surjective (original) (raw)

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:

Properties

1. 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. 1. The composition of surjective functions (when defined) is again a surjective function.
3. 1. If f:X→Y is a surjection and B⊆Y, then (see this page (http://planetmath.org/InverseImage))