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Math Help - Set Theory Problems (please help)

  1. #1
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    Question Set Theory Problems (please help)

    1. Prove that

    a) (A-B) \cap C = (A \cap C) - (B \cap C);

    b) (A symmetric difference B) = (A U B) - (A \cap B)
    Last edited by mr fantastic; January 26th 2009 at 05:47 PM. Reason: Fixed the latex
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ninano1205 View Post
    1. Prove that

    a) (A-B) \cap C = (A \cap C) - (B \cap C);

    b) (A symmetric difference B) = (A U B) - (A \cap B)
    for both, show that each side is a subset of the other. do you know how to do this?

    for instance, in the first, show that x \in (A - B) \cap C \implies x \in (A \cap C) - (B \cap C) AND x \in (A \cap C) - (B \cap C) \implies x \in (A - B) \cap C

    can you continue?
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  3. #3
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    Thank you for the response. But I know till u proceeded, my problem is after that. For the first part, I don't know how to prove LHS is a subset of RHS.
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  4. #4
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    Hello,

    (A \cap C)-(B \cap C)=(A \cap C) \cap (B \cap C)' (definition of the difference)

    (A \cap C)-(B \cap C)=(A \cap C) \cap (B' \cup C') (de Morgan's law)

    (A \cap C)-(B \cap C)=[(A \cap C) \cap C'] \cup [(A \cap C) \cap B'] (distributivity)

    (A \cap C)-(B \cap C)=[A \cap (C \cap C')] \cup [(A \cap B') \cap C] (associativity)

    But C \cap C'=\emptyset
    Hence A \cap (C \cap C')=\emptyset

    And A \cap B'=A-B (definition of the difference)

    So now you have :

    (A \cap C)-(B \cap C)=\emptyset \cup [(A-B) \cap C]=\boxed{(A-B) \cap C}
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