Question

Let $R$ be the relation on the set $R$ of all real numbers, defined by $aRb$ If $|a - b| \le 1$. Then, $R$ is

• A. reflexive and symmetric only
• B. reflexive and transitive only
• C. equivalence
• D. None of the above

Answer 1

Answer: (A)

reflexive and symmetric only

Answer 2

Since, $|a-a|=0 \leq 1$, so $a R a, \forall a \in R$
$\therefore R$ is reflexive.
Now, $a R b \Rightarrow |a-b| \leq 1$
$\Rightarrow |b-a| \leq 1$
$\Rightarrow b R a$
$\therefore R$ is symmetric.
But $R$ is not transitive as
$1 R 2,2 R 3$ but $1, R 3$
$[\because|1-3|=2>1]$