Let $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ be two relations defined as follows: $\mathrm{R}_{1}=\left\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R}^{2}: \mathrm{a}^{2}+\mathrm{b}^{2} \in \mathrm{Q}\right\}$ and $R_{2}=\left\{(a, b) \in R^{2}: a^{2}+b^{2} \notin Q\right\}$, where $Q$ is the set of all rational numbers. Then:

(A) Neither $\mathrm{R}_{1}$ nor $\mathrm{R}_{2}$ is transitive.

(B) $\mathrm{R}_{1}$ is transitive but $\mathrm{R}_{2}$ is not transitive.

(C) $R_{1}$ and $R_{2}$ are both transitive

(D) $R_{2}$ is transitive but $R_{1}$ is not transitive.

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Sets Relations and Functions

Let

R_{1}

and

R_{2}

be two relations defined as follows:

R_{1} = {(a, b) \in R^{2} : a^{2} + b^{2} \in Q}

and

R_{2} = {(a, b) \in R^{2} : a^{2} + b^{2} \in / Q}

, where

Q

is the set of all rational numbers. Then:

Answer: (A)

Test

R_{1}

, where the sum of squares is rational. Take

a = 1 + 2

b = 1 - 2

c = 8^{1/4}

a^{2} + b^{2} = 6

is rational and

b^{2} + c^{2} = (3 - 22) + 22 = 3

is rational, yet

a^{2} + c^{2} = (3 + 22) + 22 = 3 + 42

is irrational, so

R_{1}

is not transitive. Test

R_{2}

, where the sum of squares is irrational. Take

a = 1 + 2

b = 2

c = 1 - 2

a^{2} + b^{2} = 5 + 22

and

b^{2} + c^{2} = 5 - 22

are both irrational, yet

a^{2} + c^{2} = 6

is rational, so

R_{2}

is not transitive either. Neither relation is transitive.

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Single correct option