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Symmetric difference in sets

  1. Jul 13, 2011 #1
    1. The problem statement, all variables and given/known data

    There is a symmetric difference in sets X & Y, X Y is defined to be the sets of elements that are either X or Y but not both
    Prove that for any sets X,Y & Z that
    (X[itex]\oplus[/itex]Y)[itex]\oplus[/itex](Y[itex]\oplus[/itex]Z) = X[itex]\oplus[/itex]Z

    2. Relevant equations

    [itex]\oplus[/itex] = symmetic difference

    3. The attempt at a solution
    i can see it in the venn diagams, but im not good at converting what i see into set statements this is my attempt in words

    the symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C because the symmetric difference of two repeated symmetric differences is the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed.

    i dont have a clue how to show that properly in subsets etc that why i need some help
     
  2. jcsd
  3. Jul 13, 2011 #2
  4. Jul 13, 2011 #3

    micromass

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    How did you define [itex]X\oplus Y[/itex] in symbols??

    You can split the proof up in this parts:
    - Associativity: show that [itex]X\oplus (Y\oplus Z)=(X\oplus Y)\oplus Z[/itex]
    - Show that [itex]Y\oplus Y=\emptyset[/itex]
    - Show that [itex]Y\oplus \emptyset=Y[/itex]

    (this actually implies that [itex]\oplus[/itex] forms a group operation). These three things together imply what you want to show, do you see that?
     
  5. Jul 14, 2011 #4
    ok here is my revised attempt at the answer with your help:

    In order to prove that (X⊕Y)⊕(Y⊕Z) = X⊕Z you must imply that ⊕ forms a group operation this proof is split up into three parts

    The symmetric difference is associative which means that
    (X⊕Y)⊕(Y⊕Z)= (Y⊕Y)⊕(X⊕Z) --I think that is correct?--
    or X⊕(Y⊕Z ) = (X⊕Y) ⊕Z

    The symmetric difference of the same set yields an empty set, Y⊕Y= ∅

    The symmetric difference of a set and empty set yields a Y⊕∅= Y

    so (Y⊕Y) = ∅
    ∅ ⊕(X⊕Z)= X⊕Z
    hence for any sets X,Y & Z ,(X⊕Y)⊕(Y⊕Z) = X⊕Z

    THAT ALL CORRECT?
     
  6. Jul 15, 2011 #5

    micromass

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    Well, you're using commutativity here (which is fine, but you must first show that you can do that). IF you don't want to do that, then

    [tex](X\oplus Y)\oplus (Y\oplus Z)=X\oplus (Y\oplus Y)\oplus Z=X\oplus \emptyset \oplus Z=X\oplus Z[/tex]

    is also fine...

     
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