<p>From the relation \(2x + y = 41\), the number of elements in the domain of \(R = \{1, 2, 3, \ldots, 20\}\) is:</p>
Step-by-Step Solution
Key Concept: We need to find all ordered pairs (x, y) satisfying 2x + y = 41 where x ∈ R = {1, 2, 3, ..., 20} and determine how many valid x-values exist in the domain such that y is a positive integer.
<p><strong>Step 1:</strong> We have the relation defined by 2x + y = 41, which can be rewritten as y = 41 - 2x.</p><p><strong>Step 2:</strong> For (x, y) to be in the relation, x must be from the given set R = {1, 2, 3, ..., 20}.</p><p><strong>Step 3:</strong> For each x ∈ R, we calculate y = 41 - 2x and check if y is a positive integer.</p><p><strong>Step 4:</strong> When x = 1: y = 41 - 2(1) = 39 ✓</p><p>When x = 2: y = 41 - 2(2) = 37 ✓</p><p>When x = 3: y = 41 - 2(3) = 35 ✓</p><p>⋮</p><p>When x = 20: y = 41 - 2(20) = 41 - 40 = 1 ✓</p><p><strong>Step 5:</strong> For all x ∈ {1, 2, 3, ..., 20}, the corresponding y values are {39, 37, 35, ..., 3, 1}, which are all positive integers.</p><p><strong>Step 6:</strong> Since every element of R produces a valid y-value (all positive), every element in R is in the domain of the relation.</p><p><strong>Step 7:</strong> Therefore, the number of elements in the domain = 20.</p><p><strong>∴ Answer:</strong> 20</p>
Correct Answer: 20