Which Relation on \( \mathbb{Z} \) is Not an Equivalence Relation?
📺 Video Explanation
📝 Question
Let \( \mathbb{Z} \) be the set of all integers. Which of the following relations is not an equivalence relation?
- A. \(xRy \iff x\leq y\)
- B. \(xRy \iff x=y\)
- C. \(xRy \iff x-y \text{ is even}\)
- D. \(xRy \iff x\equiv y \pmod 3\)
✅ Solution
An equivalence relation must be:
- Reflexive
- Symmetric
- Transitive
🔹 Option A: \(x\leq y\)
Reflexive:
\[ x\leq x \] ✔ True
Symmetric:
If:
\[ x\leq y \]
Then must have:
\[ y\leq x \]
This is false in general.
Example:
\[ 2\leq5 \] but \[ 5\leq2 \] is false.
❌ Not symmetric.
So not equivalence.
🔹 Option B: \(x=y\)
Equality relation is always:
- ✔ Reflexive
- ✔ Symmetric
- ✔ Transitive
🔹 Option C: \(x-y\) even
This means same parity.
✔ Equivalence relation.
🔹 Option D: Congruence mod 3
Congruence modulo 3 is:
- ✔ Reflexive
- ✔ Symmetric
- ✔ Transitive
✔ Equivalence relation.
🎯 Final Answer
\[ \boxed{xRy \iff x\leq y} \]
✔ Correct option: A
🚀 Exam Shortcut
- Order relations like \( \leq \) are usually not symmetric
- Equality and congruence relations are equivalence relations
- Same parity relation is also equivalence