Question

On $R$, a relation $\rho$ is defined by $x \rho y$ 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
• C. $\rho$ is reflexive and symmetric but not transitive
• D. $\rho$ is symmetric and transitive but not reflexive

Answer 1

The Correct Option is C

On the set $R$
$x \rho y \Rightarrow x-y$ is zero or irrational number.
Now, $x \rho x$
$\Rightarrow x-x=0$
$\Rightarrow \rho$ is reflexive.
If $x \rho y \Rightarrow x-y$ is zero or irrational.
$=-(y-x)$ is zero or irrational.
$\Rightarrow y \rho x$ 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.