Question

Let R be the relation on the set R of all real numbers defined by a R b if $|a-b|\le \frac{1}{2}$. Then R is?

• A. Reflexive and symmetric but not transitive
• B. Symmetric and transitive but not reflexive
• C. Transitive but neither reflexive nor symmetric
• D. Reflexive, symmetric and transitive

Answer 1

The correct option is A Reflexive and symmetric but not transitive

R is reflexive since $|a-a|=0\le \frac{1}{2}\forall a\in R$.

Also, R is symmetric as $|a-b|\le \frac{1}{2}$

$\Rightarrow |b-a|\le \frac{1}{2}$

But R is not transitive.

e.g.: If we take three numbers $\frac{3}{4},\frac{1}{3},\frac{1}{8}$ then $|\frac{3}{4}-\frac{1}{3}|=\frac{5}{12}\le \frac{1}{2}$ and $|\frac{1}{3}-\frac{1}{8}|=\frac{5}{24}\le \frac{1}{2}$

But $|\frac{3}{4}-\frac{1}{8}|=\frac{5}{8}>\frac{1}{2}$

Thus, $\frac{3}{4}R\frac{1}{3}$ and $\frac{1}{3}R\frac{1}{8}$ but $\frac{3}{7}R\frac{1}{8}$.