If \(A'=B'\), which conclusion is necessarily true?
Answer and explanation
Correct answer: \(A=B\)
Take the complement of both sides of the given equality \(A'=B'\). Complementation is an involution, meaning that taking a complement twice returns the original set: \((A')'=A\) and \((B')'=B\). Thus \((A')'=(B')'\) implies \(A=B\). The other statements are not necessary consequences: equal sets need not be disjoint, their union need not be the universal set, and A need not be contained in \(B'\). Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
\(A=B\)
Why is this the correct answer?
Take the complement of both sides of the given equality \(A'=B'\). Complementation is an involution, meaning that taking a complement twice returns the original set: \((A')'=A\) and \((B')'=B\). Thus \((A')'=(B')'\) implies \(A=B\). The other statements are not necessary consequences: equal sets need not be disjoint, their union need not be the universal set, and A need not be contained in \(B'\). Therefore option A is correct.
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.
