Question

On the set $N$ of all natural numbers define the relation $R$ by $aRb$ 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

$aRa$ only if GCD of a and a is 2 which is clearly not the case.

Hence not reflexive.

$aRb\Rightarrow bRa$ if GCD of a and b is 2 then GCD of b and a is also 2.

The relation is symmetric.

Transitiveness is not necessary,i.e, if and b have GCD 2 and b and c also have GCD 2 a and c need not have GCD 2.

The relation is symmetric only.