Question

Let $n$ be a fixed positive integer. Let a relation $R$ be defined in $Z$ (the set of all integers) as follows: $aRb$, if $\frac{(a-b)}{n}$, that is, if $a-b$ is divisible by $n.$ Then the relation $R$, is

• A. Reflexive only
• B. Symmetric only
• C. Transitive only
• D. An equivalence relation

Answer 1

The correct option is D An equivalence relation

$R$ is reflexive as for any integer $a$ we have $a-a=0$ divisible by $n$. Hence $aRa$ for all $a\in Z$

If $R$ is symmetric, let $aRb$. Then by definition of $R$, $a-b=nk$ where $k\in Z$ and $b-a=-kn$ where $-k\in Z$ and so $bRa$.

Thus we have, $a-b=k_{1}n,b-c=k_{2}n$ where $k_1\in k_{2}Z$. then it follows that $(a-c)=(a-b)+(b-c)=k_{1}n+k_{2}n=n(k_1+k_2)$ where $k_1+k_2\in Z$