Question

Let R be the realtion on the set R of all real

numbers defined by a R b if |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

|a - a| = 0 < 1 $\therefore$ a R a ∀ a $\in$ R

$\therefore$ R is reflexive.

Again a R b $\Rightarrow$ |a-b| $\le 1\Rightarrow |b-a|\le 1\Rightarrow bRa$
$\therefore$ R is symmetric, Again $1R\frac{1}{2}$ and $\frac{1}{2}RI$ but $\frac{1}{2}\ne 1$
$\therefore$ R is not anti - symmetric
Further, $1R_2$ and $2R_3$ $[\because |1-3|=2>1]$
$\therefore$ R is not transitive.