Check Reflexive, Symmetric and Transitive
📺 Video Explanation
📝 Question
Let:
\[ A = \{0,1,2,3\} \]
\[ R = \{(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3)\} \]
Check whether \( R \) is reflexive, symmetric and transitive.
✅ Solution
🔹 Step 1: Reflexive
Reflexive requires: \[ (a,a) \in R \quad \forall a \in A \]
Check: \[ (0,0), (1,1), (2,2), (3,3) \]
All are present.
✔ Therefore, relation is Reflexive.
🔹 Step 2: Symmetric
Check pairs:
- \( (0,1) \) and \( (1,0) \) ✔
- \( (0,3) \) and \( (3,0) \) ✔
All pairs have their reverse.
✔ Therefore, relation is Symmetric.
🔹 Step 3: Transitive
Check chains:
\[ (0,1), (1,0) \Rightarrow (0,0) \in R ✔ \]
\[ (1,0), (0,3) \Rightarrow (1,3) \text{ should be in } R \]
But: \[ (1,3) \notin R \]
❌ Transitivity fails.
🎯 Final Answer
✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: No
\[ \therefore R \text{ is not transitive} \]
🚀 Exam Insight
- Reflexive → check all (a,a)
- Symmetric → check reverse pairs
- Transitive → check chains carefully
- One missing pair breaks transitivity