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Homework Help: Abstract - Prove (A-B)union(B-A)=(AunionB)-(AintersectB)

  1. Sep 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove:
    (A-B)[tex]\cup[/tex](B-A)=(A[tex]\cup[/tex]B)-(A[tex]\cap[/tex]B)

    2. Relevant equations



    3. The attempt at a solution
    We need to show (A-B)[tex]\cup[/tex](B-A)[tex]\subseteq[/tex](A[tex]\cup[/tex]B)-(A[tex]\cap[/tex]B)
    and (A[tex]\cup[/tex]B)-(A[tex]\cap[/tex]B)[tex]\supseteq[/tex](A-B)[tex]\cup[/tex](B-A).

    We begin by showing the first:
    Let x[tex]\in[/tex](A-B)[tex]\cup[/tex](B-A).
    By definition of union, x[tex]\in[/tex]A-B or x[tex]\in[/tex]B-A.
    If x[tex]\in[/tex]A-B, we know x[tex]\in[/tex]A .......


    This is where I've begun to get stuck. Not sure where to go next.
     
  2. jcsd
  3. Sep 5, 2010 #2

    Office_Shredder

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    We know x is in A, AND x is NOT in B.

    What does it mean for an element to be in [tex]A\cup B- A\cap B[/tex]? Is x inside of it in this case?
     
  4. Sep 5, 2010 #3
    An elelemnt is in A or B and not in A and B.
     
  5. Sep 5, 2010 #4

    Office_Shredder

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    So does the x that we were looking at satisfy that requirement?
     
  6. Sep 5, 2010 #5
    I'm thinking yes, but I'm having trouble visualizing that just from the x an element of A and not B.
     
  7. Sep 5, 2010 #6

    vela

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    Just break it down into simple pieces. If x is an element of A and not an element of B:

    1. Is x in A or B?
    2. Is x in A and B?

    Therefore...
     
  8. Sep 5, 2010 #7
    x is in A or B, but not in A and B.
    Therefore, we have the right side of the equation.

    Ok, but what about if x is an element of B-A?
    x is in B, bot not in A.
    Then x is in B or A, but not in B and A.
    So, therefore, we have the right side of the equation.

    Ok, and then I just work the other way to prove equality?
     
  9. Sep 5, 2010 #8

    vela

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    Yup.
     
  10. Sep 6, 2010 #9
    Thanks a lot. That makes a lot more sense now.
     
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