Definition:Injection/Definition 1 a

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Definition

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

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


That is, distinct elements of the domain are mapped to distinct elements of the codomain.


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.


Sources