Definition:Injection/Definition 6
Definition
Let $f: S \to T$ be a mapping where $S \ne \O$.
Then $f$ is an injection if and only if $f$ is left cancellable:
- $\forall X: \forall g_1, g_2: X \to S: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$
where $g_1$ and $g_2$ are arbitrary mappings from an arbitrary set $X$ to the domain $S$ 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
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections