# Is f injective or not?

• Sep 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 !
• Sep 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 \$\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?
• Sep 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
• Sep 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 \$\displaystyle g\$ is injective and \$\displaystyle g(p)=g(q)\$ the by definition \$\displaystyle p=q~.\$
• Sep 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?
• Sep 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?

\$\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.
• Sep 15th 2012, 01:50 PM
Burcin
Re: Is f injective or not?
YES I understand! Thank you very much !
• Sep 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 \$\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.
• Sep 15th 2012, 02:48 PM
emakarov
Re: Is f injective or not?
There is a typo in b).
• Sep 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 \$\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