Coding Theory Set Problem Supose that $\mathbb{A}$ is a finite set and take $\overline{u},\overline{v} \in \mathbb{A}^n$. Let:
$$X=\{\overline{x} \in \mathbb{A}^n\mid d(\overline{u},\overline{x})<d(\overline{v},\overline{x})\}$$
$$Y=\{\overline{y} \in \mathbb{A}^n\mid d(\overline{u},\overline{y})>d(\overline{v},\overline{y})\}$$
Prove that $Card(X)=Card(Y)$
I am really stuck on this problem. I thought about creation some bijection between both sets. But I don't really know how to do that, what makes me feel it's not the good path. Maybe I have to use some propiety about distances. 

It's not necessary that the underlying set is a field (or abelian group). Just show that the cardinality of the set $\{x\in A^n\mid d(u,x)=i\}$ with $i\geq 0$ fixed is independent of the choice of $u\in A^n$.
Then we can write $d_i = |\{x\in A^n\mid d(u,x)=i\}|$ for $i\geq 0$.
