Question

Let $S$ be the set of all real numbers. A relation $R$ has been defined on $S$ by $aRb⟺|a-b|\le 1$, then $R$ is

• A. reflexive and transitive but not symmetric
• B. an equivalence relation
• C. symmetric and transitive but not reflexive
• D. reflexive and symmetric but not transitive

Answer 1

Answer: (D)

Given, $aRb\iff |a-b|\le 1$
For Reflexive
$aRa=|a-a|=0\le 1$
So, it is reflexive.
For Symmetric
$aRb\iff |a-b|\le 1$
$\Rightarrow |b-a|\le 1$
i.e., $aRb\Rightarrow bRa$
So, it is symmetric.
For Transitive
Take $a=1,b=2$ and $c=3$
Now, $|a-b|=|1-2|=1$
and $|b-c|=|2-3|=1$
But $|a-c|=|1-3|=2>1$, which is not true.
i.e., $aRb,bRc\ne aRC$
So, it is not transitive.