Question

Let $R$ be the relation on the set $R$, of all real numbers defined by $aRb$ if $|a-b|\le 1$. Then, $R$ is :

• A. Reflexive and symmetric
• B. Symmetric only
• C. Transitive only
• D. Anti - symmetric only

Answer 1

Answer: (A)

Reflexive and symmetric

We know that for every $a\in R,$ $|a-a|=0<1$

$\therefore aRa$ $\forall a\in R$
$\therefore R$ is reflexive.
Let $a,b\in R$ such that $aRb$

$\Rightarrow |a-b|\le 1\Rightarrow |b-a|\le 1\Rightarrow bRa$
$\therefore R$ is symmetric

Again $|1-\frac{1}{2}|\le 1$ and $|\frac{1}{2}-1|\le 1$

Then, $1R\frac{1}{2}$ and $\frac{1}{2}R1$
but $\frac{1}{2}\ne 1$
$\therefore R$ is not anti-symmetric
Further, $1R2$ and $2R3$

i.e. $|1-2|\le 1$ and $|2-3|\le 1$ i
But $|1-3|=2>1$

i.e. $1R3$ is not true
$\therefore R$ is not transitive.