Question

On the set $N$ of all natural numbers define the relation $R$ by $a R b$ if and only if the $G.C.D.$ 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)

symmetric only

Answer 2

The relation R is defined by $aRb,$ if and only if the GCD of a and b is 2. (i) $oRa,$ 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) $aRb,\text{ }bRc\Rightarrow cRa$ 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.