Question

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

Answer 1

Answer: (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) $aRa,bRc\Rightarrow cRa$
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.