Checking Reflexive, Symmetric and Transitive Relations
📺 Video Explanation
📝 Question
Let \( A = \{a, b, c\} \). Relations are defined as:
\( R_1 = \{(a,a), (a,b), (a,c), (b,b), (b,c), (c,a), (c,b), (c,c)\} \)
\( R_2 = \{(a,a)\} \)
\( R_3 = \{(b,a)\} \)
\( R_4 = \{(a,b), (b,c), (c,a)\} \)
Check whether each relation is reflexive, symmetric, and transitive.
✅ Solution
🔹 Relation \( R_1 \)
Reflexive:
Since \( (a,a), (b,b), (c,c) \in R_1 \), ✔ Reflexive
Symmetric:
Check: \( (a,b) \in R_1 \) but \( (b,a) \notin R_1 \) ❌ Not Symmetric
Transitive:
Example: \( (b,c) \) and \( (c,a) \in R_1 \), but \( (b,a) \notin R_1 \) ❌ Not Transitive
🔹 Relation \( R_2 \)
Reflexive:
Needs \( (a,a), (b,b), (c,c) \), but only \( (a,a) \) present ❌ Not Reflexive
Symmetric:
\( (a,a) \) is symmetric with itself ✔ Symmetric
Transitive:
\( (a,a), (a,a) \Rightarrow (a,a) \) (already present) ✔ Transitive
🔹 Relation \( R_3 \)
Reflexive:
No \( (a,a), (b,b), (c,c) \) ❌ Not Reflexive
Symmetric:
\( (b,a) \in R_3 \), but \( (a,b) \notin R_3 \) ❌ Not Symmetric
Transitive:
No pairs to satisfy condition ✔ Transitive (vacuously true)
🔹 Relation \( R_4 \)
Reflexive:
No \( (a,a), (b,b), (c,c) \) ❌ Not Reflexive
Symmetric:
\( (a,b) \in R_4 \), but \( (b,a) \notin R_4 \) ❌ Not Symmetric
Transitive:
\( (a,b), (b,c) \in R_4 \), but \( (a,c) \notin R_4 \) ❌ Not Transitive
🎯 Final Answer
R₁: Reflexive ✔, Symmetric ❌, Transitive ❌
R₂: Reflexive ❌, Symmetric ✔, Transitive ✔
R₃: Reflexive ❌, Symmetric ❌, Transitive ✔
R₄: Reflexive ❌, Symmetric ❌, Transitive ❌
🚀 Exam Insight
- Reflexive → Check all diagonal elements
- Symmetric → Check reverse pairs
- Transitive → Check chain condition
- If no chain exists → relation is transitive (vacuous truth)