Relation on Set \( A=\{a,b,c\} \)
📺 Video Explanation
📝 Question
Let \[ A=\{a,b,c\} \] and relation
\[ R=\{(a,a),(b,b),(c,c),(a,b)\} \]
Then, \(R\) is:
- (a) identity relation
- (b) reflexive
- (c) symmetric
- (d) equivalence relation
✅ Solution
🔹 Check Identity Relation
Identity relation on \(A\) is:
\[ I=\{(a,a),(b,b),(c,c)\} \]
But \(R\) also contains:
\[ (a,b) \]
So:
❌ Not identity relation.
🔹 Check Reflexive
A relation is reflexive if all self-pairs are present:
\[ (a,a),(b,b),(c,c) \]
All are in \(R\).
✔ Reflexive.
🔹 Check Symmetric
Since:
\[ (a,b)\in R \]
Symmetry requires:
\[ (b,a)\in R \]
But:
\[ (b,a)\notin R \]
❌ Not symmetric.
🔹 Check Equivalence
Equivalence requires:
- reflexive ✔
- symmetric ❌
- transitive
Since symmetric fails:
❌ Not equivalence relation.
🎯 Final Answer
\[ \boxed{\text{R is reflexive}} \]
✔ Correct option: (b)
🚀 Exam Shortcut
- Reflexive needs all self-pairs
- Identity has only self-pairs
- One missing reverse pair breaks symmetry