Relation Based on Relatively Prime Numbers
📺 Video Explanation
📝 Question
Let:
\[ A = \{3,5,7\}, \quad B = \{2,6,10\} \]
Relation \( R \) is defined as:
\[ R = \{(x,y) : x \in A,\ y \in B,\ \gcd(x,y) = 1\} \]
Find \( R \) and \( R^{-1} \).
✅ Solution
🔹 Step 1: Check Relatively Prime Pairs
Two numbers are relatively prime if:
\[ \gcd(x,y) = 1 \]
- \( x = 3 \): (3,2) ✔, (3,6) ✖, (3,10) ✔
- \( x = 5 \): (5,2) ✔, (5,6) ✔, (5,10) ✖
- \( x = 7 \): (7,2) ✔, (7,6) ✔, (7,10) ✔
🔹 Step 2: Write Relation \( R \)
\[ R = \{(3,2), (3,10), (5,2), (5,6), (7,2), (7,6), (7,10)\} \]
🔹 Step 3: Find Inverse Relation \( R^{-1} \)
Swap each ordered pair:
\[ R^{-1} = \{(2,3), (10,3), (2,5), (6,5), (2,7), (6,7), (10,7)\} \]
🎯 Final Answer
\[ R = \{(3,2), (3,10), (5,2), (5,6), (7,2), (7,6), (7,10)\} \]
\[ R^{-1} = \{(2,3), (10,3), (2,5), (6,5), (2,7), (6,7), (10,7)\} \]
🚀 Exam Insight
- Check gcd = 1 carefully
- Prime numbers help simplify
- Inverse = reverse pairs
- Domain and range swap in inverse