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SETS problem

  1. Feb 15, 2007 #1
    this is one problem which i dont know how to start..im not sure which proof i should use to solve this problem
    the problem is:
    If A,B,C,and D are sets, does it follow
    (A Φ B) Φ (C Φ D) = (A Φ C) Φ (B Φ D)
    the symbol that is separting the characters is called the oplus symbol, thats the closet symbol i could find
    the oplus symbol is a circle with one line going across it and one line going down
  2. jcsd
  3. Feb 15, 2007 #2


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  4. Feb 15, 2007 #3
    no it doesnt
    the symbol im trying to illustrate there is the oplus which can be viewed at this link

    http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideSym.php [Broken]
    Last edited by a moderator: May 2, 2017
  5. Feb 15, 2007 #4


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    Does that mean the XOR operation, ie, [itex] A \oplus B = (A-B) \cup (B-A)[/itex], also known as the symmetric difference? What have you tried?
  6. Feb 15, 2007 #5
    yes that is the symbol which im referring to...symmetric difference

    i dont know where to start..i have no clue on what to do?
  7. Feb 15, 2007 #6


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    Start by trying to see if it's true. You can draw a picture, something like 4 overlapping circles, and see if you can find the regions corresponding to each side of that equation.
  8. Feb 15, 2007 #7
    i have tried to draw circles before...but i just keep getting lost..i dont know how to proof this and which would be the best proof to use in such a case..im totally lost on how to start the problem
  9. Feb 15, 2007 #8


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    Ok, then just write out the definitions. Get everything in terms of unions, intersections and complements, which should be easier to work with.
  10. Feb 15, 2007 #9
    im going to see what i can come up with..but iam clueless in what to do...
  11. Feb 16, 2007 #10


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    You want to prove that two sets, the set on the left side and the set on the right side, are equal. Typically you do that by proving that anything in one of the sets is also in the other.

    Suppose x is in [itex] (A\oplus B)\oplus (C\oplus D)[/itex]. What must be true about it? From the definition, if a point, p, is in [itex]A\oplus B[/itex] is it in A or B? exactly what is true about p from the definitions? Once you know exactly where, relative to A, B, C, D, p must be in order to be in the left set, use that information to show that it must be in the right set.

    Now reverse and do it the other way.
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