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Math Help - Is f injective or not?

  1. #1
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    Unhappy 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 !
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  2. #2
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    Re: Is f injective or not?

    Quote Originally Posted by Burcin View Post
    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?
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  3. #3
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    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
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    Re: Is f injective or not?

    Quote Originally Posted by Burcin View Post
    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~.
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    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?
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    Re: Is f injective or not?

    Quote Originally Posted by Burcin View Post
    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.
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  7. #7
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    Re: Is f injective or not?

    YES I understand! Thank you very much !
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    Re: Is f injective or not?

    Quote Originally Posted by Burcin View Post
    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.
    Last edited by Plato; September 15th 2012 at 03:55 PM.
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  9. #9
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    Re: Is f injective or not?

    There is a typo in b).
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  10. #10
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    Re: Is f injective or not?

    Quote Originally Posted by Plato View Post
    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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