How many elements are there in \(\mathcal{P}(\mathcal{P}(\{a\}))\)?
Answer and explanation
Correct answer: 4
The set \(\{a\}\) has one element, so its power set has \(2^1=2\) elements: \(\varnothing\) and \(\{a\}\). Taking the power set once more gives a set with \(2^2=4\) elements. Therefore, \(|\mathcal{P}(\mathcal{P}(\{a\}))|=4\), so option B is correct. In general, if \(|A|=n\), then \(|\mathcal{P}(\mathcal{P}(A))|=2^{2^n}\).
Frequently asked questions
What is the correct answer to this question?
4
Why is this the correct answer?
The set \(\{a\}\) has one element, so its power set has \(2^1=2\) elements: \(\varnothing\) and \(\{a\}\). Taking the power set once more gives a set with \(2^2=4\) elements. Therefore, \(|\mathcal{P}(\mathcal{P}(\{a\}))|=4\), so option B is correct. In general, if \(|A|=n\), then \(|\mathcal{P}(\mathcal{P}(A))|=2^{2^n}\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
