Definition:Injection/Definition 1
Definition
A mapping $f$ is an injection, or injective if and only if:
- $\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$
That is, an injection is a mapping such that the output uniquely determines its input.
Definition 1 a
This can otherwise be put:
- $\forall x_1, x_2 \in \Dom f: x_1 \ne x_2 \implies \map f {x_1} \ne \map f {x_2}$
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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- Weisstein, Eric W. "Injection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Injection.html