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Math Help - Prove or disprove using boundary points

  1. #1
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    Prove or disprove using boundary points

    Let A and C be subsets of R^n with boundaries B(A), B(C) respectively.
    Prove or disprove:
    1. B(A\cup C)=B(A)\cup B(C)
    2. B(A\cap C)=B(A)\cap B(C)


    I haven't gotten to look much at boundaries yet, so I could use a hand.
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  2. #2
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    Quote Originally Posted by Runty View Post
    Let A and C be subsets of R^n with boundaries B(A), B(C) respectively.
    Prove or disprove:
    1. B(A\cup C)=B(A)\cup B(C)
    2. B(A\cap C)=B(A)\cap B(C)
    For #1. In \mathbb{R}^1 let A=[0,1]~\&~C=[1,2].

    For #2. In \mathbb{R}^1 let A=[0,1)~\&~C=(1,2].
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  3. #3
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    I'm afraid that doesn't help that much, or at least I don't know how to use that information. The source material we were given relating to boundaries is very scant on details and has no practice problems to speak of.

    The question itself, frankly, is not very clear on what it's looking for.
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  4. #4
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    Quote Originally Posted by Runty View Post
    I'm afraid that doesn't help that much, or at least I don't know how to use that information. The source material we were given relating to boundaries is very scant on details and has no practice problems to speak of.
    For #1. In \mathbb{R}^1 let A=[0,1]~\&~C=[1,2].
    \beta(A)=\{0,1\}~,~\beta(C)=\{1,2\}~\&~\beta(A\cup C)=\{0,2\}

    For #2. In \mathbb{R}^1 let A=[0,1)~\&~C=(1,2].
    \beta(A)=\{0,1\}~,~\beta(C)=\{1,2\}~\&~\beta(A\cap C)=\emptyset
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  5. #5
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    Okay, so far I've gotten this through Plato's suggestions (which I'm assuming are meant to disprove each equality):

    #1: B(A\cup C)=\{0,2\} and B(A)\cup B(C)=\{0,1,2\}, so it is disproven.

    #2: B(A\cap C)=\emptyset and B(A)\cap B(C)=\{1\}, so it is disproven.

    Am I on the right track, or did I make a mistake somewhere? I really can't be sure because of how little information I've been able to obtain on boundaries.

    EDIT: Made a mistake with notation on second answer. I put in the symbols for union instead of intersection. Fixed.
    Last edited by Runty; September 19th 2010 at 12:37 PM. Reason: Error with notation
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