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Math Help - Can someone please helpe me with set theory proof?

  1. #1
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    Is my set thoery proof correct?

    Prove or disprove: For all sets A and B, (A-B) U (A intersect B) = A

    My attempt at a proof:
    This statement is true.

    Suppose that A and B are sets where (A-B) U (A intersect B) and x is an element (A-B) U (A intersect B).

    By definition of Union x is an element A-B OR x is an element of A intersect B (but not both).

    So if x is an element of (A-B) then by definition of difference x is an element of A and x is NOT an element of B. Therefore, A-B=A.

    ___________________________________

    If x is an element of A intersection B then x is an element of A and x is an element of B by definition of intersection.

    Therefore, x is an element of A intersect B.

    So, if x is an element of A intersect B then x is NOT and element of A-B and if x is an element of A-B then x is NOT an element of A intersect B.



    __________________________________________
    A-B=A because by definition of difference if x is an element of A-B then x is an element of A but it is NOT an element of B. So, x is an element of A which equals A.

    If that is NOT the case, then,


    x is must be an element of A intersect B, which means it is an element of A and an element of B, so either way if x is an element of B or not an element of B it is always an element of A and a set is always a subset of itself. Therefore, the original statement IS true and for all sets A and B, (A-B) U (A intersect B) = A
    END OF PROOF.
    Last edited by matthayzon89; April 10th 2010 at 08:50 AM.
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  2. #2
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    Quote Originally Posted by matthayzon89 View Post
    Prove or disprove: For all sets A and B, (A-B) U (A intersect B) = A

    My attempt at a proof:
    This statement is true.

    Suppose that A and B are sets where (A-B) U (A intersect B) and x is an element (A-B) U (A intersect B).

    By definition of Union x is an element A-B OR x is an element of A intersect B (but not both).

    So if x is an element of (A-B) then by definition of difference x is an element of A and x is NOT an element of B. Therefore, A-B=A.

    ___________________________________

    If x is an element of A intersection B then x is an element of A and x is an element of B by definition of intersection.

    Therefore, x is an element of A intersect B.

    So, if x is an element of A intersect B then x is NOT and element of A-B and if x is an element of A-B then x is NOT an element of A intersect B.



    __________________________________________
    A-B=A because by definition of difference if x is an element of A-B then x is an element of A but it is NOT an element of B. So, x is an element of A which equals A.

    If that is NOT the case, then,


    x is must be an element of A intersect B, which means it is an element of A and an element of B, so either way if x is an element of B or not an element of B it is always an element of A and a set is always a subset of itself. Therefore, the original statement IS true and for all sets A and B, (A-B) U (A intersect B) = A
    END OF PROOF.

    This is too long......and at least twice I saw A-B=A which, of course, is false in general.

    1) Suppose x\in (A-B)\cup(A\cap B)\Longrightarrow x\in A-B\,\,\,or\,\,\,x\in A\cap B\Longrightarrow \,\,anyway\,,\,\,x\in A \Longrightarrow (A-B)\cup(A\cap B)\subset A

    2) Suppose now x\in A ; if also x\in B\,\,\,then\,\,\,x\in A\cap B , otherwise x\notin B\Longrightarrow x\in A-B\Longrightarrow \,\,\,anyway\,\,\,x\in (A-B)\cup(A\cap B)\Longrightarrow A\subset (A-B)\cup(A\cap B)

    Tonio
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