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Number of Onto Functions

  1. May 7, 2009 #1
    # of Onto Functions

    How many onto functions are there from a set with four elements to a set with two elements?

    My math professor gave an example of a set with 50 to a set with 35 (again asking for the number of onto functions). The way to go about the problem according to him is to just do 50*49*48*...*17*16 (basically the first element in the second set has 50 choices, the next 49, etc.). For the first two problems problems I had a set of 3 elements to a set of 2 elements, so using this reasoning (& it seemed to make sense), I'd do 3*2=6. It works. The next one I had was a set of 3 to a set of 3, so 3*2*1=6. It works. Then I get the one above, 4*3=12. It does not work. The answer should be 14. But without writing out all the possibilities, I'm trying to find a better way to understand these problems and be able to solve them more efficiently.
     
  2. jcsd
  3. May 7, 2009 #2
    Re: # of Onto Functions

    I have not done this type of stuff in years but I will help out since you are not getting any replies.

    Let S(m,n) denote the number of surjective (onto) functions from a set of m elements to a set of n elements.

    Then S(m,n) = (-1)n [tex]\sum[/tex](-1)k(n choose k) km

    where the sum runs from 1 to n and (n choose k) is the binomial coefficient (I couldn't find the symbol for it).

    This may also help: http://en.wikipedia.org/wiki/Bell_number
     
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