Consider the following relations $R=\{(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w\}$; $S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q m=p m\}$. Then

Question

Consider the following relations

$R=\{(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w\}$;

$S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q m=p m\}$. Then

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

Neither $R$ nor $S$ is an equivalence relation

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

$R$ and $S$ both are equivalence relations

Solution

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