Question

On the set $N$ of all natural numbers definite 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

The correct option is B

Symmetric only

Explanation of the correct answer:

Determining the relation:

Given For $a,b\in N$, $aRb\iff GCD(a,b)=2$

For Reflexive, we should have $(a,a)\in R$, for all $a\in N$.

Now, $GCD(a,a)=a$ and for $3\in N$ $GCD(3,3)=3$. So, $(3,3)\notin R$.

Hence, $R$ is not Reflexive

For Symmetric, we should have $(b,a)\in R$, whenever $(a,b)\in R$, for all $a,b\in N$.

Now, we know that $GCD(a,b)=GCD(b,a)$. So,

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

Hence $R$ is Symmetric

For Transitive, we should have, if $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$, for all $a,b,c\in N$.

Now, $GCD(4,2)=2$ and $GCD(2,4)=2$ but $GCD(4,4)=4$.

So, $(4,2)\in R$ and $(2,4)\in R$ but $(4,4)\notin R$.

Hence, $R$ is not transitive.

Hence, option (B) is the correct answer.