If \(A=\varnothing\), how many members are there in \(\mathcal{P}(\mathcal{P}(A))\)?
Answer and explanation
Correct answer: 2
The empty set has exactly one subset, namely itself, so \(\mathcal{P}(\varnothing)=\{\varnothing\}\) and its cardinality is 1. Applying the power-set operation once more gives \(|\mathcal{P}(\mathcal{P}(A))|=2^{|\mathcal{P}(A)|}=2^1=2\). Explicitly, the two members are \(\varnothing\) and \(\{\varnothing\}\). Thus, option B is correct.
Frequently asked questions
What is the correct answer to this question?
2
Why is this the correct answer?
The empty set has exactly one subset, namely itself, so \(\mathcal{P}(\varnothing)=\{\varnothing\}\) and its cardinality is 1. Applying the power-set operation once more gives \(|\mathcal{P}(\mathcal{P}(A))|=2^{|\mathcal{P}(A)|}=2^1=2\). Explicitly, the two members are \(\varnothing\) and \(\{\varnothing\}\). Thus, option B is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
