<p>From the relation \(2x + y = 41\), the number of elements in the range of \(R = \{1, 3, 5, 7, \ldots, 37, 39\}\) is:</p>
Step-by-Step Solution
Key Concept: The range of relation R consists of y-values from the equation 2x + y = 41 where x ∈ R (the given set of odd numbers). You must solve y = 41 - 2x for each x in R and count distinct y-values that satisfy the constraint.
<p><strong>Step 1:</strong> Identify the domain. R = {1, 3, 5, 7, ..., 37, 39} contains odd numbers from 1 to 39. This set has (39-1)/2 + 1 = 20 elements.</p><p><strong>Step 2:</strong> From 2x + y = 41, express y = 41 - 2x.</p><p><strong>Step 3:</strong> For each x ∈ R, calculate y:</p><ul><li>When x = 1: y = 41 - 2 = 39</li><li>When x = 3: y = 41 - 6 = 35</li><li>When x = 5: y = 41 - 10 = 31</li><li>When x = 39: y = 41 - 78 = -37</li></ul><p><strong>Step 4:</strong> As x increases through odd values {1, 3, 5, ..., 39}, y decreases through values {39, 35, 31, ..., -37}. These are also odd numbers in arithmetic progression with common difference -4.</p><p><strong>Step 5:</strong> Count elements in range: y takes values 39, 35, 31, 27, ..., 3, -1, -5, ..., -37. Since there's a one-to-one correspondence between the 20 elements of domain R and the y-values produced, the range has exactly 20 distinct elements.</p><p>∴ Answer: <strong>20</strong></p>
Correct Answer: 20