Relations
Equivalence Classes
JEE Main 2021
Grade 11
Question:
Let $A = \{2, 3, 4, 5, \dots, 30\}$ with equivalence $\simeq$ on $A \times A$ defined by $(a, b) \simeq (c, d) \Leftrightarrow ad = bc$. The number of ordered pairs equivalent to $(4, 3)$ is:
(1) $5$
(2) $6$
(3) $7$
(4) $8$
Step-by-Step Solution
Key Concept: $(a, b) \simeq (4, 3) \Leftrightarrow a/b = 4/3$. So $a = 4k, b = 3k$ for $k \in \mathbb{N}$, with $a, b \in A$: $k = 1, 2, \dots, 7$ (since $4 \times 7 = 28 \leq 30$ and $4 \times 8 = 32 > 30$). Count: 7 pairs.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)