Consider the following relations:

R = {(x, y) | x, y are real numbers an x = wy for some rational number w};

$S=\{(\frac{m}{n},\frac{p}{q})|m,n,p$ and q are integers such that n, q ≠ 0 and qm = pn}. Then:

S is an equivalence relation but R is not an equivalence relation

xRy need not implies yRX

$S:\frac{m}{n}s\frac{p}{q}⇔qm=pn$

$\frac{m}{n}s\frac{m}{n}$ reflexive

$\frac{m}{n}s\frac{p}{n}⇔\frac{p}{q}s\frac{m}{n}$ symmetric

$\frac{m}{n}s\frac{p}{q},\frac{p}{q}s\frac{r}{s}⇒pn,ps,=rq⇒ms=m$ transitive

S is an equivalence relation.