Divisibility Relation on \( \mathbb{N} \)
📺 Video Explanation
📝 Question
Let relation \(R\) on natural numbers \( \mathbb{N} \) be defined by:
\[ nRm \iff n \text{ divides } m \]
Then, \(R\) is:
- A. Reflexive and symmetric
- B. Transitive and symmetric
- C. Equivalence relation
- D. Reflexive, transitive but not symmetric
✅ Solution
🔹 Reflexive Check
Every natural number divides itself:
\[ n\mid n \]
✔ Reflexive.
🔹 Symmetric Check
If:
\[ n\mid m \]
Does it imply:
\[ m\mid n \] ?
No.
Example:
\[ 2\mid4 \] but:
\[ 4\nmid2 \]
❌ Not symmetric.
🔹 Transitive Check
If:
\[ n\mid m \quad \text{and} \quad m\mid p \]
Then:
\[ n\mid p \]
✔ Transitive.
🎯 Final Answer
\[ \boxed{\text{Reflexive, transitive but not symmetric}} \]
✔ Correct option: D
🚀 Exam Shortcut
- Divisibility always reflexive
- Chain of divisibility gives transitivity
- Usually not symmetric unless numbers equal