<p>If \(A\) is the null set, then the number of elements in the power set \(P(P(\phi))\) is</p>
Step-by-Step Solution
Key Concept: The power set of the empty set is {∅}, and you must carefully apply the power set operation twice to count elements correctly using the formula |P(S)| = 2^|S|.
<p><strong>Step 1:</strong> Find P(φ) where φ is the null/empty set.</p><p>The power set of the empty set contains only the empty set itself: P(φ) = {φ}</p><p>So |P(φ)| = 1</p><p><strong>Step 2:</strong> Find P(P(φ)) = P({φ})</p><p>The power set of {φ} is the set of all subsets of {φ}. The subsets are:</p><ul><li>The empty set: φ</li><li>The set itself: {φ}</li></ul><p>Therefore, P(P(φ)) = P({φ}) = {φ, {φ}}</p><p><strong>Step 3:</strong> Count the elements.</p><p>|P(P(φ))| = 2</p><p>Alternatively, using the formula: |P(P(φ))| = 2^|P(φ)| = 2^1 = 2</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: B