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.

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✅ 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.

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🔹 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.

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🎯 Final Conclusion

✔ Every identity relation is reflexive
❌ Every reflexive relation is not identity

\[ \therefore \text{Identity ⇒ Reflexive, but converse is false} \]

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🚀 Exam Insight

• Identity relation = only diagonal elements
• Reflexive relation = all diagonal elements (may include extra pairs)
• Always use counterexample to disprove converse