Identity Relation is Reflexive but Converse is Not True
📺 Video Explanation
📝 Statement
Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.
✅ Proof
🔹 Part 1: Identity Relation is Reflexive
Let \( A \) be a set.
The identity relation on \( A \) is:
\[ I = \{(a, a) : a \in A\} \]
For every element \( a \in A \), the pair \( (a,a) \in I \).
Hence, by definition:
\[ (a,a) \in I \quad \forall a \in A \]
✔ Therefore, identity relation is Reflexive.
🔹 Part 2: Converse is Not Necessarily True
The converse would mean:
“Every reflexive relation is an identity relation”
This is not true.
Counterexample:
Let \( A = \{1,2\} \)
Define relation: \[ R = \{(1,1), (2,2), (1,2)\} \]
Here:
- \( (1,1), (2,2) \in R \) ⇒ Reflexive ✔
- But \( (1,2) \in R \), so extra element present
So, \( R \neq I \) (identity relation)
❌ Hence, reflexive does not imply identity.
🎯 Final Conclusion
✔ Every identity relation is reflexive
❌ Every reflexive relation is not identity
\[ \therefore \text{Identity ⇒ Reflexive, but converse is false} \]
🚀 Exam Insight
- Identity relation = only diagonal elements
- Reflexive relation = all diagonal elements (may include extra pairs)
- Always use counterexample to disprove converse