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Power set P(S) with symmetry difference.

  1. Feb 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine the orders of all the elements of the power set P(S) of a set S with symmetric difference Δ.

    2. Relevant equations



    3. The attempt at a solution

    If A,b are two elements of the power set
    the symmetric difference is

    AΔB = (A-B) U (B - A)

    How are we going to find the order of the elements?
     
  2. jcsd
  3. Feb 29, 2012 #2

    Dick

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    What's AΔA? That's A^2 with your group operation.
     
  4. Feb 29, 2012 #3
    its A Δ B not A Δ A.

    I dont know how to start this
     
  5. Feb 29, 2012 #4

    Dick

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    You would if you paid some attention to me. The order of A is the number of times you have to take the symmetric difference of A with itself to get the group identity. What's the identity of your group?
     
  6. Feb 29, 2012 #5
    the identity is the empty set.
     
  7. Mar 1, 2012 #6

    Deveno

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    ok, now: what is A-A?

    and therefore, what is (A-A) U (A-A) = AΔA?
     
  8. Mar 1, 2012 #7
    I also know that AΔB = (A-B) U (B - A)= (B-A) U (A - B)=BΔA
     
  9. Mar 1, 2012 #8

    Deveno

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    well, that's very nice, but:

    in a group, the order of an element g is the smallest possible positive integer k such that gk = e.

    what is gk? it's g*g*g*...*g (k times).

    so when calculating order of an element, you don't need to look at "other elements".

    in the group you are considering, we are using "A" instead of "g", and "Δ" instead of "*".

    so Ak = AΔAΔA....ΔA (k times).

    there's no need to bring up "B".
     
  10. Mar 1, 2012 #9
    To the answer is that we have one element A and the order of A is k?
     
  11. Mar 1, 2012 #10

    Dick

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    Once again. What is AΔA?
     
  12. Mar 1, 2012 #11
    Its A^2
     
  13. Mar 1, 2012 #12

    Dick

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    Ok, yes, but what is that? What elements are in that set?
     
  14. Mar 1, 2012 #13
    just A and A ?
     
  15. Mar 1, 2012 #14

    Dick

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    Use your definition! AΔA = (A-A) U (A-A). What is that? It's a very simple set.
     
  16. Mar 1, 2012 #15
    it says that the A-A union A-A. On sets if A-A=0 it means it is zero
     
  17. Mar 1, 2012 #16

    Dick

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    A-A isn't 0, if you mean the number zero. Do you know what A-A means? It's a set, not a number.
     
  18. Mar 1, 2012 #17
    can you please explain me
     
  19. Mar 1, 2012 #18

    Dick

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    A-B means the set of all the elements of A that are not in B. What is {1,2,3,4}-{5,4,2,7}? Now what is {1,2,3,4}-{1,2,3,4}?
     
  20. Mar 1, 2012 #19
    yes i know that thats why i told you 0. There are no elements left

    {1,2,3,4}-{5,4,2,7} 1,3

    {1,2,3,4}-{1,2,3,4} no elements
     
  21. Mar 1, 2012 #20

    HallsofIvy

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    And what do you call a set that contains no elements? That's what Dick is asking.
     
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