Definition:Surjection

Definition
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping from $S$ to $T$.

Also known as
The phrase $f$ is surjective is often used for $f$ is a surjection.

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 thence described as into.

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 Subset under 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.