Sets, Relations & Functions
Set Operations
Grade 11
Question:
<p>If <span class="math">X = \{4^n - 3n - 1 : n \in \mathbb{N}\}</span> and <span class="math">Y = \{9(n-1) : n \in \mathbb{N}\}</span>, where <span class="math">\mathbb{N}</span> is the set of natural numbers, then <span class="math">X \cup Y</span> is equal to</p>
<p>(a) <span class="math">\mathbb{N}</span></p>
<p>(b) <span class="math">Y - X</span></p>
<p>(c) <span class="math">X</span></p>
<p>(d) <span class="math">Y</span></p>
Step-by-Step Solution
Key Concept: Compute the sets X and Y explicitly by substituting natural numbers, then determine their union by checking if one is a subset of the other.
<p><strong>Solution:</strong></p><p>We have <span class="math">X = \{4^n - 3n - 1 : n \in \mathbb{N}\}</span></p><p>Putting <span class="math">n = 1, 2, 3, \ldots</span>:</p><p><span class="math">X = \{0, 9, 54, 243, \ldots\}</span></p><p>And <span class="math">Y = \{9(n-1) : n \in \mathbb{N}\}</span></p><p>Putting <span class="math">n = 1, 2, 3, \ldots</span>:</p><p><span class="math">Y = \{0, 9, 18, 27, \ldots\}</span></p><p>It is clear that <span class="math">X \subseteq Y</span>.</p><p><span class="math">\therefore X \cup Y = Y</span></p>
Correct Answer: D