A function f:XY is called surjectivePlanetmathPlanetmath or onto if, for every yY, there is an xX such that f(x)=y.

Equivalently, f:XY is onto when its image is all the codomain:



  1. 1.

    If f:XY is any function, then f:Xf(X) is a surjection. That is, by restricting the codomain, any function induces a surjection.

  2. 2.

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

  3. 3.

    If f:XY is a surjection and BY, then (see this page (

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