Question

Let the relation $\rho$ be defined on $R$ by $a\rho b$ holds if and only if $a-b$ is zero or irrational then

• A. $\rho$ is equivalence relation
• B. $\rho$ is reflextive & symmetric but is not transitive
• C. $\rho$ is reflexive and transitive but is not symmetric
• D. $\rho$ is reflexive only

Answer 1

The correct option is B $\rho$ is reflextive & symmetric but is not transitive

For $\rho$ to be reflexive
Taking element $(a,a)$
$a\rho a\Rightarrow a-a=0$
So $\rho$ is reflexive
For $\rho$ to be symmetric
Taking element $(a,b)$ such that $a\ne b$
If $a-b$ irrational
$\Rightarrow b-a$ irrational
So $\rho$ is symmetric
For $\rho$ to be transitive
Let $1,\sqrt{2},2\in R$ such that
$1-\sqrt{2}=\text{irrational};\sqrt{2}-2=\text{irrational}\Rightarrow a-c=\text{rational}$
So $\rho$ is not transitive