Is Every Symmetric and Transitive Relation Reflexive?
📺 Video Explanation
📝 Question
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
✅ Solution
🔹 Statement
The statement is False.
🔹 Counterexample
Let \( A = \{1,2\} \)
Define relation:
\[ R = \{(1,1)\} \]
🔹 Check Properties
Symmetric:
Since \( (1,1) \in R \), its reverse is itself. ✔ Symmetric
Transitive:
\[ (1,1), (1,1) \Rightarrow (1,1) \in R \]
✔ Transitive
Reflexive:
For reflexive, need: \[ (1,1), (2,2) \]
But \( (2,2) \notin R \)
❌ Not Reflexive
🎯 Final Conclusion
✔ Relation is symmetric and transitive
❌ But not reflexive
\[ \therefore \text{A relation can be symmetric and transitive but not reflexive} \]
🚀 Exam Insight
- Symmetric + Transitive does NOT guarantee reflexive
- Always check all diagonal elements for reflexive
- Use small sets like {1,2} for counterexamples