# surjective

A function $f\colon X\to Y$ is called surjective or onto if, for every $y\in Y$, there is an $x\in X$ such that $f(x)=y$.

Equivalently, $f\colon X\to Y$ is onto when its image is all the codomain:

 $\mathrm{Im}f=Y.$

## Properties

1. 1.

If $f\colon X\to Y$ is any function, then $f\colon X\to f(X)$ is a surjection. That is, by restricting the codomain, any function induces a surjection.

2. 2.

The composition of surjective functions (when defined) is again a surjective function.

3. 3.

If $f\colon X\to Y$ is a surjection and $B\subseteq Y$, then (see this page (http://planetmath.org/InverseImage))

 $ff^{-1}(B)=B.$
 Title surjective Canonical name Surjective Date of creation 2013-03-22 12:32:48 Last modified on 2013-03-22 12:32:48 Owner drini (3) Last modified by drini (3) Numerical id 7 Author drini (3) Entry type Definition Classification msc 03-00 Synonym onto Related topic TypesOfHomomorphisms Related topic InjectiveFunction Related topic Bijection Related topic Function Related topic OneToOneFunctionFromOntoFunction Defines surjection