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 & symmetric but not transitive
• D. $\rho$ is symmetric & transitive but not reflexive

Answer 1

The correct option is C $\rho$ is reflexive & symmetric but not transitive

Here, it's been given that
$x\rho y\Rightarrow x-y$ is zero or irrational
$\Rightarrow x\rho x\Rightarrow 0$
Hence, reflecxive.
Also, if $x\rho y\Rightarrow x-y$ is zero or irrational
$\Rightarrow y-x$ is zero or irrational
Hence, $y\rho x$ is symmetric too.
Also, $x\rho y\Rightarrow x-y$ is 0 or irrational
and $y\rho z\Rightarrow y-z$ is 0 or irrational
then $(x-y)+(y-z)=x-z$ may be rational
Hence, it is not transitive.
Thus, the relation is reflexive and symmetric but not transitive.
Thus, Option C. is correct.