Domain of Relation on \( \mathbb{N} \) Defined by \( x+2y=8 \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{N} \) be defined by:
\[ xRy \iff x+2y=8 \]
Find the domain of \( R \).
- (a) \(\{2,4,8\}\)
- (b) \(\{2,4,6,8\}\)
- (c) \(\{2,4,6\}\)
- (d) \(\{1,2,3,4\}\)
✅ Solution
Domain means all values of \(x\in\mathbb{N}\) for which there exists at least one natural number \(y\) satisfying:
\[ x+2y=8 \]
🔹 Express \(x\)
\[ x=8-2y \]
Since \(x,y\in\mathbb{N}\), take natural numbers:
\[ y=1,2,3 \]
🔹 Find corresponding \(x\)
For:
- \(y=1 \Rightarrow x=6\)
- \(y=2 \Rightarrow x=4\)
- \(y=3 \Rightarrow x=2\)
If:
\[ y=4 \Rightarrow x=0 \]
But 0 is not in natural numbers (in school-level convention).
🔹 Domain
\[ \{2,4,6\} \]
🎯 Final Answer
\[ \boxed{\{2,4,6\}} \]
✔ Correct option: (c)
🚀 Exam Shortcut
- Domain = first values for valid ordered pairs
- Use equation to generate possible natural numbers
- Exclude zero if \( \mathbb{N}=\{1,2,3,\dots\} \)