If \(A^c = U\), then what is \(A\) equal to?
Answer and explanation
Correct answer: \(\varnothing\)
The complement \(A^c\) consists of all elements of the universal set \(U\) that are not in \(A\). If \(A^c=U\), every element of \(U\) lies outside \(A\). Therefore, \(A\) contains no element and must be the empty set, \(A=\varnothing\). This also agrees with the standard identity \(A\cup A^c=U\) and \(A\cap A^c=\varnothing\). Option B is incorrect because \(A=U\) would imply \(A^c=\varnothing\), not \(U\).
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
The complement \(A^c\) consists of all elements of the universal set \(U\) that are not in \(A\). If \(A^c=U\), every element of \(U\) lies outside \(A\). Therefore, \(A\) contains no element and must be the empty set, \(A=\varnothing\). This also agrees with the standard identity \(A\cup A^c=U\) and \(A\cap A^c=\varnothing\). Option B is incorrect because \(A=U\) would imply \(A^c=\varnothing\), not \(U\).
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.
