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Thread: Set Proof

  1. #1
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    Set Proof

    THM: $\displaystyle (A-B) - C = (A-C) - (B -C)\ \ \forall $ sets $\displaystyle A,B,C$.

    Prove the above thm.
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  2. #2
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    $\displaystyle \begin{array}{l}
    \left( {A \cap C^c } \right) \cap \left( {B \cap C^c } \right)^c \\
    \left( {A \cap C^c } \right) \cap \left( {B^c \cup C} \right) \\
    \left( {A \cap C^c \cap B^c } \right) \\
    \end{array}
    $
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  3. #3
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    Quote Originally Posted by Plato View Post
    $\displaystyle \begin{array}{l}
    \left( {A \cap C^c } \right) \cap \left( {B \cap C^c } \right)^c \\
    \left( {A \cap C^c } \right) \cap \left( {B^c \cup C} \right) \\
    \left( {A \cap C^c \cap B^c } \right) \\
    \end{array}
    $
    I appreciate it. I believe the $\displaystyle C^c$ for instance means the compliment of C (that is everything that is not C)- is there another way to prove this? I don't quite see how it proves it. Thanks for the explanation.
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  4. #4
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    $\displaystyle \left( {A \cap C^c \cap B^c } \right) = \left( {A \cap B^c } \right) \cap C^c = \left( {A\backslash B} \right)\backslash C
    $
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