surjective
A function f:X→Y is called surjective or onto if, for every y∈Y, there is an x∈X such that f(x)=y.
Equivalently, f:X→Y is onto when its image is all the codomain:
Imf=Y. |
Properties
-
1.
If f:X→Y is any function, then f:X→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:X→Y is a surjection and B⊆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 |