Making a Relation Reflexive, Symmetric and Transitive
📺 Video Explanation
📝 Question
Given relation: \[ R = \{(1,2),(2,3)\} \text{ on } A = \{1,2,3\} \]
Add minimum number of ordered pairs so that the relation becomes reflexive, symmetric and transitive.
✅ Solution
🔹 Step 1: Make Reflexive
Reflexive requires: \[ (1,1),(2,2),(3,3) \]
Add: \[ (1,1),(2,2),(3,3) \]
🔹 Step 2: Make Symmetric
Given: \[ (1,2) \Rightarrow (2,1) \] \[ (2,3) \Rightarrow (3,2) \]
Add: \[ (2,1),(3,2) \]
🔹 Step 3: Make Transitive
Check chains:
\[ (1,2),(2,3) \Rightarrow (1,3) \]
Add: \[ (1,3) \]
Now symmetry also requires: \[ (3,1) \]
Add: \[ (3,1) \]
🔹 Step 4: Final Relation
\[ R’ = \{(1,1),(2,2),(3,3), (1,2),(2,1), (2,3),(3,2), (1,3),(3,1)\} \]
🎯 Final Answer
Minimum pairs added:
\[ (1,1),(2,2),(3,3), (2,1),(3,2), (1,3),(3,1) \]
Final relation becomes:
\[ R’ = A \times A \]
✔ Reflexive
✔ Symmetric
✔ Transitive
🚀 Exam Insight
- Reflexive → add all diagonal elements
- Symmetric → add reverse pairs
- Transitive → complete chains
- Often result becomes \( A \times A \)