Expressing equivalence relation on $\mathbb{R}$ as an union of cartesian products of the set with itself

Question

Let $R \subseteq \mathbb{R^2}$ be an equivalence relation. I am trying to prove, that there exist a family $\mathcal{B} \subseteq \mathcal{P}(\mathbb{R})$ such that we can write $$R = \bigcup_{B \in \mathcal{B}} B \times B$$ I am stuck on this problem, I don't see how to start it.

Answer 1

Let $\mathcal{B}$ be the family of equivalence classes of $R$. Then, a general property of equivalence relations says that two elements are $R$-equivalent if and only if they both lie within the same equivalence class in $\mathcal{B}$.