injective function f o g

Question

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

It is also given that $f\,\,o\,\,g\,\,$ is one to one

Let $a \in A\,$, $\,b \in B\,$,$\,c \in C\,$

Then,

If $f\,\,o\,\,g\,\,$ is one to one

if $f\,\left(\,\,g \left(a\right)\right)=f\,\left(\,\,g \left(b\right)\right) $

$\Rightarrow g \left(a\right)=g \left(b\right)$

it is valid .

My question is that based on above,

i.e $ g \left(a\right)=g \left(b\right)$

is a =b valid ..?

Answer 1

I believe you want to prove

If $f\circ g$ is one-to-one, then $g$ is also one-to-one.

Suppose $g(b) = g(a)$, then $f\circ g(b)=f(g(b)) = f(g(a))=f\circ g(a)$. Now use the fact that $f\circ g$ is one-to-one to make your conclusion.