If \(A^c=\varnothing\), then A is equal to what?
Answer and explanation
Correct answer: \(U\)
The complement \(A^c\) contains all elements of U that are not in A. If this complement is empty, there is no element of U outside A. Consequently, A must contain every element of U, so \(A=U\). This also follows from the identity \(A\cup A^c=U\): when \(A^c=\varnothing\), the union becomes A, giving A=U. Therefore option A is correct; the other choices do not follow from the definition.
Frequently asked questions
What is the correct answer to this question?
\(U\)
Why is this the correct answer?
The complement \(A^c\) contains all elements of U that are not in A. If this complement is empty, there is no element of U outside A. Consequently, A must contain every element of U, so \(A=U\). This also follows from the identity \(A\cup A^c=U\): when \(A^c=\varnothing\), the union becomes A, giving A=U. Therefore option A is correct; the other choices do not follow from the definition.
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.
