Checking Reflexive, Symmetric and Transitive Relations
📺 Video Explanation
📝 Question
Let \( A = \{1,2,3\} \)
\( R_1 = \{(1,1), (1,3), (3,1), (2,2), (2,1), (3,3)\} \)
\( R_2 = \{(2,2), (3,1), (1,3)\} \)
\( R_3 = \{(1,3), (3,3)\} \)
Check whether each relation is reflexive, symmetric and transitive.
✅ Solution
🔹 Relation \( R_1 \)
Reflexive:
Since \( (1,1), (2,2), (3,3) \in R_1 \) ✔ Reflexive
Symmetric:
Check: \( (2,1) \in R_1 \) but \( (1,2) \notin R_1 \) ❌ Not Symmetric
Transitive:
Check: \( (2,1) \) and \( (1,3) \in R_1 \) ⇒ should have \( (2,3) \), but not present ❌ Not Transitive
🔹 Relation \( R_2 \)
Reflexive:
Missing \( (1,1) \) and \( (3,3) \) ❌ Not Reflexive
Symmetric:
\( (1,3) \) and \( (3,1) \) both present, \( (2,2) \) is self-pair ✔ Symmetric
Transitive:
\( (1,3) \) and \( (3,1) \) ⇒ need \( (1,1) \), not present ❌ Not Transitive
🔹 Relation \( R_3 \)
Reflexive:
Missing \( (1,1), (2,2) \) ❌ Not Reflexive
Symmetric:
\( (1,3) \in R_3 \) but \( (3,1) \notin R_3 \) ❌ Not Symmetric
Transitive:
Check: \( (1,3) \) and \( (3,3) \) ⇒ need \( (1,3) \) (already present) ✔ Transitive
🎯 Final Answer
R₁: Reflexive ✔, Symmetric ❌, Transitive ❌
R₂: Reflexive ❌, Symmetric ✔, Transitive ❌
R₃: Reflexive ❌, Symmetric ❌, Transitive ✔
🚀 Exam Insight
- Reflexive → all diagonal elements must be present
- Symmetric → reverse pairs must exist
- Transitive → check chain condition carefully
- Duplicate pairs do not matter