Why Relation is Not Transitive
📺 Video Explanation
📝 Question
Let relation \( R \) on set \( \{1,2,3\} \) be:
\[ R = \{(1,2), (2,1)\} \]
State the reason why \( R \) is not transitive.
✅ Solution
🔹 Definition of Transitive Relation
A relation \( R \) is transitive if:
\[ (a,b) \in R \text{ and } (b,c) \in R \Rightarrow (a,c) \in R \]
🔹 Check Given Relation
We have:
\[ (1,2) \in R \text{ and } (2,1) \in R \]
So, by transitivity:
\[ (1,1) \in R \text{ must be present} \]
But:
\[ (1,1) \notin R \]
🎯 Final Answer
Since:
\[ (1,2), (2,1) \in R \text{ but } (1,1) \notin R \]
❌ Therefore, \( R \) is not transitive.
🚀 Exam Insight
- Always look for chain: (a,b), (b,c)
- If (a,c) missing ⇒ not transitive
- Very common exam trick