Question

R is relation over the set of integers and it is given by (x, y) $ϵ$ R $\iff$ R |x - y| $\le$ 1. Then, R is

• A. Reflexive and transitive
• B. reflexive and symmetric
• C. Symmetric and transitive
• D. an equivalence relation

Answer 1

The correct option is B reflexive and symmetric

As (x,x) $ϵ$ R $\Rightarrow$ $|x-x|\le$ 1
$\Rightarrow$ 0 $\le$ 1 (True),
Thus, reflexive.
As (x,y) $ϵ$ R $\Rightarrow$ $|x-y|\le$ 1
$\Rightarrow |y-x||\le 1\Rightarrow$ (y,x) $ϵ$ R,
Thus, symmetric.
Again, (x, y) $ϵ$ R and (y, z) $ϵ$ R
$\Rightarrow |x-y|\le$ 1 and $|y-z|1\text{/}\Rightarrow |x-z|\le$ 1
$\therefore$ Not transitive