Question

Let $ R $ be a relation on $ \mathbb{N} $ defined as $ x R y $ iff $ x + 2y = 8 $, the domain of $ R $ is

• A. {2, 4, 8}
• B. {2, 4, 6, 8}
• C. {2, 4, 6}
• D. {1, 2, 3, 4}

Hint:

When solving for the domain of a relation, make sure to identify the values of \( x \) for which there exists a corresponding \( y \) that satisfies the given relation.

Answer 1

The Correct Option is B

Step 1: Understanding the Relation
The relation is defined by the equation \( x + 2y = 8 \).
The domain of the relation consists of all values of \( x \) for which there exists a corresponding value of \( y \) in \( \mathbb{N} \) (natural numbers) that satisfies the relation.
Step 2: Solving the Equation for Possible \( x \) and \( y \)
We can solve for different values of \( x \) and \( y \):
For \( x = 2 \), \( 2 + 2y = 8 \Rightarrow y = 3 \) (valid)
For \( x = 4 \), \( 4 + 2y = 8 \Rightarrow y = 2 \) (valid)
For \( x = 6 \), \( 6 + 2y = 8 \Rightarrow y = 1 \) (valid)
For \( x = 8 \), \( 8 + 2y = 8 \Rightarrow y = 0 \) (valid)
Thus, the domain of \( R \) is \( \{2, 4, 6, 8\} \).
Step 3: Conclusion
The domain of \( R \) is \( \{2, 4, 6, 8\} \).