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Math Help - Proof help please

  1. #1
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    Proof help please

    Let A, B, and C be sets. Prove the following claim, or give a counterexample:

    (A - B) - C = (A - C) - (B - C)
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  2. #2
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    Set Proofs

    Hello thehollow89
    Quote Originally Posted by thehollow89 View Post
    Let A, B, and C be sets. Prove the following claim, or give a counterexample:

    (A - B) - C = (A - C) - (B - C)
    (A-C)-(B-C) = (A\cap C')\cap(B\cap C')'

     = (A\cap C')\cap(B'\cup C) De Morgan's Law

    = (A \cap C' \cap B')\cup(A \cap C' \cap C) Distributive Law

    = (A \cap B' \cap C')\cup(A \cap \oslash) Commutative Law, Complement Law

    = (A \cap B' \cap C')\cup \oslash Identity Law

    = (A \cap B') \cap C' Identity Law

    =(A-B)-C

    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello thehollow89 (A-C)-(B-C) = (A\cap C')\cap(B\cap C')'

     = (A\cap C')\cap(B'\cup C) De Morgan's Law

    = (A \cap C' \cap B')\cup(A \cap C' \cap C) Distributive Law

    = (A \cap B' \cap C')\cup(A \cap \oslash) Commutative Law, Complement Law

    = (A \cap B' \cap C')\cup \oslash Identity Law

    = (A \cap B') \cap C' Identity Law

    =(A-B)-C

    Grandad
    Is A-B = A n B? When n is the upside down u sign.
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  4. #4
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    Quote Originally Posted by thehollow89 View Post
    Is A-B = A n B? When n is the upside down u sign.
    A\backslash B means A intersect B complement.
    That is, all of A that is not in B.
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  5. #5
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    Set notation

    Hello thehollow89
    Quote Originally Posted by thehollow89 View Post
    Is A-B = A n B? When n is the upside down u sign.
    A-B, also written (as Plato has done) A\backslash B, is the set of elements that are in A and not in B; that is, the set A \cap B' (notice the prime character ', after the B). You may find the notes at Discrete mathematics/Set theory - Wikibooks, collection of open-content textbooks helpful (which I have written under the name Nigeltn35).

    Grandad
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