Q. On the set $N$ of all natural numbers define the relation $R$ by $aRb$if and only if the GCD of $a$ and $b$ is 2, then $R$ is
A) Reflexive, but not symmetric
B) Symmetric only
C) Reflexive, and transitive
D) Reflexive, symmetric and transitive
Explanation
(b)
(i) $aRa$, then GCD of $a$ and $a$ is $a$.
$\therefore R$ is not reflexive.
(ii) $aRb \Rightarrow bRa$
If GCD of $a$ and $b$ is 2, then GCD of $b$ and $a$ is 2.
$\therefore R$ is symmetric.
(iii) $aRa,\ bRc \nRightarrow cRa$
If GCD of $a$ and $b$ is 2 and GCD of $b$ and $c$ is 2, then it is need not to be GCD of $c$ and $a$ is 2.
$\therefore R$ is not transitive.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.