Question

Consider the following relations:
R = {(x, y) | x, y are real numbers and 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 $\ne$ 0 and qm = pn}.
Then

• A. Neither R nor S is an equivalence relation
• B. S is an equivalence relation but R is not an equivalence relation.
• C. R and S both are equivalence relations
• D. R is an equivalence relation but S is not an equivalence relation

Answer 1

The correct option is B S is an equivalence relation but R is not an equivalence relation.

X Ry need not implies yRx
$\therefore$ R is not symmetric and hence not an equivalence relation.
$S:\frac{m}{n}s\frac{p}{q}$
Given $qm=pn\Rightarrow \frac{p}{q}=\frac{m}{n}\therefore \frac{m}{n}s\frac{m}{n}\left(reflexive\right)\frac{m}{n}s\frac{p}{q}\Rightarrow \frac{p}{q}s\frac{m}{n}\left(symmetric\right)\frac{m}{n}s\frac{p}{q},\frac{p}{q}s\frac{r}{s}\Rightarrow qm=pm,ps=rq\Rightarrow \frac{p}{q}=\frac{m}{n}=\frac{r}{s}\Rightarrow ms=rn\left(transitive\right)$
S is an equivalence relation.