- Thread starter
- #1

- Apr 14, 2013

- 4,036

For the functions $f:A\rightarrow B$, $g:B\rightarrow A$ and $h:B\rightarrow A$ it holds that $(g\circ f)(x)=x, \ \forall x\in A$ and $(f\circ h)(x)=x, \ \forall x\in B$. Show that it holds that $g\equiv h$.

I don't really have an idea how to show that.

Let $x\in A$. Then $(g\circ f)(x)=x \Rightarrow g(f(x))=x$. Then $f(x)\in B$.

We have that $(f\circ h)(x)=x, \ \forall x\in B$. Therefore $f(x)=(f\circ h)(f(x))$.

Is this the correct way to start?