Question

On the set $N$ of all natural numbers define the relation $R$ by $a$ and $b$ if and only if G.C.D. of $a$ and $b$ is $2$, then $R$ is

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

Answer 1

Answer: (B)

symmetric only

G.C.D of $a$ and $a$ is $a\ne 2$.

Hence, $R$ is not reflexive.

G.C.D. of $a$ and $b$ is $2$ and G.C.D of $b$ and $a$ is also $2$.

Hence, $R$ is symmetric.

For transitive, take an example: $a=4,b=6,c=16$

G.C.D of $a$ and $b$ is $2$ and G.C.D. of $b$ and $c$ is $2$. But, G.C.D. of $a$ and $c$ is $4\ne 2$

Hence, $R$ is not transitive.