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Math Help - Can someone verify/correct this proof?

  1. #1
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    Can someone verify/correct this proof?

    Can you conclude that A=B if A,B, and C are sets such that:
    A U C = B U C and A intersection C = B intersection C

    I know its true but I'm not sure of the proof:

    pf: Let A,B,C be sets.
    Suppose x be in AUC. Then x is in BUC.
    Then x is in A and x is in B or x is in.
    Suppose x is in A intersect C. Then x is in B intersect C.
    Then x is A, x is in B, and x is in C.
    Since x is in A and x is in then x is in AUC, BUC.
    Since x is in C, then x is in AUC, BUC.
    Thus, from these last two lines, x in A
    intersect C and x is in B intersect C.
    Therefore, if x is in AUC=BUC then x is automatically in A and B or in C.
    If x is in A
    intersect C = B intersect C, then x is A, B, and C.
    A=B since x is in the intersection of A and C and
    intersection of B and C and also since x appears in the AUC and BUC regardless of C, because of the statement x is in A and x is in B or x is in C.

    I really don't think this proves anything...anyone know of a more obvious or quick approach?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathgirl13 View Post
    Can you conclude that A=B if A,B, and C are sets such that:
    A U C = B U C and A intersection C = B intersection C

    I know its true but I'm not sure of the proof:

    pf: Let A,B,C be sets.
    Suppose x be in AUC. Then x is in BUC.
    Then x is in A and x is in B or x is in.
    Suppose x is in A intersect C. Then x is in B intersect C.
    Then x is A, x is in B, and x is in C.
    Since x is in A and x is in then x is in AUC, BUC.
    Since x is in C, then x is in AUC, BUC.
    Thus, from these last two lines, x in A intersect C and x is in B intersect C.
    Therefore, if x is in AUC=BUC then x is automatically in A and B or in C.
    If x is in A intersect C = B intersect C, then x is A, B, and C.
    A=B since x is in the intersection of A and C and intersection of B and C and also since x appears in the AUC and BUC regardless of C, because of the statement x is in A and x is in B or x is in C.

    I really don't think this proves anything...anyone know of a more obvious or quick approach?
    Let x\in A, clearly then x\in A\cup C\implies x\in B\cup C. We can see then that either x\in B or x\in C or it's in both. If the first or third is true we are done, so assume then that x\in C,x\notin B. Then x\notin B\cap C\implies x\notin A\cap C but this is a contradiction since both x\in A,x\in C. Thus x\in B and A\subseteq B. Use similar logic to show that B\subseteq A
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  3. #3
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    uhh, thank you!! That makes much more sense!!
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