# Metric spaces that don't match

• Apr 3rd 2011, 04:01 PM
Connected
Metric spaces that don't match
Consider \$\displaystyle X(n)\$ the set of every ordered \$\displaystyle n-\$tuples of zeroes and ones. Prove that \$\displaystyle X(n)\$ is a metric space with the metric \$\displaystyle d(x,y)\$ is the number of entries where \$\displaystyle 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 \$\displaystyle d(x,y)\$ explicitly.
• Apr 3rd 2011, 05:14 PM
Drexel28
Quote:

Originally Posted by Connected
Consider \$\displaystyle X(n)\$ the set of every ordered \$\displaystyle n-\$tuples of zeroes and ones. Prove that \$\displaystyle X(n)\$ is a metric space with the metric \$\displaystyle d(x,y)\$ is the number of entries where \$\displaystyle 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 \$\displaystyle d(x,y)\$ explicitly.

This is often referred to as the 'weight' space. Namely, if one takes \$\displaystyle X_n=\{0,1\}^n\$ then one can define a norm on \$\displaystyle X_n\$ (yes, it is a vector space) by \$\displaystyle \|(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 \$\displaystyle 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 \$\displaystyle 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 \$\displaystyle \mathbb{Z}_2^n\$.