CBSE · Class 11 · Mathematics · Relations and FunctionsLet $R$ be a relation on the set $\mathbb{R}$ of real numbers defined by $R = \{(a, b) : a \le b^2\}$. Is $R$ reflexive?
Step-by-Step Solution
A relation is reflexive if $(a, a) \in R$ for all $a \in \mathbb{R}$, which means $a \le a^2$ for all real numbers $a$. Let's test with $a = 1/2$. Then $1/2 \le (1/2)^2 \implies 1/2 \le 1/4$, which is false. Therefore, $R$ is not reflexive. Option C is correct.
Detailed Options Breakdown
Option : Yes, always
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.
Option 1: Only for positive numbers
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.
Option 2: No (Correct Answer)
Correct choice. Refer to the step-by-step verified solution guidelines above for details.
Option 3: Only for integers
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.
💡 Study Guide: This question tests core syllabus concepts from Relations and Functions. For formulas, key summaries, and mock exam reference guides, read the full Relations and Functions Revision Notes.