Relations
Building Equivalence Relations
JEE Main 2025
Grade 11
Question:
Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation on $\{1, 2, 3, 4\}$. The minimum number of elements that must be added to $R$ so that $R$ becomes an equivalence relation is _____.
Step-by-Step Solution
Key Concept: Add diagonal: $(1, 1), (2, 2), (4, 4)$ ($(3, 3)$ present). Add symmetric: $(2, 1), (3, 2)$. Transitivity: $(1, 2), (2, 3) \to (1, 3)$; need $(3, 1)$ for symmetry. Final relation has 10 elements; original had 3; elements added $= 7$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 7