Relation: m is a Multiple of n
📺 Video Explanation
📝 Question
A relation \( R \) on integers is defined as:
\[ (m, n) \in R \iff m \text{ is a multiple of } n \]
Check whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
🔹 Step 1: Understanding the Relation
“m is a multiple of n” means:
\[ m = kn \quad \text{for some integer } k \]
🔹 Step 2: Reflexive
A relation is reflexive if: \[ (m, m) \in R \]
Check: \[ m = 1 \cdot m \]
✔ True for all integers
✔ Therefore, the relation is Reflexive.
🔹 Step 3: Symmetric
A relation is symmetric if: \[ (m, n) \in R \Rightarrow (n, m) \in R \]
If \( m = kn \), does it imply \( n = k’m \)?
Example: \[ m = 6,\ n = 3 \Rightarrow 6 = 2 \cdot 3 \]
But: \[ 3 \neq k \cdot 6 \]
❌ Therefore, the relation is Not Symmetric.
🔹 Step 4: Transitive
A relation is transitive if: \[ (m, n) \in R \text{ and } (n, p) \in R \Rightarrow (m, p) \in R \]
Given: \[ m = k_1 n,\quad n = k_2 p \]
Substitute: \[ m = k_1(k_2 p) = (k_1 k_2)p \]
So: \[ (m, p) \in R \]
✔ Therefore, the relation is Transitive.
🎯 Final Answer
✔ Reflexive: Yes
✔ Symmetric: No
✔ Transitive: Yes
\[ \therefore R \text{ is reflexive and transitive but not symmetric} \]
🚀 Exam Insight
- “Multiple of” relations are always transitive
- They are usually not symmetric (one-way relation)
- Check reflexive using \( m = 1 \cdot m \)