Question

Consider the following relations:
$R=\left\{\right(x,y$) $|x,y$ 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

Answer: (B)

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

$xRy$ need not implies $yRx$.
$S:s\iff qm=pn$
$\frac{m}{n}s\frac{m}{n}$ reflexive.
$\frac{m}{n}s\frac{p}{q}=\frac{p}{q}s\frac{m}{n}$ symmetric.
$\frac{m}{n}s\frac{p}{q},\frac{p}{q}s\frac{r}{s}=qm=pn,ps=rq$$S$ is an equivalence relation.
$=ms=rn$ transitive.