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Symmetric difference proof

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Let % be the symmetric difference.
    Prove that for any sets A, B, C;

    A%B=C iff B%C=A iff A%C=B

    2. Relevant equations

    (I will use forwardslash as I cannot find the backslash on my keyboard.)

    3. The attempt at a solution

    Take x in A%B. Then x is either in A/B or in B/A and in C. Choose the former. Then x is in A and x is in C which implies x is in AnC. This is what I know.
    Now, using these I will prove that B%C=A.
    Take x in (B/C)U(C/B). Then x is either in the first or in the second. Actually it cannot be in the first because that would imply that x is not in C, which contradicts that x is in C. So x is in C/B. But I already know that x is in A, so nothing to prove in that direction. Take x in A... I don't know how to proceed.
  2. jcsd
  3. Nov 14, 2012 #2


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    Homework Helper

    hi bedi! :smile:
    (to prove from left to right …)

    you need to start "B%C = B%(A%B) = … " :wink:
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