# Is f injective or not?

• September 15th 2012, 12:33 PM
Burcin
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 !
• September 15th 2012, 12:43 PM
Plato
Re: Is f injective or not?
Quote:

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

Suppose that $f(a)=f(b)$.
Does it follow that $f\circ f(a)=f\circ f(b)~?$ WHY?

You know that $f\circ f$ in injective. SO?
• September 15th 2012, 01:00 PM
Burcin
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
• September 15th 2012, 01:05 PM
Plato
Re: Is f injective or not?
Quote:

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 $g$ is injective and $g(p)=g(q)$ the by definition $p=q~.$
• September 15th 2012, 01:14 PM
Burcin
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?
• September 15th 2012, 01:21 PM
Plato
Re: Is f injective or not?
Quote:

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

$f\circ f$ is injective so $f\circ f(a)=f\circ f(b)$ implies $a=b$.

So if $f(a)=f(b)$ then $a=b$, proving $f$ is injection.
• September 15th 2012, 01:50 PM
Burcin
Re: Is f injective or not?
YES I understand! Thank you very much !
• September 15th 2012, 02:16 PM
Plato
Re: Is f injective or not?
Quote:

Originally Posted by Burcin
YES I understand! Thank you very much !

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

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

c) if $g\circ f$ is bijective then $f$ is injective and $g$ is surjective.
• September 15th 2012, 02:48 PM
emakarov
Re: Is f injective or not?
There is a typo in b).
• September 15th 2012, 11:22 PM
Burcin
Re: Is f injective or not?
Quote:

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

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

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

We haven't had this at school yet but it will help me to solve more problems! thank you