Question

Let $R$ be a relation on $R$, given by $R=\{(a,b):3a-3b+\sqrt{7}$ is an irrational number $\}$. Then $R$ is

• A. reflexive and symmetric but not transitive
• B. reflexive and transitive but not symmetric
• C. reflexive but neither symmetric nor transitive
• D. an equivalence relation

Answer 1

Answer: (C)

Check for reflexivity:
As $3(a-a)+\sqrt{7}=\sqrt{7}$
which belongs to relation so relation is reflexive
Check for symmetric:
Take $a=\frac{\sqrt{7}}{3},b=0$
Now (a, b) $\in R$ but $(b,a)\notin R$
As $3(b-a)+\sqrt{7}=0$
which is rational so relation is not symmetric.
Check for Transitivity:
Take (a, b) as $\left(\frac{\sqrt{7}}{3},1\right)$
$\&(b,c)$ as $\left(1,\frac{2\sqrt{7}}{3}\right)$
So now (a, b) $\in R\&(b,c)\in R$ but $(a,c)\notin R$ which means relation is not transitive