If $R$ is transitive relation, is $RoR$ also transitive?

Question

It is true that equivalence relations are closed under composition with themselves, i.e., if $R$ is an equivalence relation $RoR$ is also so because $RoR =R$

But this does not imply that any transitive relation is also closed under composition with themselves. I wonder if this is the case or not, i.e, if $R$ is transitive, is $RoR$ transitive too?

Thanks.

Answer 1

$R$ is transitive iff $R\circ R\subseteq R$.

If that is the case then: $$\left(R\circ R\right)\circ\left(R\circ R\right)\subseteq R\circ R$$ so $R\circ R$ is indeed transitive.