If $R$ is an equivalence relation, does $R = R^3$ ?
I tried for about 40minutes to construct a relation $R$ that is an equivalence relation that when multiplied with itself twice, it will make $R = R^3$ false and haven't succeeded.
Would love to see a counter example
If $R$ is a reflexive relation, then $R\subseteq R^2$. In fact, if $(a,b)\in R$, then $(a,c)\in R$ and $(c,b)\in R$ is satisfied by $c=a$. Therefore $(a,b)\in R^2$.
If $R$ is transitive, $R^2\subseteq R$. If you think about it, it is exactly the definition of transitivity.
For a relation $R$ satisfying both (for instance, equivalences and non-strict orderings) it holds $R^2=R$, hence $R^n=R$ for all $n\ge1$.