Definition:Injection

Definition
A mapping $$f$$ is an injection, or injective, or one-one, or one-to-one 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.

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

The symbol $$f: S \rightarrowtail T$$ is sometimes used to indicate that $$f$$ is an injection from $$S$$ to $$T$$.

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.

Basic Properties of an Injection

 * 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 Mapping it is shown that a mapping $$f$$ is an injection iff the inverse mapping 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 Unique 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 Preimage of Image of 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.

Also see

 * Surjection
 * Bijection