prove or disprove reflexive of $R$

Question

How to prove or disprove that if $R^2$ is reflexive then also $R$ is reflexive ?

I tried to prove $R^2 \supseteq (x,x)\forall x \in R\implies R_{rex}$ but without success, maybe I have to find counetrexample?

Answer 1

Let $R$ be the relation in $\mathbb{R} \times \mathbb{R}$ defined by $R=\{(x,y) : \vert x-y \vert =1\}$. It is easy to see that $R^2$ is reflexive and $R$ is not.