Among the relations S= \(a,b): a,bin R-\0\,\ 2+frac ab>0 \ and T= \(a,b): a,bin R,\ a^2-b^2in Z \ ,

Sets, Relations and Function: Among the relations S= \(a,b): a,bin R-\0\,\ 2+frac ab>0 \ and T=… | JEE Main 2023

JEE Main 2023 (Online) 31st January Evening ShiftQuestion 1

Among the relations $S=\left\{(a,b): a,b\in R-\{0\},\ 2+\frac ab>0\right\}$ and $T=\left\{(a,b): a,b\in R,\ a^{2}-b^{2}\in Z\right\}$,

Sets, Relations and Function

$S$ is transitive but $T$ is notboth $S$ and $T$ are symmetricneither $S$ nor $T$ is transitive$T$ is symmetric but $S$ is not✓ Correct answer

Solution

$S$: $a=1,b=-\frac25$ gives $2+\frac ab<0$ but $2+\frac ba>0$, so $S$ is not symmetric (it is not transitive either: $a/b=10,b/c=-1.5$ give $a/c=-15$). $T$: $a^2-b^2\in\mathbb Z\Rightarrow b^2-a^2\in\mathbb Z$, so $T$ is symmetric (and transitive). $T$ is symmetric but $S$ is not.

Sets, Relations and Function - JEE Mathematics question (JEE Main 2023 (Online) 31st January Evening Shift)

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