Relation \( R=\{(1,2),(2,3),(1,3)\} \) on Set \( A=\{1,2,3\} \)
📺 Video Explanation
📝 Question
Let \[ A=\{1,2,3\} \] and:
\[ R=\{(1,2),(2,3),(1,3)\} \]
Then, \(R\) is:
- (a) neither reflexive nor transitive
- (b) neither symmetric nor transitive
- (c) transitive
- (d) none of these
✅ Solution
🔹 Reflexive Check
For reflexive relation:
\[ (1,1),(2,2),(3,3) \] must be present.
These are missing.
❌ Not reflexive.
🔹 Symmetric Check
Since:
\[ (1,2)\in R \]
Then:
\[ (2,1)\in R \]
must be present for symmetry.
But missing.
❌ Not symmetric.
🔹 Transitive Check
Transitive means:
If:
\[ (a,b)\in R,\ (b,c)\in R \]
then:
\[ (a,c)\in R \]
Here:
\[ (1,2),(2,3)\in R \]
So:
\[ (1,3) \] must be in \(R\).
It is present.
✔ Transitive.
No other chain violates transitivity.
🎯 Final Answer
\[ \boxed{\text{R is transitive}} \]
✔ Correct option: (c)
🚀 Exam Shortcut
- Reflexive needs all self-pairs
- Symmetric needs reverse pairs
- Transitive depends on chain completion