The relation R in R (set of real numbers) is defined by R = (a, b): a…

The correct answer is Option (4) → Neither reflexive nor symmetric nor transitive

Each property fails for a suitable counterexample.

Reflexive: it needs \(a \le a^3\) for every real a. For \(a = \frac{1}{2}\):

    \( \displaystyle \frac{1}{2} \le \frac{1}{8} \) is false

So \(\left(\frac{1}{2}, \frac{1}{2}\right) \notin R\), and R is not reflexive.

Symmetric: \((1, 2) \in R\) since \(1 \le 8\), but \((2, 1) \notin R\) since \(2 \le 1\) is false. So R is not symmetric.

Transitive: take a = 10, b = 3, \(c = \frac{3}{2}\):

    \((10, 3) \in R\), since \(10 \le 27\)

    \( \displaystyle \left(3, \frac{3}{2}\right) \in R, \text{ since } 3 \le \frac{27}{8} \)

    \( \displaystyle \left(10, \frac{3}{2}\right) \notin R, \text{ since } 10 \le \frac{27}{8} \text{ is false} \)

So R is not transitive.

NCERT (2026-27 edition):
Types of Relations