<p>Let \(S = \{1, 2, 3, 4\}\). The total number of unordered pairs of disjoint subsets of \(S\) is equal to</p>
Step-by-Step Solution
Key Concept: Each element has 3 independent choices (first set, second set, or neither), giving \(3^n\) ordered pairs. Convert to unordered by accounting for symmetry.
<p><strong>Solution:</strong> For each element in \(S\), there are 3 choices: it belongs to the first subset, second subset, or neither.</p><p>Total ordered pairs of disjoint subsets = \(3^4 = 81\)</p><p>This includes the pair (∅, ∅) once.</p><p>Unordered pairs = \(\frac{81 + 1}{2} = 41\)</p><p>The +1 accounts for the symmetric pair (∅, ∅).</p>
Correct Answer: D