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

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✅ Solution

🔹 Reflexive Check

Every natural number divides itself:

\[ n\mid n \]

✔ Reflexive.

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🔹 Symmetric Check

If:

\[ n\mid m \]

Does it imply:

\[ m\mid n \] ?

No.

Example:

\[ 2\mid4 \] but:

\[ 4\nmid2 \]

❌ Not symmetric.

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🔹 Transitive Check

If:

\[ n\mid m \quad \text{and} \quad m\mid p \]

Then:

\[ n\mid p \]

✔ Transitive.

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🎯 Final Answer

\[ \boxed{\text{Reflexive, transitive but not symmetric}} \]

✔ Correct option: D

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🚀 Exam Shortcut

• Divisibility always reflexive
• Chain of divisibility gives transitivity
• Usually not symmetric unless numbers equal