Question

Let $n$ be a fixed positive integer. Define a relation $R$ in the set $Z$ of integers by $aRb$ if and only if $\frac{n}{a-b}$. The relation $R$ is

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

Answer 1

The correct option is B Symmetric

Given $aRb$ such that $R:\frac{n}{a-b}ϵZ$

$\left(A\right)aRa:$ $\frac{n}{a-a}=\infty$ and $\infty$ does not belongs to $Z$

So not a reflexive relation and hence (A) and (D) options are ruled out.

$\left(B\right)aRbϵZ$ we need to check whether $bRaϵZ$ or not.

$\frac{n}{a-b}ϵZ$ Now, $\frac{n}{b-a}\to -\frac{n}{a-b}$

As $\frac{n}{a-b}$ is integer so negative of it also integer

$\therefore bRaϵZ$

Hence relation $R$ is Symmetric.

$\left(C\right)$ Now $aRbϵZ$ and $bRcϵZ$ but we can't say anything about $aRcϵZ.$ Hence not a transitive relation.