<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 range of this relation.</p>
Step-by-Step Solution
Key Concept: For each valid natural number x, there is exactly one corresponding y = 41 - 2x that must also be a natural number. The range consists of all such valid y values, determined by the constraint y ∈ ℕ which gives 41 - 2x ≥ 1, so x ≤ 20.
<p><strong>Step 1:</strong> Write the relation constraint: 2x + y = 41, which gives y = 41 - 2x</p><p><strong>Step 2:</strong> For R to contain (x,y), both x and y must be natural numbers (x, y ∈ ℕ, typically x ≥ 1, y ≥ 1)</p><p><strong>Step 3:</strong> Apply constraint on x: Since x ∈ ℕ, we need x ≥ 1</p><p><strong>Step 4:</strong> Apply constraint on y: Since y = 41 - 2x ∈ ℕ, we need 41 - 2x ≥ 1, so 2x ≤ 40, giving x ≤ 20</p><p><strong>Step 5:</strong> Therefore x ∈ {1, 2, 3, ..., 20}</p><p><strong>Step 6:</strong> Calculate the range set:</p><p>• When x = 1: y = 41 - 2 = 39</p><p>• When x = 2: y = 41 - 4 = 37</p><p>• When x = 3: y = 41 - 6 = 35</p><p>• ...</p><p>• When x = 20: y = 41 - 40 = 1</p><p><strong>Step 7:</strong> The range is {39, 37, 35, 33, ..., 5, 3, 1} (odd numbers from 1 to 39)</p><p><strong>Step 8:</strong> Number of odd numbers from 1 to 39 = (39 - 1)/2 + 1 = 20</p><p>∴ <strong>Answer: 20</strong></p>
Correct Answer: 20