# Surjection

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(Redirected from Surjectivity)

Grade: A*

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- Surjective is onto - for [math]f:A\rightarrow B[/math] every element of [math]B[/math] is mapped onto from at least one thing in [math]A[/math]

## Contents

## Definition

Given a function [ilmath]f:X\rightarrow Y[/ilmath], we say [ilmath]f[/ilmath] is *surjective* if:

- [math]\forall y\in Y\exists x\in X[f(x)=y][/math]
- Equivalently [math]\forall y\in Y[/math] the set [math]f^{-1}(y)[/math] is non-empty. That is [math]f^{-1}(y)\ne\emptyset[/math]

## Theorems

The composition of surjective functions is surjective

Let [ilmath]f:X\rightarrow Y[/ilmath] and [ilmath]g:Y\rightarrow Z[/ilmath] be surjective maps, then their composition, [ilmath]g\circ f=h:X\rightarrow Z[/ilmath] is surjective.

- We wish to show that [math]\forall z\in Z\exists x\in X[h(x)=z][/math]

- Let [ilmath]z\in Z[/ilmath] be given
- Then [ilmath]\exists y\in Y[/ilmath] such that [ilmath]g(y)=z[/ilmath]
- Of course also [ilmath]\exists x\in X[/ilmath] such that [ilmath]f(x)=y[/ilmath]
- We now know [ilmath]\exists x\in X[/ilmath] with [ilmath]f(x)=y[/ilmath] and [ilmath]g(y)=g(f(x))=h(x)=z[/ilmath]

- Thus it is shown that:
- [ilmath]\forall z\in Z\exists x\in X[h(x)=z][/ilmath]

- as required.
^{[1]}