Definition:Surjection

Definition
A mapping $f: S \to T$ is described as 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.

Also known as
Authors who prefer to limit the jargon of mathematics tend to use the term an onto mapping for a surjection, and onto for surjective.

A mapping which is not surjective is described as into.

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

Also see

 * Definition:Injection
 * Definition:Bijection


 * 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.