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Math Help - Metric spaces that don't match

  1. #1
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    Metric spaces that don't match

    Consider X(n) the set of every ordered n-tuples of zeroes and ones. Prove that X(n) is a metric space with the metric d(x,y) is the number of entries where x,y don't match.

    I have no idea what to do here... I know what a metric space verifies, but I don't get how to define d(x,y) explicitly.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    Consider X(n) the set of every ordered n-tuples of zeroes and ones. Prove that X(n) is a metric space with the metric d(x,y) is the number of entries where x,y don't match.

    I have no idea what to do here... I know what a metric space verifies, but I don't get how to define d(x,y) explicitly.
    This is often referred to as the 'weight' space. Namely, if one takes X_n=\{0,1\}^n then one can define a norm on X_n (yes, it is a vector space) by \|(x_1,\cdots,x_n)\|=\#\{x_k:x_k=1\} then your metric is just the one induced by that norm. So, just prove that it is a norm.
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    if one takes X_n=\{0,1\}^n
    What is this, explicitly?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    What is this, explicitly?
    The set of all n-tuples with one or zero entries. Maybe it'd be easier, since this is what you need to think of it as a vector space, as \mathbb{Z}_2^n.
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