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Math Help - Union of sets in R^n

  1. #1
    Senior Member I-Think's Avatar
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    Union of sets in R^n

    Question about a question
    The question states:
    For A,B\subset{{R^n}}, prove that \overline{A\cup{B}}=\overline{A}\cup{\overline{B}}

    Shouldn't it be \overline{A}\cap{\overline{B}} by De Morgan's Laws?
    Also, I thought of this counter example in {R{^2}}

    Let A,B\subset{{R{^2}}}
    x\in{A} if x_1 even, x_2=3k, k\in{\mathbb{Z}}
    x\in{B} if x_1 even, x_2=4k, k\in{\mathbb{Z}}

    Now consider y=(4,8). y\in{\overline{A}}, thus y\in{\overline{A}\cup{\overline{B}}}.
    But y not in \overline{A\cup{B}}
    Thus \overline{A\cup{B}}\neq{\overline{A}\cup{\overline  {B}}}

    Where is my mistake or is this the question's mistake?
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  2. #2
    MHF Contributor

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    Re: Union of sets in R^n

    Quote Originally Posted by I-Think View Post
    The question states:
    For A,B\subset{{R^n}}, prove that \overline{A\cup{B}}=\overline{A}\cup{\overline{B}}

    Shouldn't it be \overline{A}\cap{\overline{B}} by De Morgan's Laws?
    I think you are miss-reading the question.
    \overline{A} is not the complement of A. It is the closure of A.

    The theorem is: The closure of a union is the union of closures.
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