Composite and one-one functions.

Question

I wanted to know if $g \circ f$ is defined and is one-one then under what conditions will both $g$ and $f$ be one-one? Particularly is it possible if $f:A\to B$ and $g:B\to A$?

Answer 1

If $g\circ f$ is one-to-one, then $f$ is also one-to-one (it's a good exercise to prove this), but $g$ need not be. For example, if $f\colon\{x\}\to B$ and $g\colon B\to\{x\}$ (here $\{x\}$ is any singleton set), then $f$ and $g\circ f$ are automatically one-to-one, but $g$ itself is not when $\#B>1$.