Question

Prove that if f: A -->B and g: B-->C and g(f(x)): A -->C which is one to one then f: A --> B is one to one

Answer 1

work directly from the definition of one to one

Answer 2

if f(x)=f(y) then x=y

Answer 3

that is, start with \(g(f(a))=g(f(b))\implies a=b\)

Answer 4

now suppose \(f\) is not one one one
then there exists \(a\neq a'\in A\) such that \(f(a)=f(a')\)

Answer 5

Riighhhttt

Answer 6

but if this is true, then since \(f(a)=f(a')\) it must be true that \(g(f(a))=g(f(a'))\) since \(g\) is a function

Answer 7

contradicting the assumption

Answer 8

alrighty gotchhhaaaaaaaa

Answer 9

Thanks :)