Definition:Surjection

Definition
A mapping $$f$$ is described as onto, or a surjection, or surjective, iff:
 * $$\forall y \in \operatorname{Rng} \left({f}\right): \exists x \in \operatorname{Dom} \left({f}\right): f \left({x}\right) = y$$.

That is, every element in the range of $$f$$ is mapped to by at least one element in the domain.

If $$f$$ is not a surjection, then $$f$$ is described as into.

Basic Properties of a Surjection

 * In Surjection iff Image equals Range, it is shown that a mapping $$f$$ is a surjection iff its image equals its range.


 * In Surjection iff Right Cancellable it is shown that a mapping $$f$$ is a surjection iff it is right cancellable.


 * In Right Inverse Mapping it is shown that a mapping $$f$$ is a surjection iff it has a right inverse.


 * In Preimages All Exist iff Surjection, it is shown that a mapping $$f$$ is a surjection iff the preimage of every element is guaranteed not to be empty.

Also see

 * Injection
 * Bijection