Relations
Equivalence Relations
JEE Main 2024
Grade 11
Question:
Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a, b) \, R \, (c, d)$ iff $ad - bc$ is divisible by $5$. Then $R$ is:
(1) reflexive and symmetric but not transitive
(2) reflexive and transitive but not symmetric
(3) reflexive, symmetric, and transitive
(4) symmetric and transitive but not reflexive
Step-by-Step Solution
Key Concept: Reflexive: $ad - bc = ab - ba = 0 \equiv 0 \pmod 5\checkmark$. Symmetric: if $5 \mid (ad - bc)$ then $5 \mid (bc - ad) = -(ad - bc)\checkmark$. Transitive: requires a careful check — $(ad - bc) \equiv 0$ and $(cf - de) \equiv 0 \pmod 5$; show $(af - be) \equiv 0$ using $5 \mid d(af - be) - f(ad - bc) + b(cf - de) = 0$. Transitive$\checkmark$. Answer: (3).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)