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Math Help - Set Theory Help.

  1. #1
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    Set Theory Help.

    Ok guys, for any 2 non-empty sets X and Y drawn from the same univserse, U, let Z be defined
    Z = (A \cap B) \cup (A \cap B)

    Therefore, Z = A \Delta B

    I need to prove Z could equal .
    Also, Z can equal U.

    Thanks for any help.
    Last edited by mr fantastic; April 7th 2011 at 05:16 PM. Reason: Title.
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  2. #2
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    Quote Originally Posted by Kostapoulous View Post
    Ok guys, for any 2 non-empty sets X and Y drawn from the same univserse, U, let Z be defined
    Z = (A \cap B) \cup (A \cap B)
    Therefore, Z = A \Delta B
    I need to prove Z could equal .
    Also, Z can equal U.
    What if A=B~?.
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  3. #3
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    A\vartriangle B=\emptyset iff A = B.

    A\vartriangle B=U iff A\cup B = U and A\cap B=\emptyset.
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  4. #4
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    Quote Originally Posted by Plato View Post
    What if A=B~?.
    I don't quite follow what you mean, I was under the assumption that A could not equal B. I am quite confused.
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  5. #5
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    Quote Originally Posted by Kostapoulous View Post
    I don't quite follow what you mean, I was under the assumption that A could not equal B. I am quite confused.
    If both are non-empty and A\ne B then A\Delta B\ne\emptyset.

    If A\cap B=\emptyset then A\Delta B=A\cup B
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  6. #6
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    Quote Originally Posted by Plato View Post
    If both are non-empty and A\ne B then A\Delta B\ne\emptyset.

    If A\cap B=\emptyset then A\Delta B=A\cup B
    But if Z = A\Delta B , i.e. the symmetrical difference, which is what is in set A but not B, and what is in set B but not A, how can it ever equal if they are non-empty? Would they not always have some elements in the sets?
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  7. #7
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    Quote Originally Posted by Kostapoulous View Post
    But if Z = A\Delta B , i.e. the symmetrical difference, which is what is in set A but not B, and what is in set B but not A, how can it ever equal if they are non-empty? Would they not always have some elements in the sets?
    A\Delta B=\emptyset{\text{ if and only if }A=B.

    A\Delta B\ne\emptyset{\text{ if and only if }A<br />
\ne B.
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  8. #8
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    Quote Originally Posted by Plato View Post
    A\Delta B=\emptyset{\text{ if and only if }A=B.

    A\Delta B\ne\emptyset{\text{ if and only if }A<br />
\ne B.
    Yes! It's finally clicked in my head! I feel like such an idiot now for not picking this up earlier!
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