Q72 / 155 · WBJEE 2018 · Mathematics
On $R$, a relation $\rho$ is defined by xpy if and only if $x-y$ is zero or irrational. Then.
- A$\rho$ is equivalence relation
- B$\rho$ is reflexive but neither symmetric nor transitive
- Correct answerC$\rho$ is reflexive and symmetric but not transitive
- D$\rho$ is symmetric and transitive but not reflexive
Solution
On the set $R$ $x p y \Rightarrow x-y$ is zero or irrational number.
Now, $x p x$
$\Rightarrow x-x=0$
$\Rightarrow \rho$ is reflexive.
If $x \rho y \Rightarrow x-y$ is zero or irrational.
$=-(y-x)$ is zero or irrational.
$\Rightarrow$ yox is zero or irrational.
$\Rightarrow \rho$ is symmetric.
And if $x \rho y \Rightarrow x-y$ is 0 or irrational.
$y \rho z \Rightarrow y-z$ is 0 or irrational.
Then, $(x-y)+(y-z)=x-z$ may be 0 or rational.
$\Rightarrow \rho$ is not transitive.
Now, $x p x$
$\Rightarrow x-x=0$
$\Rightarrow \rho$ is reflexive.
If $x \rho y \Rightarrow x-y$ is zero or irrational.
$=-(y-x)$ is zero or irrational.
$\Rightarrow$ yox is zero or irrational.
$\Rightarrow \rho$ is symmetric.
And if $x \rho y \Rightarrow x-y$ is 0 or irrational.
$y \rho z \Rightarrow y-z$ is 0 or irrational.
Then, $(x-y)+(y-z)=x-z$ may be 0 or rational.
$\Rightarrow \rho$ is not transitive.
Hence, the correct answer is option: 'C'.