Q39 / 90 · JEE Main 2010 · Mathematics

Consider the following relations: $R=\{(x, y) \mid x, y$ are real numbers and $x=$ wy 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 $\left.q m=p n\right\}$. Then

  • A
    neither $\mathrm{R}$ nor $\mathrm{S}$ 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
  • Correct answer
    D
    $\mathrm{R}$ is an equivalence relation but $\mathrm{S}$ is not an equivalence relation

Solution

xRy need not implies $y R x$ $\mathrm{S}: \frac{\mathrm{m}}{\mathrm{n}} \mathrm{s} \frac{\mathrm{p}}{\mathrm{q}} \Leftrightarrow \mathrm{qm}=\mathrm{pn}$ $\frac{m}{n} s \frac{m}{n}$ reflexive $$ \begin{array}{ll} \frac{m}{n} s \frac{p}{q} & \Rightarrow \frac{p}{q} s \frac{m}{n} \text { symmetric } \\ \frac{m}{n} s \frac{p}{q}, \frac{p}{q} s \frac{r}{s} & \Rightarrow q m=p n, p s=r q \quad \Rightarrow m s=r n \text { transitive. } \end{array} $$ $S$ is an equivalence relation.

Hence, the correct answer is option: 'D'.

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