If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A^c=\{2,4,6,8\}\), what is the value of \(A\cap A^c\)?
Answer and explanation
Correct answer: \(\varnothing\)
By definition, \(A^c\) contains precisely the elements of the universal set that are not in \(A\). Consequently, no element can belong to both \(A\) and \(A^c\) at the same time. Their intersection is therefore empty: \(A\cap A^c=\varnothing\). Although the given complement allows us to identify \(A=\{1,3,5,7\}\), it does not change this identity.
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
By definition, \(A^c\) contains precisely the elements of the universal set that are not in \(A\). Consequently, no element can belong to both \(A\) and \(A^c\) at the same time. Their intersection is therefore empty: \(A\cap A^c=\varnothing\). Although the given complement allows us to identify \(A=\{1,3,5,7\}\), it does not change this identity.
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.
