Identity Relation on Set \( A=\{1,2,3\} \)
📺 Video Explanation
📝 Question
Let:
\[ A=\{1,2,3\} \]
and relation:
\[ R=\{(1,1),(2,2),(3,3)\} \]
Then, \(R\) is:
- (a) reflexive
- (b) symmetric
- (c) transitive
- (d) all the three options
✅ Solution
This relation is the identity relation on set \(A\).
🔹 Reflexive
A relation is reflexive if all self-pairs are present:
\[ (1,1),(2,2),(3,3) \]
All are in \(R\).
✔ Reflexive.
🔹 Symmetric
For every:
\[ (a,b)\in R \]
we must have:
\[ (b,a)\in R \]
Since all pairs are of form \((a,a)\), condition is satisfied.
✔ Symmetric.
🔹 Transitive
If:
\[ (a,b)\in R,\ (b,c)\in R \]
then:
\[ (a,c)\in R \]
Identity relation always satisfies this.
✔ Transitive.
🎯 Final Answer
\[ \boxed{\text{All three properties hold}} \]
✔ Correct option: (d)
🚀 Exam Shortcut
- Identity relation is always reflexive
- Identity relation is always symmetric
- Identity relation is always transitive
- So identity relation is an equivalence relation