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Thread: injection,surjection, bijection and compositions

  1. #1 injection,surjection, bijection and compositions 
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    Hello!

    First, Happy New 2010.


    Now. I need some examples.

    Could you possibly give me example of two functions f and g that are not injection (or at least one of them is not injection) but their composition fg(x) or gf(x) is injection?

    Thanks in advance.


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  3. #2 Re: injection,surjection, bijection and compositions 
    . DrRocket's Avatar
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    Quote Originally Posted by scientist91
    Hello!

    First, Happy New 2010.


    Now. I need some examples.

    Could you possibly give me example of two functions f and g that are not injection (or at least one of them is not injection) but their composition fg(x) or gf(x) is injection?

    Thanks in advance.
    Here is not injective but is. But pay attention to the domains of each.


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  4. #3 Re: injection,surjection, bijection and compositions 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by scientist91
    Hello!

    First, Happy New 2010.


    Now. I need some examples.

    Could you possibly give me example of two functions f and g that are not injection (or at least one of them is not injection) but their composition fg(x) or gf(x) is injection?

    Thanks in advance.
    Here is not injective but is. But pay attention to the domains of each.
    Thanks for the reply.

    The domain of gf is all real numbers. And the composition is |x|. And |x| is not injection.

    The domain of fg is [0,+Infinity]. So the composition fg would be |x|, but with the restriction [0,+Infinity], it would be just x for x>=0. (Just the right part of the graph |x|.)

    Thanks a lot.

    And do you have some example for two functions that are not bijection but their composition is bijection?
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  5. #4 Re: injection,surjection, bijection and compositions 
    . DrRocket's Avatar
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    Quote Originally Posted by scientist91

    And do you have some example for two functions that are not bijection but their composition is bijection?
    I just gave you one. Look at those examples a little closer.
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  6. #5  
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    But isn't the definition of bijection:
    "If a function is injection and surjection then it is bijection".

    So the first function is not injection, but the implication it follows that it is true.

    FALSE ^ TRUE => (T or F) is TRUE so f is bijection.
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