Sets, Relations & Functions
Range of a Relation
Grade 11
Question:
<p>Given \(N\) is the set of all natural numbers. Let \(R_1 = \{(x, y) \in N \times N : 2x + y = 10\}\) and \(R_2 = \{(x, y) \in N \times N : x + 2y = 10\}\). The range of \(R_2\) is:</p>
<p>(A) \(\{1, 2, 3, 4\}\)</p>
<p>(B) \(\{2, 4, 6, 8\}\)</p>
<p>(C) \(\{1, 2, 4, 8\}\)</p>
<p>(D) \(\{1, 3, 5, 9\}\)</p>
Step-by-Step Solution
Key Concept: The range of a relation consists of all possible y-values that actually appear in the ordered pairs. For R₂, substitute each valid natural number x into x + 2y = 10, solve for y, and include only those y-values that are natural numbers.
<p><strong>Step 1:</strong> Identify valid pairs in R₂ = {(x, y) ∈ ℕ × ℕ : x + 2y = 10}</p><p>Rearrange: x = 10 - 2y. For x to be a natural number, we need x ≥ 1.</p><p><strong>Step 2:</strong> Find constraints on y:<br/>10 - 2y ≥ 1<br/>9 ≥ 2y<br/>y ≤ 4.5<br/>Since y ∈ ℕ, we have y ∈ {1, 2, 3, 4}</p><p><strong>Step 3:</strong> Verify each y gives a valid natural number x:<br/>• y = 1: x = 10 - 2 = 8 ✓ → (8, 1)<br/>• y = 2: x = 10 - 4 = 6 ✓ → (6, 2)<br/>• y = 3: x = 10 - 6 = 4 ✓ → (4, 3)<br/>• y = 4: x = 10 - 8 = 2 ✓ → (2, 4)<br/>• y = 5: x = 10 - 10 = 0 ✗ (not natural)</p><p><strong>Step 4:</strong> Extract all y-values from valid pairs.</p><p>∴ Range of R₂ = {1, 2, 3, 4}</p>
Correct Answer: A