If n(U) = 300, n(A') = 120, n(B') = 150, and n(A' ∪ B') = 210, what is n(A ∪ B)?
Answer and explanation
Correct answer: 240
First use inclusion–exclusion for the complements: n(A′ ∩ B′) = n(A′) + n(B′) − n(A′ ∪ B′) = 120 + 150 − 210 = 60. De Morgan’s law gives (A ∪ B)′ = A′ ∩ B′, so the complement of A ∪ B has 60 elements. Therefore n(A ∪ B) = n(U) − n((A ∪ B)′) = 300 − 60 = 240. Option A is correct; 210 is the given union of complements, not the requested union.
Frequently asked questions
What is the correct answer to this question?
240
Why is this the correct answer?
First use inclusion–exclusion for the complements: n(A′ ∩ B′) = n(A′) + n(B′) − n(A′ ∪ B′) = 120 + 150 − 210 = 60. De Morgan’s law gives (A ∪ B)′ = A′ ∩ B′, so the complement of A ∪ B has 60 elements. Therefore n(A ∪ B) = n(U) − n((A ∪ B)′) = 300 − 60 = 240. Option A is correct; 210 is the given union of complements, not the requested union.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
