Question

Let S be the set of all real numbers and let R be a relation on S, defined by $a$ R $b\iff |a-b|\le 1$. Then, R is?

• A. Reflexive and symmetric but not transitive
• B. Reflexive and transitive but not symmetric
• C. Symmetric and transitive but not reflexive
• D. An equivalence relation

Answer 1

The correct option is A Reflexive and symmetric but not transitive

(I) Reflexive

$|a-a|=0\le 1$ is always true.

(ii) Symmetric

$aRb\Rightarrow |a-b|\le 1\Rightarrow |-(a-b)|\le 1\Rightarrow |b-a|\le 1\Rightarrow bRa.$

(iii) Transitive

$2R1$ and $1R\frac{1}{2}$.

But, $2$ is not related to $\frac{1}{2}$.

So, R is not transitive.