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Homework Help: Help with surjective maps

  1. Jun 17, 2011 #1
    1. The problem statement, all variables and given/known data

    Let S = {1,2,3,...,n}

    How many surjective maps are there from S to S?

    2. Relevant equations


    3. The attempt at a solution

    The book's answer is n!

    However, I thought that total number of surjective maps = n^n because 1-1 isn't required. Where am I wrong?
  2. jcsd
  3. Jun 17, 2011 #2


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    This is understandable as a surjective map from itself to itself, and this is essential just the set S being rearranged, so the answer will be n!.

  4. Jun 17, 2011 #3
    1-1 is implicitly required, because each element of the domain can only map to one element of the range - so to map to every element of the range you'll need a 1-1 mapping.
  5. Jun 17, 2011 #4


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    But since S is finite, if the map wasn't 1-1 it couldn't be onto, could it?
  6. Jun 17, 2011 #5
    Got it. Thanks.
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