Union of Two Transitive Relations
📺 Video Explanation
📝 Question
If \( R \) and \( S \) are transitive relations on a set \( A \), show that:
\[ R \cup S \text{ may not be transitive} \]
✅ Solution (By Counterexample)
🔹 Step 1: Take a Set
Let: \[ A = \{0,1,2\} \]
🔹 Step 2: Define Relations
Let: \[ R = \{(0,1)\}, \quad S = \{(1,2)\} \]
🔹 Step 3: Check Transitivity of R and S
In \( R \), there is no pair of the form: \[ (a,b), (b,c) \]
So, \( R \) is transitive (vacuously true).
Similarly, \( S \) is also transitive.
✔ Hence, both \( R \) and \( S \) are transitive.
🔹 Step 4: Find Union
\[ R \cup S = \{(0,1), (1,2)\} \]
🔹 Step 5: Check Transitivity of Union
We have: \[ (0,1) \in R \cup S,\quad (1,2) \in R \cup S \]
For transitivity, we must have: \[ (0,2) \in R \cup S \]
But: \[ (0,2) \notin R \cup S \]
❌ So, transitivity condition fails.
✔ Therefore, \( R \cup S \) is not transitive.
🎯 Final Conclusion
Even if \( R \) and \( S \) are transitive,
\[ R \cup S \text{ may NOT be transitive} \]
🚀 Exam Insight
- Always use counterexample for “may not” questions
- Small set (3 elements) is enough
- Key idea: missing link like \( (0,2) \)
- Union does NOT preserve transitivity (important property)