# Thread: Is f injective or not?

1. ## Is f injective or not?

Hey guys I can't figure out this question.

let f:A->A be a function fof is injective
Prove wether f is injective or not

Can someone show me what's the proof
THANKS !

2. ## Re: Is f injective or not?

Originally Posted by Burcin
let f:A->A be a function fof is injective
Prove wether f is injective or not
Suppose that $\displaystyle f(a)=f(b)$.
Does it follow that $\displaystyle f\circ f(a)=f\circ f(b)~?$ WHY?

You know that $\displaystyle f\circ f$ in injective. SO?

3. ## Re: Is f injective or not?

yes it follows because f(a)=f(b) so I see that fof(a)=fof(b)
I cant see the next step

4. ## Re: Is f injective or not?

Originally Posted by Burcin
yes it follows because f(a)=f(b) so I see that fof(a)=fof(b)
I cant see the next step
If $\displaystyle g$ is injective and $\displaystyle g(p)=g(q)$ the by definition $\displaystyle p=q~.$

5. ## Re: Is f injective or not?

oww I see it so a is not the same as b so f isn't injective is it right?

6. ## Re: Is f injective or not?

Originally Posted by Burcin
oww I see it so a is not the same as b so f isn't injective is it right?
$\displaystyle f\circ f$ is injective so $\displaystyle f\circ f(a)=f\circ f(b)$ implies $\displaystyle a=b$.

So if $\displaystyle f(a)=f(b)$ then $\displaystyle a=b$, proving $\displaystyle f$ is injection.

7. ## Re: Is f injective or not?

YES I understand! Thank you very much !

8. ## Re: Is f injective or not?

Originally Posted by Burcin
YES I understand! Thank you very much !
This one of a group of three part theorem.
Given any functions $\displaystyle f~\&~g~:$
a) if $\displaystyle g\circ f$ is injective then $\displaystyle f$ is injective,

b) if$\displaystyle g\circ f$ is surjective then $\displaystyle g$ is surjective,

c) if $\displaystyle g\circ f$ is bijective then $\displaystyle f$ is injective and $\displaystyle g$ is surjective.

9. ## Re: Is f injective or not?

There is a typo in b).

10. ## Re: Is f injective or not?

Originally Posted by Plato
This one of a group of three part theorem.
Given any functions $\displaystyle f~\&~g~:$
a) if $\displaystyle g\circ f$ is injective then $\displaystyle f$ is injective,

b) if$\displaystyle g\circ f$ is surjective then $\displaystyle g$ is surjective,

c) if $\displaystyle g\circ f$ is bijective then $\displaystyle f$ is injective and $\displaystyle g$ is surjective.
We haven't had this at school yet but it will help me to solve more problems! thank you