Definition:Injection/Definition 3
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 or one-to-one map 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
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Functions
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.3$: Definition $1.9 \ \text{(a)}$