Relations
Types of Relations
JEE Main 2023
Grade 11
Question:
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. The relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is:
(1) reflexive but not symmetric
(2) symmetric but neither reflexive nor transitive
(3) an equivalence relation
(4) transitive but not symmetric
Step-by-Step Solution
Key Concept: $R = \{(1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)\}$. Symmetric: $x + y = y + x\checkmark$. Not reflexive: $(x, x) \in R \Rightarrow 2x = 7$ (impossible for integers). Not transitive: $(1, 6), (6, 1) \in R$ but $(1, 1) \notin R$. Answer: (2).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)