Consider the following relations. R={(x, y)|x, y are real numbers and x = wy for some rational number w} S={mn,pq m,n,p and q are integers such that n,q≠0 and qm = pn}, then

xRy need not implies yRx  since 0,5∈R for taking  w=0 but 5,0∉R for any rational number w.R is not symmetric and hence it is not an equivalence relation.S is reflexive since  mnSmn⇔mn=nmmnSpq⇒pqSmn is symmetric.and mnSpq,pqSrt⇒qm=pn,pt=qr⇒mt=nr⇒mnSrt is transitive.∴S is an equivalence relation