JEE Main 2011 — Sets and Relations Question with Solution

$\mathrm{x}-\mathrm{y}$ is an integer $x-x=0$ is an integer $\Rightarrow A$ is Reflexive $x-y$ is an integer $\Rightarrow y-x$ is an integer $\Rightarrow A$ is symmetric $x-y,y-z$ are integers As sum of two integers is an integer. $\Rightarrow (x-y)+(y-z)=x-z$ is an integer $\Rightarrow \mathrm{A}$ is transitive. Hence statement $-1$ is true. Also $\frac{\mathrm{x}}{\mathrm{x}}=1$ is a rational number $\Rightarrow B$ is reflexive $\frac{\mathrm{x}}{\mathrm{y}}=\alpha$ is rational $\Rightarrow \frac{\mathrm{y}}{\mathrm{x}}$ need not be rational i.e., $\frac{0}{1}$ is rational $\Rightarrow \frac{1}{0}$ is not rational Hence B is not symmetric $\Rightarrow B$ is not an equivalence relation.