Let a relation $R = \{(a, b)$: a is a factor of b, $a, b ∈ N\}$. Then, R is _____.
Answer & explanation
Correct answer: option 3
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.