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Math Help - Infimum of subsets

  1. #1
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    Infimum of subsets

    Suppose that A and B are non-empty subsets of \mathbb{R}^2. Define the distance d(A,B) between A and B by the formula:

    d(A,B)=inf\{|\mathbf{a}-\mathbf{b}|:\mathbf{a}\in A, \mathbf{b}\in B\}

    1) Explain why this infimum always exists

    So because,

    |a-b|\geq 0 \Rightarrow there must be a greatest lower bound

    2) Suppose that A=\{(x,y) : x^2+y^2<1\} and B=\{(x,y) : x^2-y^2>9\}. Find d(A,B)

    Just from drawing a picture, I see that the smallest distance between the points on the graph is 2.

    |a-b|=|x^2+y^2-(x^2-y^2)|

    =|2y^2|

    Now, I'll try to plug in different values of y, and so the infimum of this is y=0

    When y=0\Rightarrow x^2<1, x^2>9

    x<1, x>3

    inf\{|a-b\}=inf\{|1+0-3-0|\}=2


    3) Find 2 disjoint sets A and B such that d(A,B)=0

    Is there any sets at all?



    Not too sure if I did it correct?

    Yeah I'm just edited my post to show what I've done
    Last edited by acevipa; September 16th 2010 at 07:38 AM.
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  2. #2
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    Quote Originally Posted by acevipa View Post
    Suppose that A and B are non-empty subsets of \mathbb{R}^2. Define the distance d(A,B) between A and B by the formula:

    d(A,B)=inf\{|\mathbf{a}-\mathbf{b}|:\mathbf{a}\in A, \mathbf{b}\in B\}

    1) Explain why this infimum always exists

    2) Suppose that A=\{(x,y) : x^2+y^2<1\} and B=\{(x,y) : x^2-y^2>9\}. Find d(A,B)

    3) Find 2 disjoint sets A and B such that d(A,B)=0
    I see that you have over two-hundred postings.
    By now you should understand that this is not a homework service
    So you need to either post some of your work on a problem or you need to explain what you do not understand about the question.

    Here is a hint on #1. |a-b|\ge 0.

    For #3. Consider A=\{(x,y) : x^2+y^2<1\} \text{ and } B=\{(x,y) : x^2-y^2>1\}
    Last edited by Plato; September 16th 2010 at 08:10 AM.
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