Making a Relation Reflexive and Transitive
📺 Video Explanation
📝 Question
Let \( A = \{a,b,c\} \) and
\[ R = \{(a,a),(b,c),(a,b)\} \]
Find the minimum number of ordered pairs to be added so that \( R \) becomes reflexive and transitive.
✅ Solution
🔹 Step 1: Make Reflexive
Reflexive requires: \[ (a,a),(b,b),(c,c) \]
Already present: \( (a,a) \)
Missing: \[ (b,b),(c,c) \]
➡️ Add: \[ (b,b),(c,c) \]
🔹 Step 2: Make Transitive
Check chains:
\[ (a,b),(b,c) \Rightarrow (a,c) \]
But \( (a,c) \notin R \)
➡️ Add: \[ (a,c) \]
🔹 Step 3: Verify Transitivity
New relation: \[ R’ = \{(a,a),(b,c),(a,b),(b,b),(c,c),(a,c)\} \]
Check chains:
- \( (a,b),(b,c) \Rightarrow (a,c) \) ✔
- \( (a,a),(a,b) \Rightarrow (a,b) \) ✔
- \( (b,b),(b,c) \Rightarrow (b,c) \) ✔
✔ All conditions satisfied.
🎯 Final Answer
Minimum pairs added:
\[ (b,b),(c,c),(a,c) \]
Total = 3 pairs
🚀 Exam Insight
- Reflexive → add missing diagonal elements
- Transitive → complete chains like (a,b),(b,c)
- Always recheck after adding pairs