The question requires me to show that the following statement is true or false.
Let $A,B$ be sets, and $f:A\to B$, $g:B\to A$ be functions. Suppose $g\circ f\circ g$ is surjective, and $f\circ g\circ f$ is injective. Then $f\circ g$ is bijective.
So basically I apply the fact that “if $g\circ f$ is surjective, then $g$ is surjective” and “if $g\circ f$ is injective, then $f$ is injective”.
But the problem is, can I read $f\circ g\circ f$ as both $f\circ (g\circ f)$ and $(f\circ g)\circ f$?
The purpose of doing this is to get “$g\circ f$ is surjective” and “$g\circ f$ is injective”. But I am not sure if it is legal.
