Which pair of properties is most useful for checking whether a set claimed to be A^c is actually the complement of A?
Answer and explanation
Correct answer: \(A\cap A^c=\varnothing\) and \(A\cup A^c=U\)
A set and its complement must satisfy two fundamental conditions. First, they are disjoint, so they have no common element and A∩A^c=∅. Second, together they contain every element of the universal set, so A∪A^c=U. These two identities provide a complete and reliable check. The other options contradict standard complement laws or state unrelated claims.
Frequently asked questions
What is the correct answer to this question?
\(A\cap A^c=\varnothing\) and \(A\cup A^c=U\)
Why is this the correct answer?
A set and its complement must satisfy two fundamental conditions. First, they are disjoint, so they have no common element and A∩A^c=∅. Second, together they contain every element of the universal set, so A∪A^c=U. These two identities provide a complete and reliable check. The other options contradict standard complement laws or state unrelated claims.
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.
