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Math Help - Discrete metric

  1. #1
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    Discrete metric

    Let (E,d) be the discrete space.

    Compute S(a,r) for r>1, and S(a,r) for r\le1.

    I know that d(x,y) is defined to be 1 for x\ne y and 0 for x=y, but I don't know exactly how to use that to solve the problem.
    Last edited by Connected; March 26th 2011 at 05:42 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    Let (E,d) be the discrete space.

    Compute S(a,r) for r\ge1, and S(a,r) for r\le1.

    I know that d(x,y) is defined to be 1 for x\ne y and 0 for x=y, but I don't know exactly how to use that to solve the problem.
    I assume that S(a,r) is the 'sphere' (that is an uncommon noation) of radius r centered at a. I think you're overthinking your problem. S(a,r) has a simple formulation. Think about fixing this one point a then one can think (purely heuristically) as the situation being analgous to a being the origin in \mathbb{R}^2 and E-\{a\} being the unit circle. With this in mind, is the solution clear?
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  3. #3
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    No, I don't get it well.

    I understand a bit your reasoning, but is there a way to make it analytically?

    Thanks for your help.
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  4. #4
    Senior Member Tinyboss's Avatar
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    If S(a,r)=\{x\in E\mid d(x,a)<r\}, then what happens in the two cases r<1 and r\ge1? In particular, what do you know if d(x,y)<1 in this metric?
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