Relations
Equivalence Relations
JEE Main 2022
Grade 11
Question:
A relation $R$ on $\mathbb{N}$ is given by $R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}$. $R$ is an equivalence relation if and only if $\alpha$ equals:
(1) $1$
(2) $4$
(3) $2$
(4) $5$
Step-by-Step Solution
Key Concept: Reflexive: $3x + \alpha x = x(3 + \alpha) \equiv 0 \pmod 7$ for all $x$ requires $3 + \alpha \equiv 0 \pmod 7 \Rightarrow \alpha = 4$. With $\alpha = 4$: $3x + 4y \equiv 0 \pmod 7 \Leftrightarrow x \equiv y \pmod 7$ (check: $3 + 4 = 7$). This is congruence modulo 7, which is an equivalence relation.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)