Definition:Surjection

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

That is, if it is right-total, i.e. every element in the codomain of $f$ is mapped to by at least one element in the domain.

That is, a surjection is a relation which is:
 * Left-total
 * Many-to-one
 * Right-total.

The notation $f : S \twoheadrightarrow T$ is sometimes used to emphasize surjectivity.

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

Basic Properties of a Surjection

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


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


 * In Surjection iff Right Inverse 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.


 * In Image of Preimage of Surjection, it is shown that a mapping $f$ is a surjection iff the image of the preimage of every subset of its codomain equals that subset.

Also see

 * Injection
 * Bijection