Question

Let $R$ be a relation in the set $N$ of natural numbers defined by the relation $nRm\iff n$ is a factor of $m$. The relation $R$ is _____

• A. reflexive and symmetric only
• B. symmetric and transitive only
• C. reflexive and transitive only
• D. an equivalence relation

Answer 1

The correct option is C reflexive and transitive only

Let $n\in N,\therefore$ $n$ is a factor of $n$
$\therefore nRn.\therefore$ $R$ is reflexive
$3R6$ because $3$ is a factor of $6$.
Also $6$ is not a factor of $3$.
$\therefore (6,3)\text{/}\in R.\therefore R$ is not symmetric.
Let $aRb$ and $bRc$.
$\therefore b=a{k}_1$ and $c=b{k}_2$ for $k_1,k_2\in N$.
$\therefore c=\left(a{k}_1\right)k_2=a\left(k_1{k}_2\right)$ or a is a factor of $c$.
$\therefore aRc,\therefore$ $R$ is transitive.
$\therefore$ The correct answer is $C$