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Math Help - set proof

  1. #1
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    set proof

    Hello

    Let
    A, B and C be 3 sets. Prove that:
    if
    x belongs to (A-B)-C then x belongs to A-C using:
    a)direct proof.

    b)proof by contraposition

    thank you
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by qwerty321 View Post
    Hello

    Let
    A, B and C be 3 sets. Prove that:
    if
    x belongs to (A-B)-C then x belongs to A-C using:
    a)direct proof.

    b)proof by contraposition

    thank you
    a) x \in A \cap B^c \cap C^c \Rightarrow x \in A, x \notin B, x \notin C \Rightarrow x \in A, x \notin C \Rightarrow x \in A\ C

    b) x \notin A\ C \Rightarrow x \notin A, x \in C \Rightarrow x \in A^c \cup C \Rightarrow x \in A^c \cup B \cup C \Rightarrow x \in (A \cap B^c \cap C^c)^c \Rightarrow x \notin (A\ B)\ C

    Thus x \notin A\ C\Rightarrow x \notin (A\ B)\ C

    Which is proof of the contrapositive.
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  3. #3
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    when you pu the little c above A or B, what do u mean?
    and for the contrapositivity how did u pass from x belongs to A^cUC to A^cUBUC?

    thank you
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  4. #4
    Lord of certain Rings
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    Quote Originally Posted by qwerty321 View Post
    when you pu the little c above A or B, what do u mean?
    and for the contrapositivity how did u pass from x belongs to A^cUC to A^cUBUC?

    thank you
    A^c stands for A complement.

    Since x \in C, x \in A^c \cup B \cup C.
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