Thread: Metric spaces that don't match

1. 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.

2. 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.

3. Originally Posted by Drexel28
if one takes $\displaystyle X_n=\{0,1\}^n$
What is this, explicitly?

4. 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 $\displaystyle \mathbb{Z}_2^n$.