Definition:Injection

Definition
A mapping $f$ is an injection, or injective iff:
 * $\forall x_1, x_2 \in \operatorname{Dom} \left({f}\right): f \left({x_1}\right) = f \left({x_2}\right) \implies x_1 = x_2$.

That is, it is a mapping such that the output uniquely determines its input.

This can otherwise be put:
 * $\forall x_1, x_2 \in \operatorname{Dom} \left({f}\right): x_1 \ne x_2 \implies f \left({x_1}\right) \ne f \left({x_2}\right)$.

It can be seen that this definition is consistent with that of a one-to-one relation.

Thus an injection is a relation which is both one-to-one and left-total.

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, and 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 recommended that the rather more technical-sounding term injection is used instead.

An injective mapping is sometimes written $f: S \rightarrowtail T$ or $f: S \hookrightarrow T$.

Also see

 * Definition:Surjection
 * Definition:Bijection


 * In Injection iff Left Cancellable it is shown that a mapping $f$ is an injection iff it is left cancellable.


 * In Injection iff Inverse of Image is Mapping it is shown that a mapping $f$ is an injection iff the inverse of $f$ from its image is itself a mapping.


 * In Injection iff Left Inverse it is shown that a mapping $f$ is an injection iff it has a left inverse.


 * In Preimages all Singletons iff Injection, it is shown that a mapping $f$ is an injection iff the preimage of every element of the codomain is guaranteed to have no more than one element.


 * In Subset equals Preimage of Image iff Mapping is Injection, it is shown that a mapping $f$ is an injection iff the preimage of the image of every subset of its domain equals that subset.