Relations
Equivalence Classes
JEE Main 2021
Grade 11
Question:
Let $R = \{(P, Q) \mid P, Q \in \mathbb{R}^2, |OP| = |OQ|\}$ where $O$ is the origin. The equivalence class of $(1, -1)$ is the set:
(1) $\{(x, y) : x^2 + y^2 = 1\}$
(2) $\{(x, y) : x^2 + y^2 = 2\}$
(3) $\{(x, y) : x + y = 0\}$
(4) $\{(x, y) : x^2 - y^2 = 0\}$
Step-by-Step Solution
Key Concept: $R$ is an equivalence relation (equality of distances from origin). The equivalence class of $(1, -1)$ is the set of all points equidistant from $O$ as $(1, -1)$: $|(1, -1)| = \sqrt{2}$, so class $= \{(x, y) : x^2 + y^2 = 2\}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)