Domain of Relation Defined by Relatively Prime Numbers
📺 Video Explanation
📝 Question
A relation \( R \) is defined from:
\[ A=\{2,3,4,5\}, \quad B=\{3,6,7,10\} \]
by:
\[ xRy \iff x \text{ is relatively prime to } y \]
Find the domain of \( R \).
- (a) \(\{2,3,5\}\)
- (b) \(\{3,5\}\)
- (c) \(\{2,3,4\}\)
- (d) \(\{2,3,4,5\}\)
✅ Solution
Domain = set of all first elements \(x\in A\) for which at least one \(y\in B\) satisfies relation.
That means:
\[ \gcd(x,y)=1 \] for some \(y\in B\).
🔹 Check each element of A
For \(x=2\)
Check with:
- \(\gcd(2,3)=1\) ✔
So 2 is in domain.
For \(x=3\)
- \(\gcd(3,7)=1\) ✔
So 3 is in domain.
For \(x=4\)
- \(\gcd(4,3)=1\) ✔
So 4 is in domain.
For \(x=5\)
- \(\gcd(5,3)=1\) ✔
So 5 is in domain.
🔹 Domain
\[ \{2,3,4,5\} \]
🎯 Final Answer
\[ \boxed{\{2,3,4,5\}} \]
✔ Correct option: (d)
🚀 Exam Shortcut
- Domain means first elements having at least one valid pair
- Check gcd = 1 for any one pair
- If one match exists, include the number in domain