Question

On a set $N$ of all natural numbers is defined the relation $R$ by $aRb$ iff the $GCD$ of $a$ and $b$ is $2$, then $R$ is

• A. Reflexive and Transitive
• B. Symmetric and Transitive
• C. Symmetric only
• D. Not reflexive, not symmetric, not transitive

Answer 1

The correct option is C

Symmetric only

Explanation of the Correct Option:

The correct option is C: Symmetric only

We will verify whether the relation is reflexive , symmetric , transitive or equivalence

Reflexive: Let $a\in N$.
$(x,x)\notin R$, as the $GCD$ of $\text{'}a\text{'}$and $\text{'}a\text{'}$ is $\text{'}a\text{'}$ not $2$
$\Rightarrow R$ is not reflexive.

Symmetric: Let $a,b\in N$. Then,
$(a,b)\in R\Rightarrow GCD$ of $\text{'}a\text{'}$and $\text{'}b\text{'}$ is $2$
$\Rightarrow GCDof\text{'}b\text{'}and\text{'}a\text{'}is2$
$\Rightarrow (b,a)\in R$
$\Rightarrow R$ is symmetric.

Transitive: Let $a,b,c\in N$. Then,
$(a,b)\in Nand(b,c)\in N$
$\Rightarrow GCD$ of $\text{'}a\text{'}and\text{'}b\text{'}$ is $2$ and $GCD$ of $\text{'}b\text{'}and\text{'}c\text{'}is2$
$\Rightarrow GCD$of $\text{'}a\text{'}and\text{'}c\text{'}isnotnecessarily2$.
$\Rightarrow R$ is not transitive

For example if $a=4,b=6,c=8$

Hence, the given relation is Symmetric but not Reflexive and Transitive

Therefore , the correct option is (C)