# Composite of Surjections is Surjection

## Theorem

A composite of surjections is a surjection.

That is:

If $g$ and $f$ are surjections, then so is $g \circ f$.

## Proof

Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be surjections.

Then:

 $\displaystyle \forall z \in S_3: \exists y \in S_2: \map g y$ $=$ $\displaystyle z$ Definition of Surjection $\displaystyle \leadsto \ \$ $\displaystyle \exists x \in S_1: \map f x$ $=$ $\displaystyle y$ Definition of Surjection

By definition of a composite mapping:

$\map {g \circ f} x = \map g {\map f x} = \map g y = z$

Hence $g \circ f$ is surjective.

$\blacksquare$