If \(A=\{1,2,3,4\}\) and \(B=\{3,4,5,6\}\), what is \(\mathcal P(A\cap B)\cap\mathcal P(A\setminus B)\)?
Answer and explanation
Correct answer: \(\{\varnothing\}\)
We have \(A\cap B=\{3,4\}\) and \(A\setminus B=\{1,2\}\). These two sets are disjoint. A set that belongs to both power sets must be a subset of both \(\{3,4\}\) and \(\{1,2\}\). The only common subset of disjoint sets is the empty set. Since the empty set is an element of every power set, the intersection of the two power sets is \(\{\varnothing\}\), not \(\varnothing\).
Frequently asked questions
What is the correct answer to this question?
\(\{\varnothing\}\)
Why is this the correct answer?
We have \(A\cap B=\{3,4\}\) and \(A\setminus B=\{1,2\}\). These two sets are disjoint. A set that belongs to both power sets must be a subset of both \(\{3,4\}\) and \(\{1,2\}\). The only common subset of disjoint sets is the empty set. Since the empty set is an element of every power set, the intersection of the two power sets is \(\{\varnothing\}\), not \(\varnothing\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
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