Question

Let $S$ be the set of integers. For $a,b\in S,aRb$ if and only if $|a-b|<1$,then

• A. $R$ is not a reflexive relation
• B. $R$ is not a symmetric relation
• C. $R$ is an equivalence relation
• D. $R$ is not an equivalence relation

Answer 1

Answer: (C)

$R$ is an equivalence relation

Relation : $aRb$
Here $(a,a)\in R,(a-a)=0<1$
So it is reflexive.
Also $(a,b)\in R$
the $(b,a)\in R$ $\mid a-b\mid <1$
So it is symmetric also.

Also as $S$ is a set of integers, for the relation to be present, $a$ should be equal to $b$.

$\therefore aRb$ is transistive $\Rightarrow aRb$ is an equivalence relation.