Let a relation $R = \{(a, b)$: a is a factor of b, $a, b ∈ N\}$. Then, R is _____.

Reflexive and Transitive but NOT Symmetric.

The correct answer is Option (3) → Reflexive and Transitive but NOT Symmetric.

Given the relation:

$R=\{(a,b): a \text{ is a factor of } b,\; a,b\in \mathbb{N}\}$

Reflexive

For every $a\in \mathbb{N}$, $a$ is a factor of $a$. Therefore $(a,a)\in R$.

Symmetric

If $a$ is a factor of $b$, it is not necessary that $b$ is a factor of $a$. Example: $2$ is a factor of $6$ but $6$ is not a factor of $2$.

Hence $R$ is not symmetric.

Transitive

If $a$ is a factor of $b$ and $b$ is a factor of $c$, then $a$ is a factor of $c$.

Hence $R$ is transitive.

Final Answer

Reflexive and Transitive but NOT Symmetric.