Smallest Equivalence Relation Extension
📺 Video Explanation
📝 Question
Let:
\[ A = \{1,2,3\} \]
Given relation:
\[ R = \{(1,1), (2,2), (3,3), (1,3)\} \]
Write the ordered pairs to be added to make the smallest equivalence relation.
✅ Solution
🔹 Step 1: Check Reflexive
All elements: \[ (1,1), (2,2), (3,3) \] are already present.
✔ Reflexive satisfied
🔹 Step 2: Check Symmetric
Given: \[ (1,3) \in R \]
For symmetry, we need: \[ (3,1) \]
❌ Missing → must add \( (3,1) \)
🔹 Step 3: Check Transitive
Now consider: \[ (1,3), (3,1) \]
Then transitivity requires: \[ (1,1) \] (already present)
Also: \[ (3,1), (1,3) \Rightarrow (3,3) \] (already present)
✔ No new pairs required for transitivity
🎯 Final Answer
Pairs to be added:
\[ \boxed{\{(3,1)\}} \]
🚀 Exam Insight
- First ensure reflexive
- Then symmetric (most common missing pairs)
- Finally check transitive closure
- Goal = smallest addition