(c) S is an equivalence relation but R is not an equivalence relationR = {(x, y) | x, y ∈ R, x = wy, w is a rational number} Reflexive: x R x ⇒ x = wx ⇒ w = 1, (a rational number) Hence R is reflexive. Symmetric: x R y \( ot\Rightarrow\) y R x as 0 R 1 ⇒ 0 = (0) . 1 where 0 is a rational number but 1 R 0 ⇒ 1 = (w) 0 which is not true for any rational number. ∴ R is not an equivalence relationS = \(\bigg\{\bigg(rac{m}{n},rac{p}{q}\bigg)\bigg|\) m, n, p, q, ∈ I, n, q, ≠ 0 and qm = pn\(\bigg\}\)Reflexive \(rac{m}{n}Srac{m}{n}\) ⇒ mn = nm (True)Symmetric \(rac{m}{n}Srac{p}{q}\) ⇒ mq = pn ⇒ pn = mq ⇒ \(rac{p}{q}Srac{m}{n}\) (True)Transitive \(rac{m}{n}Srac{p}{q}\) and \(rac{p}{q}Srac{r}{s}\)⇒ mq = pn and ps = qr ⇒ mq.ps = pn.qr ⇒ ms = nr⇒ \(rac{m}{n}\) = \(rac{r}{s}\) ⇒ \(rac{m}{n}\) S \(rac{r}{s}\) (True)∴ S is an equivalence relation.