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Homesurjective

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# 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 $f(x)=y$.

# 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)

$ff^{{-1}}(B)=B.$

Defines:

surjection

Related:

TypesOfHomomorphisms,InjectiveFunction, Bijection, Function, OneToOneFunctionFromOntoFunction

Synonym:

onto

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03-00*no label found*

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