Proof: Permutations and Surjective Functions

  1. 1. The problem statement, all variables and given/known data
    Let X and Y be finite nonempty sets, |X|=m, |Y|=n≤m. Let f(n, m) denote the number of partitions of X into n subsets. Prove that the number of surjective functions X→Y is n!*f(n,m).

    2. Relevant equations
    I know a function is onto if and only if every element of Y is mapped to by an element of X. That is, for all y in Y, there is an x in X such that f(x)=y. Clearly if f is a function and n≤m, a function from X to Y can be onto (but it doesn't have to be, for instance, all of X could map to the same Y).

    3. The attempt at a solution
    I tried by induction, but got lost moving from n=k to n=k+1, so I'm not sure if that works. I think that f(n, m) has something to do with the number of ways to permute the elements of X... but not sure. This is before the lecture on combinations, so I'm not sure if we need to use that method.

    Thanks in advance!
  2. jcsd
  3. Could it have something to do with creating a function from a partitioned version of X to Y?
  4. No takers?
  5. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    Sure it could. You split X into subsets, each one of which maps into a unique element of Y. Then given a split (of which there are f(n,m)) you figure out how many ways there are to assign each subset to an element of Y. I don't think induction is really necessary here. Just explain it in words.
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