# Mapping Composed with Bijection forming Bijection is Bijection

## Theorem

Let $A$, $B$ and $C$ be sets.

Let $f: A \to B$ and $g: B \to C$ be mappings.

Let the composite mapping $g \circ f$ be a bijection.

Let either $f$ or $g$ be a bijection.

Then both $f$ and $g$ are bijections.

## Proof

We note that $g \circ f$ is both an injection and a surjection by definition of bijection.

We are given that:

- $g \circ f$ is a bijection.

From Injection if Composite is Injection it follows that $f$ is an injection.

From Surjection if Composite is Surjection it follows that $g$ is a surjection.

$\Box$

First suppose that $g$ is a bijection.

We already have that $f$ is an injection.

Aiming for a contradiction, suppose $f$ is not a surjection.

Then:

- $\exists y \in B: \neg \exists x \in A: \map f x = y$

But then $g$ is a surjection and so:

- $\forall z \in C: \exists y \in B: z = \map g y$

and so:

- $\exists z \in C: \neg \exists x \in A: \map g {\map f x} = z$

and so $g \circ f$ is not a surjection.

This contradicts our assertion that $g \circ f$ is a bijection and so a surjection.

Hence by Proof by Contradiction $f$ must be a surjection

So $f$ is both an injection and a surjection, and so a bijection.

$\Box$

Now suppose that $f$ is a bijection.

Aiming for a contradiction, suppose $g$ is not an injection.

Then:

- $\exists y_1, y_2 \in B: \map g {y_1} = \map g {y_2}$

But then $f$ is a bijection and so:

- $\exists x_1, x_2 \in A: y_1 = \map f {x_1} \ne \map f {x_2} = y_2$

and so:

- $\exists x_1, x_2 \in A: \map g {\map f {x_1} } = \map g {\map f {x_2} }$

and so $g \circ f$ is not an injection.

This contradicts our assertion that $g \circ f$ is a bijection and so an injection.

Hence by Proof by Contradiction $g$ must be an injection.

So $g$ is both an injection and a surjection, and so a bijection.

$\Box$

Hence the result.

$\blacksquare$