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XOR and the null set

  1. Sep 26, 2007 #1
    1. The problem statement, all variables and given/known data
    What does it mean if A ⊕ B = ∅? Explain

    2. Relevant equations
    From what I know the null set is part of every subset and A XOR B would be true if and only if exactly one of A or B would be true.

    3. The attempt at a solution
    If someone could please help me in formulating an explantation on A XOR B = null set
  2. jcsd
  3. Sep 26, 2007 #2


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    I would have thought of XOR as a logical operator than a set operator but there is no reason why it can't be: I interpret A XOR B as meaning "those elements that are in A or in B but not in both: [itex]A\cup B- A\cap B[/itex]. More often (to me at least) called the "symmetric difference" of A and B. If A XOR B is empty, then there must NOT be in points that are in A or in B but not in their intersection. What does that imply about [itex]A\cap B[/itex] and so A and B themselves?
  4. Sep 26, 2007 #3
    This is part b of a question. I did not list down the entire question because I thought it is not important (and it may still not be), but here it is:

    The symmetric difference of two sets A and B is defined as A XOR B = (A - B) U (B -A).

    b. What does it mean if A XOR B = null set?

    Can I say it it this way:

    Let A = {1,2,3,4,5} and B = {1,2,3,4,5} then A XOR B = null set? This is because there are no elements that are exclusive to either set.
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