Question

On the set $N$ of all natural numbers, define the relation $R\times R$ be such that the greatest common divisor 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

If $R$ is a relation on set $N$, then for it to be reflexive,

$(a,a)\in R$ for all $a\in N$. But $(4,4)\notin R$. Hence not reflexive.
If greatest common divisor of $a$ and $b$ is $2$,

then greatest common divisor of $b$ and $a$ is also $2$.
So, the given relation is symmetric. $(\because aRb\Rightarrow bRa)$.
The given relation is not transitive as $(6,10)\in R$ and $(10,18)\in R$, but $(6,18)\notin R$ (greatest common divisor of $6$ and $18$ is $6$)