We will show that the reflexive closure of transitive closure of a relation is equal to the transitive closure of its reflexive closure through a series of simpler results:
(1) let r1 be a transitive relation on a set s, show that its reflexive closure rr1 is also transitive.
(2) let r2 be a reflexive relation on a set s, show that its transitive closure tr2 is also symmetric.
(3) using the previous results or otherwise, show that r(tr) = t(rr) for any relation r on a set. hint: you may fine the fact that transitive (resp. reflexive) closures of r are the smallest transitive (resp. reflexive) relation containing r useful.