If \(A=\{\emptyset\}\), then what is \(n(\mathcal{P}(A))\)?
Answer and explanation
Correct answer: 2
The set \(A=\{\emptyset\}\) contains exactly one element: the empty set \(\emptyset\). It is important to distinguish \(\emptyset\), which has no elements, from \(\{\emptyset\}\), which has one element. A set with \(n\) elements has \(2^n\) subsets, so its power set has cardinality \(n(\mathcal{P}(A))=2^{n(A)}=2^1=2\). The two subsets are \(\emptyset\) and \(\{\emptyset\}\). Therefore, option C is correct.
Frequently asked questions
What is the correct answer to this question?
2
Why is this the correct answer?
The set \(A=\{\emptyset\}\) contains exactly one element: the empty set \(\emptyset\). It is important to distinguish \(\emptyset\), which has no elements, from \(\{\emptyset\}\), which has one element. A set with \(n\) elements has \(2^n\) subsets, so its power set has cardinality \(n(\mathcal{P}(A))=2^{n(A)}=2^1=2\). The two subsets are \(\emptyset\) and \(\{\emptyset\}\). 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.
