Definition:Injection/Definition 4

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Definition

Let $f$ be a mapping.

$f$ is an injection if and only if:

$\forall y \in \Img f: \card {\map {f^{-1} } y} = \card {\set {f^{-1} \sqbrk {\set y} } } = 1$

where:

$\Img f$ denotes the image set of $f$
$\card {\, \cdot \,}$ denotes the cardinality of a set
$\map {f^{-1} } y$ is the preimage of $y$
$f^{-1} \sqbrk {\set y}$ is the preimage of the subset $\set y \subseteq \Img f$.


That is, if and only if the preimage of $y$ is a singleton for all $y$ in 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.


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

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