Relations
Types of Relations
JEE Main 2021
Grade 11
Question:
Let $R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : x^3 - 3x^2y - xy^2 + 3y^3 = 0\}$. Then $R$ is:
(1) reflexive and symmetric but not transitive
(2) reflexive but neither symmetric nor transitive
(3) symmetric and transitive but not reflexive
(4) an equivalence relation
Step-by-Step Solution
Key Concept: Factor: $(x - 3y)(x - y)(x + y) = 0$. Since $x, y \in \mathbb{N}$, $x + y > 0$, so $x = y$ or $x = 3y$. Reflexive $(x = x)\checkmark$. Not symmetric: $(3, 1) \in R$ but $(1, 3) \notin R$. Not transitive: $(9, 3) \in R$ and $(3, 1) \in R$ but $(9, 1) \notin R$ (since $9 \neq 1$ and $9 \neq 3$). Answer: (2).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)