For subsets \(A\) and \(B\) of a universal set \(U\), which of the following statements is always true?
Answer and explanation
Correct answer: \((A\cup B)'=A'\cap B'\)
The correct De Morgan identity is \((A\cup B)'=A'\cap B'\). The complement of a union contains precisely those elements that belong to neither \(A\) nor \(B\), so they must lie in both complements. Statement A is incorrect because \((A\cap B)'=A'\cup B'\). Also, \((A')'=A\), not the empty set, and \(A\cup A'=U\), not the empty set. Hence option B is the only always-true statement.
Frequently asked questions
What is the correct answer to this question?
\((A\cup B)'=A'\cap B'\)
Why is this the correct answer?
The correct De Morgan identity is \((A\cup B)'=A'\cap B'\). The complement of a union contains precisely those elements that belong to neither \(A\) nor \(B\), so they must lie in both complements. Statement A is incorrect because \((A\cap B)'=A'\cup B'\). Also, \((A')'=A\), not the empty set, and \(A\cup A'=U\), not the empty set. Hence option B is the only always-true statement.
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.
