Relation \( a – b \) divisible by 13 on \( \mathbb{Z} \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{Z} \) be defined as:
\[ (a, b) \in R \iff a – b \text{ is divisible by } 13 \]
Check whether \( R \) defines an equivalence relation.
✅ Solution
🔹 Step 1: Reflexive
For reflexive, we need: \[ (a,a) \in R \quad \forall a \in \mathbb{Z} \]
\[ a – a = 0 \]
Since 0 is divisible by 13, \[ (a,a) \in R \]
✔ Therefore, the relation is Reflexive.
🔹 Step 2: Symmetric
Assume: \[ (a,b) \in R \Rightarrow a – b = 13k \]
\[ b – a = -13k = 13(-k) \]
So, \[ (b,a) \in R \]
✔ Therefore, the relation is Symmetric.
🔹 Step 3: Transitive
Assume: \[ (a,b) \in R,\ (b,c) \in R \]
\[ a – b = 13m,\quad b – c = 13n \]
Adding: \[ a – c = (a – b) + (b – c) = 13m + 13n = 13(m+n) \]
So, \[ (a,c) \in R \]
✔ Therefore, the relation is Transitive.
🎯 Final Answer
✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes
\[ \therefore R \text{ is an equivalence relation} \]
🚀 Exam Insight
- \( a – b \) divisible by \( n \) → always an equivalence relation
- This represents congruence: \( a \equiv b \ (\text{mod } 13) \)
- Integers are divided into 13 equivalence classes