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Set question

  1. Oct 19, 2009 #1
    Hi,

    I have a simple question i'd like some help with

    let [tex]B = \{1,2,3,4,5\}[/tex]

    How many functions from B -> B are onto ?

    A kick in the right direction would be nice
     
  2. jcsd
  3. Oct 19, 2009 #2

    lanedance

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    say you have a function defined on
    [tex] f : a \in A \rightarrow f(a) = b \in B [/tex]

    A function is onto if for every b in B, there exists an a, such that f(a) = b

    in this case A = B
    [tex] f : b \in B \rightarrow f(b) = b' \in B [/tex]
    and there must be a b for every b'

    so what else can you say about f?
     
  4. Oct 19, 2009 #3

    HallsofIvy

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    There are, of course, 55 functions from B to B.

    In order to be onto, a function from two sets of the same size (which, of course, includes functions from one set to itself) must also be one-to-one. I think that makes the problem simpler.

    Choose a number to map "1" to - you have 5 choices. Once that is done, you cannot map anything else to that so you now have 4 choices to map "2" to, 3 to map "3" to, etc. See the point?
     
  5. Oct 19, 2009 #4
    I see the point, so an element can map to itself?
    So is the function also one-to-one ?
     
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