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Math Help - one prove question

  1. #1
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    one prove question

    Let A and C be subset of R^n with boundaries B(A), B(C) respectively. Prove or disprove:
    (1) B(AUC) = B(A)UB(C)
    (2) B(A \capC) = B(A) \capB(C)
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  2. #2
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    Quote Originally Posted by wopashui View Post
    Let A and C be subset of R^n with boundaries B(A), B(C) respectively. Prove or disprove:
    (1) B(AUC) = B(A)UB(C)
    (2) B(A \capC) = B(A) \capB(C)
    I haven't studied this formally per se, but both (1) and (2) seem to fail when we consider the simple case of two intersecting spheres in R^3, or even just intersecting discs in R^2.

    Edit: Actually easier to write in \,\mathbb{R},

    let A = [0,3], C = [1,2],

    so

    B(A) = \{0, 3\},

    B(C) = \{1, 2\},

    B(A \cup C) = \{0, 3\},

    B(A) \cup B(C) = \{0,1,2,3\},

    B(A \cap C) = \{1,2\},

    B(A) \cap B(C) = \emptyset
    Last edited by undefined; September 14th 2010 at 11:13 PM.
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  3. #3
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    Quote Originally Posted by undefined View Post
    I haven't studied this formally per se, but both (1) and (2) seem to fail when we consider the simple case of two intersecting spheres in R^3, or even just intersecting discs in R^2.

    Edit: Actually easier to write in \,\mathbb{R},

    let A = [0,3], C = [1,2],

    so

    B(A) = \{0, 3\},

    B(C) = \{1, 2\},

    B(A \cup C) = \{0, 3\},

    B(A) \cup B(C) = \{0,1,2,3\},

    B(A \cap C) = \{1,2\},

    B(A) \cap B(C) = \emptyset
    is this situation only in R^1 or applied in R^n
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by wopashui View Post
    is this situation only in R^1 or applied in R^n
    The edit was written just for \mathbb{R}, but in order to disprove a statement you only need one counterexample.

    Like I said though I haven't studied this formally but it seems all right.
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