Question

On the set $N$ of all natural numbers, define the relation $R$ by $aRb$ iff the $\text{G.C.D.}$ of $a$ and $b$ is $2$. Then the relation $R$ is

• A. reflexive and symmetric
• B. symmetric only
• C. reflexive and transitive
• D. equivalence relation

Answer 1

The correct option is B symmetric only

$\text{G.C.D.}$ of $a$ and $a$ is '$a$'.
If $a\ne 2$, then $\text{G.C.D.}\ne 2$.
$\therefore R$ is not reflexive.

Let $aRb$
$\Rightarrow \text{G.C.D.}$ of $a,b=2$
i.e., $(a,b)=2$
$\Rightarrow (b,a)=2$
$\Rightarrow \text{G.C.D.}$ of $b,a=2$
$\therefore R$ is symmetric.

Again, let $aRb$ and $bRc$
Let $a=4,b=2,c=8$
$\text{G.C.D.}$ of $a,b=2$
$\text{G.C.D.}$ of $b,c=2$
But $\text{G.C.D.}$ of $a,c=4$
$\therefore R$ is not transitive.