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) $|a-a|=0\le 1$ is always true
(ii) $aRb\Rightarrow |a-b|\le 1\Rightarrow |-(a-b)|\le 1\Rightarrow |b-a|\le 1\Rightarrow bRa$. So, R is symmetric
(iii) 2R 1 and $1R\frac{1}{2}$
But, 2 is not related to $\frac{1}{2}$. So, R is not transitive.