If A is a subset of the universal set U, which statement is always true for A and its complement A′?
Answer and explanation
Correct answer: A ∩ A′ = ∅
By definition, A′ contains the elements of U that are not in A. Consequently, no element can belong to A and A′ at the same time. Their intersection is therefore empty: A ∩ A′ = ∅. The complementary identity for the union is A ∪ A′ = U, not the empty set. Also, A′ need not be a subset of A, so option A is the only statement that is always true.
Frequently asked questions
What is the correct answer to this question?
A ∩ A′ = ∅
Why is this the correct answer?
By definition, A′ contains the elements of U that are not in A. Consequently, no element can belong to A and A′ at the same time. Their intersection is therefore empty: A ∩ A′ = ∅. The complementary identity for the union is A ∪ A′ = U, not the empty set. Also, A′ need not be a subset of A, so option A is the only statement that is always true.
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.
