Constructing Relations with Given Properties
📺 Video Explanation
📝 Question
Let \( A = \{1,2,3,4\} \). Construct relations on \( A \) which satisfy:
- (i) Reflexive and transitive but not symmetric
- (ii) Symmetric but neither reflexive nor transitive
- (iii) Reflexive, symmetric and transitive
✅ Solution
🔹 (i) Reflexive and Transitive but Not Symmetric
Take: \[ R_1 = \{(1,1),(2,2),(3,3),(4,4),(1,2)\} \]
Reflexive: All \( (a,a) \) present ✔
Not Symmetric: \( (1,2) \in R_1 \) but \( (2,1) \notin R_1 \) ❌
Transitive: No chain violating condition ✔
✔ Hence, satisfies required condition.
🔹 (ii) Symmetric but Neither Reflexive nor Transitive
Take: \[ R_2 = \{(1,2),(2,1)\} \]
Symmetric: Reverse pairs present ✔
Not Reflexive: Missing \( (1,1),(2,2),(3,3),(4,4) \) ❌
Not Transitive: \( (1,2),(2,1) \Rightarrow (1,1) \notin R_2 \) ❌
✔ Hence, satisfies required condition.
🔹 (iii) Reflexive, Symmetric and Transitive
Take identity relation: \[ R_3 = \{(1,1),(2,2),(3,3),(4,4)\} \]
Reflexive: All diagonal elements ✔
Symmetric: \( (a,a) \) symmetric ✔
Transitive: \( (a,a),(a,a) \Rightarrow (a,a) \) ✔
✔ Hence, satisfies all three properties.
🎯 Final Answer
(i) \( R_1 = \{(1,1),(2,2),(3,3),(4,4),(1,2)\} \)
(ii) \( R_2 = \{(1,2),(2,1)\} \)
(iii) \( R_3 = \{(1,1),(2,2),(3,3),(4,4)\} \)
🚀 Exam Insight
- Add all diagonal elements → reflexive
- Add reverse pairs → symmetric
- Avoid chains or control them → transitive
- Identity relation always satisfies all three