Class JEE Sets, Relations, and Functions Question

Question

Let $R_1$ and $R_2$ be two relation defined as follows: $R_1 = {(a, b) \in R^2 : a^2 + b^2 \in Q}$ and $R_2 = {(a, b) \in R^2 : a^2 + b^2 
otin Q}$, where $Q$ is the set of all rational numbers. Then :

Accepted Answer

Correct Answer: A. Explanation: For $R_1$:  Let $a = 1 + \sqrt{2}$, $b = 1 - \sqrt{2}$, $c = \sqrt[4]{8}$.  $aR_1b : a^2 + b^2 = 6 \in Q$  $bR_1c : b^2 + c^2 = 3 - 2\sqrt{2} + 2\sqrt{2} = 3 \in Q$  $aR_1c : a^2 + c^2 = 3 + 2\sqrt{2} + 2\sqrt{2} 
otin Q$  $	herefore$ $R_1$ is not transitive.  For $R_2$:  Let $a = 1 + \sqrt{2}$, $b = \sqrt{2}$, $c = 1 - \sqrt{2}$  $aR_2b : a^2 + b^2 = 5 + 2\sqrt{2} 
otin Q$  $bR_2c : b^2 + c^2 = 5 - 2\sqrt{2} 
otin Q$  $aR_2c : a^2 + c^2 = 3 + 2\sqrt{2} + 3 - 2\sqrt{2} = 6 \in Q$  $	herefore$ $R_2$ is not transitive.

AI Tutor Explanation

More from this Chapter