Q50 / 155 · WBJEE 2017 · Mathematics
On $R$, the relation $\rho$ be defined by 'xpy holds if and only if $x-y$ is zero or irrational'. Then,
- A$\rho$ is reflexive and transitive but not symmetric
- Correct answerB$\rho$ is reflexive and symmetric but not transitive
- C$\rho$ is symmetric and transitive but not reflexive
- D$\rho$ is equivalence relation
Solution
We have, $x p y \Rightarrow x-y$ is zero or irrational. Now, $x-x=0$
$\Rightarrow(x, x) \in \rho$
$\therefore \rho$ is reflexive.
Again, if $x-y$ is either zero or irrational, then $y-x$ will also be either zero or inational.
$(x, y) \in \rho$
$\Leftrightarrow \quad(y, x) \in \rho$
$\therefore \rho$ is symmetric.
Again, $(2, \sqrt{3}) \in \rho$ and $(\sqrt{3}, 4) \in \rho$
$But (2,4) \notin \rho$
$\therefore p$ is not transitive.
Hence, the correct answer is option: 'B'.