# Metric spaces that don't match

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• Apr 3rd 2011, 04:01 PM
Connected
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.
• Apr 3rd 2011, 05:14 PM
Drexel28
Quote:

Originally Posted by Connected
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.
• Apr 3rd 2011, 05:24 PM
Connected
Quote:

Originally Posted by Drexel28
if one takes $X_n=\{0,1\}^n$

What is this, explicitly?
• Apr 3rd 2011, 05:35 PM
Drexel28
Quote:

Originally Posted by Connected
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 $\mathbb{Z}_2^n$.