1. ## 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.

2. Originally Posted by Connected
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?

3. No, I don't get it well.

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

4. If $S(a,r)=\{x\in E\mid d(x,a), 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?