Count Relations on \( A=\{1,2,3\} \) That Are Reflexive, Symmetric but Not Transitive
📺 Video Explanation
📝 Question
Let \[ A=\{1,2,3\} \]
Find the number of relations on \( A \) which:
- contain \((1,2)\) and \((1,3)\)
- are reflexive
- are symmetric
- but are not transitive
✅ Solution
🔹 Step 1: Reflexive condition
For reflexive relation, all diagonal pairs must be present:
\[ (1,1),(2,2),(3,3) \]
🔹 Step 2: Given pairs and symmetry
Given:
\[ (1,2),(1,3) \]
Since relation is symmetric, must also include:
\[ (2,1),(3,1) \]
So fixed pairs are:
\[ (1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1) \]
🔹 Step 3: Remaining possible pairs
Only undecided pairs:
\[ (2,3),(3,2) \]
By symmetry:
- either both included
- or both excluded
So total possible symmetric reflexive relations:
\[ 2 \]
🔹 Step 4: Check transitivity
Case 1: Exclude \((2,3),(3,2)\)
Since:
\[ (2,1),(1,3)\in R \]
Transitivity requires:
\[ (2,3)\in R \]
But missing.
❌ Not transitive.
Case 2: Include \((2,3),(3,2)\)
Now all needed chains work.
✔ Transitive.
🎯 Final Answer
Only one relation satisfies:
\[ \boxed{1} \]
✔ Correct option: (a)
🚀 Exam Shortcut
- Reflexive fixes diagonal pairs
- Symmetry forces reverse pairs
- Check only remaining free pairs
- One missing link breaks transitivity