Question

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

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

Answer 1

The correct option is A Reflexive and symmetric

Given,
$aRb$ if $|a-b|\le 1$

$\left(i\right)$ Now, $|a-a|=0<1$
$\therefore aRa,\forall a\in R$
Hence, $R$ is reflexive.

$\left(ii\right)$ If $aRb\Rightarrow |a-b|\le 1$
$\Rightarrow |b-a|\le 1$ (using property of modulus)
$\Rightarrow bRa$
Hence, $R$ is symmetric.

$\left(iii\right)$ We know that $1R\frac{1}{2}$ and $\frac{1}{2}R1,$ but $\frac{1}{2}\ne 1$
Hence, $R$ is not anti-symmetric.

$\left(iv\right)1R2$ i.e. $|1-2|=1$ and $2R3$ i.e. $|2-3|=1,$ but $1R3$ is not possible as $|1-3|=2>1$
Hence, $R$ is not transitive.