# R is a relation on the set Z of integers and it is given by

Question:

R is a relation on the set Z of integers and it is given by

$(x, y) \in R \Leftrightarrow|x-y| \leq 1$. Then, $R$ is

(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) an equivalence relation

Solution:

(b) reflexive and symmetric

Reflexivity: Let $x \in R$. Then,

$x-x=0<1$

$\Rightarrow|x-x| \leq 1$

$\Rightarrow(x, x) \in R$ for all $x \in Z$

So, $R$ is reflexive on $Z$.

Symmetry: Let $(x, y) \in R$. Then,

$|x-y| \leq 0$

$\Rightarrow|-(y-x)| \leq 1$

$\Rightarrow|y-x| \leq 1$                                     [Since $|x-y|=|y-x|]$

$\Rightarrow(y, x) \in R$ for all $x, y \in Z$

So, $R$ is symmetri $c$ on $Z$.

Transitivity: Let $(x, y) \in R$ and $(y, z) \in R$. Then,

$|x-y| \leq 1$ and $|y-z| \leq 1$

$\Rightarrow$ It is not always true that $|x-y| \leq 1 .$

$\Rightarrow(x, z) \notin R$

So, $R$ is not transitive on $Z .$