If \(|\mathcal{P}(A)|=64\), what is the value of \(|\mathcal{P}(\mathcal{P}(A))|\)?
Answer and explanation
Correct answer: \(2^{64}\)
The given information says that the set \(\mathcal{P}(A)\) itself has 64 elements. For any finite set \(S\), the power set \(\mathcal{P}(S)\) has \(2^{|S|}\) elements. Taking \(S=\mathcal{P}(A)\), we obtain \(|\mathcal{P}(\mathcal{P}(A))|=2^{|\mathcal{P}(A)|}=2^{64}\). It is not \(64^2\), because forming a power set counts all subsets. Therefore, option C is correct.
Frequently asked questions
What is the correct answer to this question?
\(2^{64}\)
Why is this the correct answer?
The given information says that the set \(\mathcal{P}(A)\) itself has 64 elements. For any finite set \(S\), the power set \(\mathcal{P}(S)\) has \(2^{|S|}\) elements. Taking \(S=\mathcal{P}(A)\), we obtain \(|\mathcal{P}(\mathcal{P}(A))|=2^{|\mathcal{P}(A)|}=2^{64}\). It is not \(64^2\), because forming a power set counts all subsets. Therefore, option C is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
