Which of the Following is Not an Equivalence Relation on \( \mathbb{Z} \)?
📺 Video Explanation
📝 Question
Which of the following is not an equivalence relation on \( \mathbb{Z} \)?
- (a) \( aRb \iff a+b \text{ is an even integer} \)
- (b) \( aRb \iff a-b \text{ is an even integer} \)
- (c) \( aRb \iff a < b \)
- (d) \( aRb \iff a=b \)
✅ Solution
An equivalence relation must satisfy:
- Reflexive
- Symmetric
- Transitive
🔹 Option (a): \( a+b \) is even
If \( a+b \) is even, then both numbers have same parity.
This relation is:
- ✔ Reflexive: \( a+a=2a \) is even
- ✔ Symmetric: if \( a+b \) is even, then \( b+a \) is even
- ✔ Transitive: same parity property holds
✔ This is an equivalence relation.
🔹 Option (b): \( a-b \) is even
This means \( a \) and \( b \) have same parity.
- ✔ Reflexive: \( a-a=0 \) even
- ✔ Symmetric: if \( a-b \) even, then \( b-a \) even
- ✔ Transitive: parity preserved
✔ This is an equivalence relation.
🔹 Option (c): \( a < b \)
- ❌ Not reflexive because \( a
- ❌ Not symmetric because if \( a
❌ This is NOT an equivalence relation.
🔹 Option (d): \( a=b \)
- ✔ Reflexive
- ✔ Symmetric
- ✔ Transitive
✔ Equality relation is an equivalence relation.
🎯 Final Answer
\[
\boxed{\text{Option (c) } :\ a
✔ Correct option: (c)
🚀 Exam Shortcut