Definition:Injection/Definition 1 a

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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 image.

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.

Technical Note

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 .