Consider the following relations: R=\ (x, y) x, y are real numbers and x= wy for some rational number w \ ; S=\ ((m)/(n), (p)/(q)) m, n, p. and q are integers such that n, q ≠ 0 and .q m=p n\ . Then

Correct answer: D. R is an equivalence relation but S is not an equivalence relation xRy need not implies y R xS: m n s p q qm=pn (m)/(n) s (m)/(n) reflexive array ll (m)/(n) s (p)/(q)& (p)/(q) s (m)/(n) symmetric (m)/(n) s (p)/(q), (p)/(q) s (r)/(s)& q m=p n, p s=r q m s=r n transitive. array S is an equivalence relation.

JEE Main 2010

Exam: JEE Main
Year: 2010
Subject: Mathematics
Chapter: Sets and Relations

JEE Main 2010 · Mathematics · Sets and Relations

Consider the following relations: R=\ (x, y) x, y are real numbers...

Consider the following relations:

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

;

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

. Then

Options

Aneither $\mathrm{R}$ 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
D $\mathrm{R}$ is an equivalence relation but $\mathrm{S}$ is not an equivalence relation

Correct answer

D. $\mathrm{R}$ is an equivalence relation but $\mathrm{S}$ is not an equivalence relation

Step-by-step 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.

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