Relations
Types of Relations
JEE Main 2023
Grade 11
Question:
Let $R$ be a relation on $\mathbb{R}$ given by $R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is irrational}\}$. Then $R$ is:
(1) an equivalence relation
(2) reflexive and transitive but not symmetric
(3) reflexive but not symmetric
(4) transitive but not reflexive
Step-by-Step Solution
Key Concept: Reflexive: $3a - 3a + \sqrt{7} = \sqrt{7}$ (irrational)$\checkmark$. Symmetric: take $a = \sqrt{7}/3, b = 0$: $3(\sqrt{7}/3) - 0 + \sqrt{7} = 2\sqrt{7}$ (irrational), so $(a, b) \in R$. But $3(0) - 3(\sqrt{7}/3) + \sqrt{7} = 0$ (rational), so $(b, a) \notin R$. Not symmetric.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)