Is f injective or not?

**Burcin** · September 15th 2012, 12:33 PM

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 !

**Plato** · September 15th 2012, 12:43 PM

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?

**Burcin** · September 15th 2012, 01:00 PM

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

**Plato** · September 15th 2012, 01:05 PM

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 $p=q~.$

**Burcin** · September 15th 2012, 01:14 PM

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

**Plato** · September 15th 2012, 01:21 PM

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.

**Burcin** · September 15th 2012, 01:50 PM

YES I understand! Thank you very much !

**Plato** · September 15th 2012, 02:16 PM

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.

**emakarov** · September 15th 2012, 02:48 PM

There is a typo in b).

**Burcin** · September 15th 2012, 11:22 PM

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

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