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.

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.

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