtail T$ is f: S \rightarrowtail T .

The $\LaTeX$ code for $f: S \hookrightarrow T$ is f: S \hookrightarrow T .

## Also see

• Results about injections can be found here.