<p>Let \(R\) be a relation defined on the set of all natural numbers as \(R = \{(x, y) : x \in \mathbb{N}, 2x + y = 41\}\). Find the number of elements in the set domain of this relation.</p>
Step-by-Step Solution
Key Concept: For a relation R = {(x, y) : x ∈ ℕ, 2x + y = 41}, the domain consists of all natural numbers x for which there exists at least one natural number y satisfying the equation. From 2x + y = 41, we get y = 41 - 2x, and y must be a natural number, so 41 - 2x ≥ 1.
<p><strong>Step 1:</strong> Write the constraint equation: 2x + y = 41, which gives y = 41 - 2x</p><p><strong>Step 2:</strong> Since x ∈ ℕ, we need x ≥ 1</p><p><strong>Step 3:</strong> Since y must also be a natural number, y ≥ 1, so: 41 - 2x ≥ 1</p><p>This gives: 40 ≥ 2x, or x ≤ 20</p><p><strong>Step 4:</strong> Therefore, the domain is {1, 2, 3, ..., 20}, where each x value produces a valid (x, y) pair:</p><p>• When x = 1: y = 39 ✓</p><p>• When x = 2: y = 37 ✓</p><p>• ...</p><p>• When x = 20: y = 1 ✓</p><p>∴ Number of elements in domain = 20</p>
Correct Answer: 20