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Thread: 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
    $\displaystyle Z = (A \cap B) \cup (A \cap B)$

    Therefore, $\displaystyle Z = A \Delta B$

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

    Thanks for any help.
    Last edited by mr fantastic; Apr 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
    $\displaystyle Z = (A \cap B) \cup (A \cap B)$
    Therefore, $\displaystyle Z = A \Delta B$
    I need to prove Z could equal .
    Also, Z can equal U.
    What if $\displaystyle A=B~?$.
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  3. #3
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    $\displaystyle A\vartriangle B=\emptyset$ iff $\displaystyle A = B$.

    $\displaystyle A\vartriangle B=U$ iff $\displaystyle A\cup B = U$ and $\displaystyle A\cap B=\emptyset$.
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  4. #4
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    Quote Originally Posted by Plato View Post
    What if $\displaystyle A=B~?$.
    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 $\displaystyle A\ne B$ then $\displaystyle A\Delta B\ne\emptyset.$

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

    If $\displaystyle A\cap B=\emptyset$ then $\displaystyle A\Delta B=A\cup B$
    But if Z = $\displaystyle A\Delta B$ , 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 = $\displaystyle A\Delta B$ , 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?
    $\displaystyle A\Delta B=\emptyset{\text{ if and only if }A=B.$

    $\displaystyle A\Delta B\ne\emptyset{\text{ if and only if }A
    \ne B.$
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  8. #8
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    Quote Originally Posted by Plato View Post
    $\displaystyle A\Delta B=\emptyset{\text{ if and only if }A=B.$

    $\displaystyle A\Delta B\ne\emptyset{\text{ if and only if }A
    \ne B.$
    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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