<p>The function \(f: \mathbb{N} \to \mathbb{N}\) defined by \(f(x) = x - 5\left[\dfrac{x}{5}\right]\), where \(\mathbb{N}\) is the set of natural numbers and \([x]\) denotes the greatest integer less than or equal to \(x\), is</p>
<p>one–one and onto.</p>
<p>onto but not one–one.</p>
<p>neither one–one nor onto.</p>
<p>one–one but not onto.</p>
Step-by-Step Solution
Key Concept: Recognize that f(x) = x - 5⌊x/5⌋ computes the remainder when x is divided by 5, producing a cyclic pattern {0,1,2,3,4} that repeats. Analyze whether this maps ℕ→ℕ and determine if it's injective/surjective based on this remainder structure.
<p><strong>Step 1:</strong> Understand the function structure. For any x ∈ ℕ, we have f(x) = x - 5⌊x/5⌋. This is the remainder when x is divided by 5.</p><p><strong>Step 2:</strong> Calculate values: f(1)=1, f(2)=2, f(3)=3, f(4)=4, f(5)=0, f(6)=1, f(7)=2, ..., f(10)=0, etc.</p><p><strong>Step 3:</strong> Identify the range. The function cycles through {0,1,2,3,4}. So Range(f) = {0,1,2,3,4} ⊂ ℕ.</p><p><strong>Step 4:</strong> Check injectivity: f(5) = f(10) = f(15) = 0, so f is NOT injective (not one-to-one).</p><p><strong>Step 5:</strong> Check surjectivity: Since Range(f) = {0,1,2,3,4} and codomain is ℕ, not every natural number ≥5 is in the range, so f is NOT surjective (not onto).</p><p><strong>Conclusion:</strong> f is neither injective nor surjective.</p><p>∴ Answer: B</p>
Correct Answer: B