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Math Help - Combinations?

  1. #1
    Super Member fardeen_gen's Avatar
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    Combinations?

    If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of these functions are one-to-one?
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by fardeen_gen View Post
    If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of these functions are one-to-one?
    For every element in X, I just need to associate an element of Y(for which I have m choices). Thus m^n functions.

    For one-one functions, For every element in X, I need to associate only one element of Y. Thus for the first one I have, m choices. Then for the second element, I have m-1 choices and so on. So it is \frac{m!}{(m-n)!}
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  3. #3
    Moo
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    And for your information, there are m\times (m-1)\times \cdots\times (m-n+1) injective functions from X to Y.
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    Lord of certain Rings
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    Quote Originally Posted by Moo View Post
    And for your information, there are m\times (m-1)\times \cdots\times (m-n+1) injective functions from X to Y.
    Isnt that the same as \frac{m!}{(m-n)!}?
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  5. #5
    Moo
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    Yes, it's the same.
    But that's what I had in my notes. (they're from last year, so I don't have the proofs...)


    I'm wondering... If m\neq n, is it possible to have one-to-one functions from X to Y ?

    We may consider that we're dealing with functions whose domain is X, and whose range is Y. In which case, there is no one-to-one functions


    If m=n, the number of bijective functions is n!
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  6. #6
    Senior Member pankaj's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of these functions are one-to-one?
    O.K fardeen_gen.
    Do you know the number of ONTO functions from X to Y
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