Consider the following relations:R = {(x, y) | x, y are real numbers and x = wy forsome rational number w};`S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"}`. Then(1)neither R nor Sis an equivalence relation(2)S is anequivalence relation but R is not an equivalence relation(3)R and S both areequivalence relations(4)R is anequivalence relation but S is not an equivalence relationA. R and S both are equivalence relations.B. R is an equivalence relation but S is not an equivalence relation.C. Neither R nor S is an equivalence relation.D. S is an equivalence relation but R is not an equivalence relation.

Consider the following relations:R = {(x, y) | x, y are real numbers a

1.	Consider the following relations:R = {(x, y) \| x, y are real numbers and x = wy forsome rational number w};`S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"}`. Then(1)neither R nor Sis an equivalence relation(2)S is anequivalence relation but R is not an equivalence relation(3)R and S both areequivalence relations(4)R is anequivalence relation but S is not an equivalence relationA. R and S both are equivalence relations.B. R is an equivalence relation but S is not an equivalence relation.C. Neither R nor S is an equivalence relation.D. S is an equivalence relation but R is not an equivalence relation.
Answer» Correct Answer - D xRy need not imply yRx. `(m)/(n) S(p)/(q)hArrqm=pn` `(m)/(n)S(m)/(n)impliesS` is reflexive `(m)/(n)S(p)/(q)` `implies (p)/(q) S(m)/(n)implies S` is symmetric `(m)/(n)S(p)/(q),(p)/(q) S(r)/(s)` `implies qm=pn,ps=rqimpliesms=rn` `implies (m)/(n) S(r)/(s)implies R ` is transitive. Hence, S is an equivalence relation.