Sets, Relations and Function: Let R_1 and R_2 be two relations defined as follows :… | JEE Main 2020

JEE Main 2020 (Online) 3rd September Evening ShiftQuestion 1

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

Sets, Relations and Function

$R_1$ and $R_2$ are both transitive.$R_1$ is transitive but $R_2$ is not transitive.$R_2$ is transitive but $R_1$ is not transitive.Neither $R_1$ nor $R_2$ is transitive.✓ Correct answer

Solution

$R_1$ not transitive: take $a^2=\sqrt2,\ b^2=2-\sqrt2,\ c^2=\sqrt2$. Then $a^2+b^2=2$, $b^2+c^2=2$ rational, but $a^2+c^2=2\sqrt2$ irrational. $R_2$ not transitive: $a=0,\ b=2^{1/4},\ c=0$: $(a,b),(b,c)\in R_2$ (sum $\sqrt2$) but $(a,c)$ has $0\in\mathbf Q$, so $\notin R_2$. Neither is transitive. Option (D).

Sets, Relations and Function - JEE Mathematics question (JEE Main 2020 (Online) 3rd September Evening Shift)

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