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Math Help - Number of everywhere defined and one to one functions? Help please?

  1. #1
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    Number of everywhere defined and one to one functions? Help please?

    Let A have 5 elements and let B have 4 elements.

    1) How many everywhere defined functions are there from A to B?
    For the function to be everywhere defined it must contain all 5 elements of A and each can have 4 possible partners so there are 4*4*4*4*4=4^5 possible everywhere defined functions?Is this correct?If not how do I do it?

    2) How many one to one functions are there from A to B?
    Logically the function can have a maximum of 4 elements but I don't know how to count the possibilities here.

    Any help would be appreciated.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by chocaholic View Post
    Let A have 5 elements and let B have 4 elements.

    1) How many everywhere defined functions are there from A to B?
    For the function to be everywhere defined it must contain all 5 elements of A and each can have 4 possible partners so there are 4*4*4*4*4=4^5 possible everywhere defined functions?Is this correct?If not how do I do it?


    It is correct


    2) How many one to one functions are there from A to B?
    Logically the function can have a maximum of 4 elements but I don't know how to count the possibilities here.


    [color=red]There is none since at least one element of A must be mapped to some element of B which is already the image of some other element, because there are 5 elements in A but only 4 in .

    Tonio


    Any help would be appreciated.

    Thanks in advance.
    .
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  3. #3
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    Thanks with 1 but with 2 the function does not have to be everywhere defined just one to one so it can consist of 1 or 2 or 3 or 4 elements of A and not necessarily all 5 as long as it is related to only one element of B and the elements of B are not repeated?
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  4. #4
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    Quote Originally Posted by chocaholic View Post
    Thanks with 1 but with 2 the function does not have to be everywhere defined just one to one so it can consist of 1 or 2 or 3 or 4 elements of A and not necessarily all 5 as long as it is related to only one element of B and the elements of B are not repeated?

    When you say "functions from A to B" it is usually understood that the functions are defined in the whole A, otherwise there's no point in mentioning A , is it? That's what I meant.

    Tonio
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