Relation \( a = b \) on Set \( A \)
📺 Video Explanation
📝 Question
Let \( A = \{x \in \mathbb{Z} : 0 \le x \le 12\} \). Define relation \( R \) as:
\[ R = \{(a,b) : a = b\} \]
Show that \( R \) is an equivalence relation. Also find the set of all elements related to 1.
✅ Solution
🔹 Step 1: Reflexive
For reflexive, we need: \[ (a,a) \in R \quad \forall a \in A \]
Since every element is equal to itself, \[ a = a \]
✔ Therefore, \( (a,a) \in R \) and the relation is Reflexive.
🔹 Step 2: Symmetric
Assume: \[ (a,b) \in R \Rightarrow a = b \]
Then: \[ b = a \]
So, \[ (b,a) \in R \]
✔ Therefore, the relation is Symmetric.
🔹 Step 3: Transitive
Assume: \[ (a,b) \in R,\ (b,c) \in R \]
\[ a = b,\quad b = c \]
Thus, \[ a = c \]
So, \[ (a,c) \in R \]
✔ Therefore, the relation is Transitive.
🎯 Final Conclusion
✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes
\[ \therefore R \text{ is an equivalence relation} \]
🔹 Elements Related to 1
We need all elements \( x \in A \) such that: \[ (1,x) \in R \]
Since relation is defined by \( a = b \), \[ 1 = x \Rightarrow x = 1 \]
So, the equivalence class of 1 is:
\[ [1] = \{1\} \]
🚀 Exam Insight
- This is called Identity Relation
- Every element forms its own equivalence class
- Total classes = number of elements in the set