Relation \( R=\{(a,b),(b,a),(a,a)\} \) on Set \( A=\{a,b,c,d\} \)
📺 Video Explanation
📝 Question
Let \[ A=\{a,b,c,d\} \]
and:
\[ R=\{(a,b),(b,a),(a,a)\} \]
Then \(R\) is:
- A. symmetric and transitive only
- B. reflexive and transitive only
- C. symmetric only
- D. transitive only
✅ Solution
🔹 Reflexive Check
For reflexive relation, all self-pairs must be present:
\[ (a,a),(b,b),(c,c),(d,d) \]
Only \((a,a)\) is present.
❌ Not reflexive.
🔹 Symmetric Check
Since:
\[ (a,b)\in R \]
and:
\[ (b,a)\in R \]
Reverse pair exists.
Also:
\[ (a,a) \] is symmetric itself.
✔ Symmetric.
🔹 Transitive Check
Need:
If:
\[ (x,y)\in R,\ (y,z)\in R \]
then:
\[ (x,z)\in R \]
Check chain:
\[ (a,b),(b,a)\in R \]
Then transitivity requires:
\[ (a,a)\in R \]
Present ✔
Check:
\[ (b,a),(a,b)\in R \]
Then must have:
\[ (b,b)\in R \]
But:
\[ (b,b)\notin R \]
❌ Not transitive.
🎯 Final Answer
\[ \boxed{\text{R is symmetric only}} \]
✔ Correct option: C
🚀 Exam Shortcut
- Reflexive needs all diagonal pairs
- Symmetric needs reverse pairs
- For transitivity, check all possible chains carefully