The relation R on the set of real numbers defined by R = (a, b): a ≤b…

The correct answer is Option (3) → (B) and (C) only

(B) Not symmetric
(C) Neither reflexive nor transitive

Given: Relation \( R = \{(a, b) : a \leq b^2\} \) on the set of real numbers \( \mathbb{R} \)

(A) Reflexive:

For reflexivity, we need: \( a \leq a^2 \) for all \( a \in \mathbb{R} \)

Counterexample: Take \( a = \frac{1}{2} \Rightarrow \frac{1}{2} \not\leq \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)

So the relation is not reflexive.

(B) Symmetric:

We need: If \( a \leq b^2 \), then \( b \leq a^2 \)

Take \( a = 1, b = 2 \Rightarrow 1 \leq 4 \), but \( 2 \not\leq 1 \)

So the relation is not symmetric.

(C) Transitive:

We need: If \( a \leq b^2 \) and \( b \leq c^2 \), then \( a \leq c^2 \)

Take \( a = 5, b = 3, c = 2 \):

• \( 5 \leq 9 \) ✔️
• \( 3 \leq 4 \) ✔️
• \( 5 \leq 4 \) ❌

So the relation is not transitive.