Smallest Equivalence Relation on Set \( A = \{1,2,3\} \)
📺 Video Explanation
📝 Question
Write the smallest equivalence relation on the set:
\[ A = \{1,2,3\} \]
✅ Solution
🔹 Definition
An equivalence relation must be:
- Reflexive
- Symmetric
- Transitive
The smallest such relation contains only the minimum required pairs.
🔹 Step 1: Reflexive Requirement
We must include:
\[ (1,1), (2,2), (3,3) \]
🔹 Step 2: Check Other Properties
– No additional pairs are needed for symmetry or transitivity – Because there are no cross pairs like (1,2), (2,3), etc.
✔ So this set already satisfies all three properties
🎯 Final Answer
\[ \boxed{R = \{(1,1), (2,2), (3,3)\}} \]
🚀 Exam Insight
- Smallest equivalence relation = identity relation
- Contains only (a,a) pairs
- No extra pairs needed