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Thread: Infimum of subsets

  1. #1
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    Infimum of subsets

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

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

    1) Explain why this infimum always exists

    So because,

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

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

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

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

    $\displaystyle =|2y^2|$

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

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

    $\displaystyle x<1, x>3$

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


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

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

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

    1) Explain why this infimum always exists

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

    3) Find 2 disjoint sets $\displaystyle A$ and $\displaystyle B$ such that $\displaystyle d(A,B)=0$
    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. $\displaystyle |a-b|\ge 0$.

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