Relation on Set \( A=\{1,2,3\} \)
📺 Video Explanation
📝 Question
Let:
\[ A=\{1,2,3\} \]
and:
\[ R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\} \]
Then, \(R\) is:
- A. reflexive but not symmetric
- B. reflexive but not transitive
- C. symmetric and transitive
- D. neither symmetric nor transitive
✅ Solution
🔹 Reflexive Check
A relation is reflexive if:
\[ (1,1),(2,2),(3,3) \]
are present.
All are in \(R\).
✔ Reflexive.
🔹 Symmetric Check
Since:
\[ (1,2)\in R \]
Symmetry requires:
\[ (2,1)\in R \]
But:
\[ (2,1)\notin R \]
❌ Not symmetric.
🔹 Transitive Check
Need:
If:
\[ (x,y)\in R,\ (y,z)\in R \]
then:
\[ (x,z)\in R \]
Check important chain:
\[ (1,2),(2,3)\in R \]
Then:
\[ (1,3) \]
is present ✔
Self-pairs also satisfy transitivity automatically.
No chain violates the condition.
✔ Transitive.
🎯 Final Answer
\[ \boxed{\text{Reflexive but not symmetric}} \]
✔ Correct option: A
🚀 Exam Shortcut
- Diagonal pairs → reflexive
- Missing reverse pair → not symmetric
- Check chains carefully for transitivity