If \(\mathcal{P}(A\cap B)=\{\varnothing\}\), what is \(A\cap B\)?
Answer and explanation
Correct answer: \(\varnothing\)
The power set of any set contains the empty set. For the empty set itself, the power set is \(\mathcal{P}(\varnothing)=\{\varnothing\}\), which has exactly one element. Therefore, if \(\mathcal{P}(A\cap B)=\{\varnothing\}\), the original set must be empty. Hence, \(A\cap B=\varnothing\). Notice that \(\{\varnothing\}\) is a set containing the empty set, whereas \(\varnothing\) contains no elements.
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
The power set of any set contains the empty set. For the empty set itself, the power set is \(\mathcal{P}(\varnothing)=\{\varnothing\}\), which has exactly one element. Therefore, if \(\mathcal{P}(A\cap B)=\{\varnothing\}\), the original set must be empty. Hence, \(A\cap B=\varnothing\). Notice that \(\{\varnothing\}\) is a set containing the empty set, whereas \(\varnothing\) contains no elements.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: The Empty Set, Finite and Infinite Sets, Equal Sets.
