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Math Help - interior proofs

  1. #1
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    interior proofs

    For sets A,B subsets of the Reals

    a) int(A) intersect int(B) = int(A intersect B)
    b) bd(A U B) is subset of bd(A) U bd(B)
    c) Give examples of sets A and B where bd(A U B) = empty set and bd(A) = R = bd(B)
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Jdg6057 View Post
    For sets A,B subsets of the Reals

    a) int(A) intersect int(B) = int(A intersect B)
    b) bd(A U B) is subset of bd(A) U bd(B)
    c) Give examples of sets A and B where bd(A U B) = empty set and bd(A) = R = bd(B)
    a) Let x\in int(A)\cap int(B). Then x\in int(A) and x\in int(B). So \exists~\epsilon_1,\epsilon_2 such that N(x,\epsilon_1)\subset A and N(x,\epsilon_2)\subset B. Taking \epsilon=\min\{\epsilon_1,\epsilon_2\} guarantees that N(x,\epsilon)\subset A\cap B. So x\in int(A\cap B) and (int(A)\cap int(B))\subseteq int(A\cap B). See if you can prove the other direction.

    b) Let x\in bd(A\cup B). Then \forall~\epsilon>0, N(x,\epsilon)\cap(A\cup B)^c\neq\emptyset. Note that (A\cup B)^c=A^c\cap B^c.
    Spoiler:
    So N(x,\epsilon)\cap A^c\cap B^c\neq\emptyset. In particular, N(x,\epsilon)\cap A^c\neq\emptyset and N(x,\epsilon)\cap B^c\neq\emptyset, so x\in bd(A)\cup bd(B).


    c) Let X=\{z\in\mathbb{C}:\Im(z)\geq 0\} and Y=\{z\in\mathbb{C}:\Im(z)\leq0\}.

    bd(X) and bd(Y) consist of all z with \Im(z)=0; in other words, the real line. However, X\cup Y=\mathbb{C}, and bd(\mathbb{C})=\emptyset.
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