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. only symmetric
• C. reflexive and transitive
• D. reflexive, symmetric and transitive

Answer 1

The correct option is B

only symmetric

Explanation of correct option:

Determine the relation:
Given, $aRb\Rightarrow GCD$ of $a$ and $b$ is $2$

To find if the relation is reflexive, the following conditions should satisfy

$(a,a)\in R$

$aRa\Rightarrow GCD(a,a)=a$

Let's say,

$(3,3)\notin R$

$(4,4)\notin R$

Hence, it is not reflexive

For Symmetric

$(a,b)\in R\Rightarrow GCD(a,b)=2$

$(b,a)\in R\Rightarrow GCD(b,a)=2$

Hence, it is symmetric.

To find if the relation is transitive, the following conditions should satisfy

$$\begin{gathered} (a,b)\in R\Rightarrow GCD(a,b)=2 \\ (b,c)\in R\Rightarrow GCD(b,c)=2 \\ (a,c)\in R\Rightarrow GCD(a,c)=2 \end{gathered}$$

Say

$$\begin{gathered} (4,6)\in R \\ GCD(4,6)=2 \\ GCD(6,8)=2 \\ GCD(4,8)=4 \end{gathered}$$

Hence, it is not transitive.

Therefore, R is a symmetric relation.

Hence, the correct answer is option (B).