Definition:Injection/Definition 3

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Definition

Let $f$ be a mapping.

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

$f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping

where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to 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.


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