Question

Let $R$ be a relation on $R$, given by $R=\{(a, b): 3 a-3 b+\sqrt{7}$ 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