Relation Based on Even and Odd Numbers
📺 Video Explanation
📝 Question
Let \( A = \{1,2,3,4,5,6,7\} \). Define relation \( R \) as:
\[ (a,b) \in R \iff \text{both } a \text{ and } b \text{ are either odd or even} \]
Show that \( R \) is an equivalence relation. Also analyze the subsets.
✅ Solution
🔹 Step 1: Reflexive
Every number is either odd or even, and has the same parity as itself.
So, \[ (a,a) \in R \quad \forall a \in A \]
✔ Therefore, the relation is Reflexive. :contentReference[oaicite:0]{index=0}
🔹 Step 2: Symmetric
If: \[ (a,b) \in R \]
Then both \( a \) and \( b \) have the same parity (both even or both odd).
So, \[ (b,a) \in R \]
✔ Therefore, the relation is Symmetric. :contentReference[oaicite:1]{index=1}
🔹 Step 3: Transitive
If: \[ (a,b) \in R \text{ and } (b,c) \in R \]
Then \( a, b, c \) all have the same parity.
So, \[ (a,c) \in R \]
✔ Therefore, the relation is Transitive. :contentReference[oaicite:2]{index=2}
🎯 Final Conclusion
✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes
\[ \therefore R \text{ is an equivalence relation} \]
🔹 Given Subsets Analysis
✔ Odd Set: \( \{1,3,5,7\} \)
All elements are odd ⇒ same parity.
✔ Every pair is related.
So, all elements are related to each other.
✔ Even Set: \( \{2,4,6\} \)
All elements are even ⇒ same parity.
✔ Every pair is related.
So, all elements are related to each other.
❌ Between Odd and Even Sets
If one number is odd and the other is even:
They do NOT have same parity.
So, \[ (a,b) \notin R \]
✔ No element of \( \{1,3,5,7\} \) is related to any element of \( \{2,4,6\} \).
🚀 Exam Insight
- This is called same parity relation
- It splits set into two equivalence classes
- Classes: Odd numbers and Even numbers