If A ∪ Aᶜ = U and A ∩ Aᶜ = ∅, then n(A) + n(Aᶜ) is equal to what?
Answer and explanation
Correct answer: n(U)
A set and its complement are disjoint, as shown by A ∩ Aᶜ = ∅, and their union is the entire universal set, as shown by A ∪ Aᶜ = U. For two disjoint finite sets, the cardinality of their union equals the sum of their cardinalities. Therefore, n(A ∪ Aᶜ) = n(A) + n(Aᶜ). Since A ∪ Aᶜ = U, the result is n(A) + n(Aᶜ) = n(U).
Frequently asked questions
What is the correct answer to this question?
n(U)
Why is this the correct answer?
A set and its complement are disjoint, as shown by A ∩ Aᶜ = ∅, and their union is the entire universal set, as shown by A ∪ Aᶜ = U. For two disjoint finite sets, the cardinality of their union equals the sum of their cardinalities. Therefore, n(A ∪ Aᶜ) = n(A) + n(Aᶜ). Since A ∪ Aᶜ = U, the result is n(A) + n(Aᶜ) = n(U).
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.
