Let f: A-->B, g:B--->C be functions. Prove that if g o f is one to one then f is one to one.
Proof by contrapositive
Assume that f is not one to one then f(a_{1})≠f(a_{2}) where a_{1},a_{2} belong to set A. We also assume a_{1}≠a_{2}. Let f(a_{1})=r and f(a_{2})=s where r and s belong to set B. Then that means r≠s. Hence (g o f)(a_{1})≠(g o f)(a_{2})=g(f(a_{1})≠g(f(a_{2})=
g(r)≠g(s). Thus g o f is not one to one.
Did I do this right? When a function is not one to one do i get to assume that f(x)≠f(y) where x,y belong to domain and furthermore x≠y?
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Remark
I did this wrong. I forgot that when you negate the one to one definition it becomes A function f: A--->B is not one one when f(x)=f(y) where x,y are elements of A and x≠y
Edit to proof
Assume that f is not one to one then f(a_{1})=f((a_{2}) where a_{1},a_{2} belong to set A. We also assume a_{1}≠a_{2}. Let f(a_{1})=r and f(a_{2})=s where r and s belong to set B. Then that means f(a_{1})=f(a_{2}). Hence (g o f)(a_{1})=(g o f)(a_{2})<=>g(f(a_{1})=g(f(a_{2})<=>
g(r)=g(s) but r≠s. Thus g o f is not one to one.
is it right now?