On the set N of natural numbers, define the relation R by $a\, R\, b$ if the GCD of a and b is 2, then R is

symmetric only

For any $a ∈ N$, we have

(GCD of a and a) = a

So, R is not reflexive.

Let $(a, b) ∈ R$. Then,

$a\, R\, b$

⇒ GCD of a and b is 2

⇒ GCD of b and a is also 2

$⇒ b\,R\,a$

So, R is symmetric.

We observe that: GCD of 6 and 4 is 2 and GCD of 4 and 18 is also 2.

But, GCD of 6 and 18 is 6.

i.e. 6 R 4 and 4 R 18 but 6 R 18.

So, R is not transitive.