Let $A = \mathbb{N} \times \mathbb{N} $, and let $R$ be an equivalence relation on $A$ such that:
$$R = \left\{\big((m,n),(h,k)\big) \in A \times A \mid m + k = n + h\right\}.$$
Prove that each equivalence class of $R$ contains exactly one element $(m,n)$ such that at least one of $m$ or $n$ is $0$.