The relation R in R (set of real numbers) is defined by $R = \{(a, b): a≤ b^3\}$, then R is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Neither reflexive nor symmetric nor transitive
Given: $R = \{(a,b) : a \le b^3\}$
Reflexive: For $(a,a)$, $a \le a^3$ must hold for all $a \in \mathbb{R}$.
Counterexample: $a = \frac{1}{2}$, $\frac{1}{2} \leq \left(\frac{1}{2}\right)^3 = \frac{1}{8}$. Not reflexive.
Symmetric: If $a \le b^3$, then $b \le a^3$ must hold.
Counterexample: $a = 1$, $b = 2$, $1 \le 8$ true, but $2 \le 1$ false. Not symmetric.
Transitive: If $a \le b^3$ and $b \le c^3$, then $a \le c^3$ must hold for all real numbers.
Counterexample: $a=2$, $b=1.5$, $c=1.1$: $2 \le (1.5)^3=3.375$ true, $1.5 \le (1.1)^3 \approx 1.331$ false. Not transitive.
Conclusion: Neither reflexive nor symmetric nor transitive.