Let the universal set be \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{1,3,5,7\}\). If \(C=A^c\), what is the value of \(C^c\)?
Answer and explanation
Correct answer: \(\{1,3,5,7\}\)
Since \(C=A^c\), first find the complement of \(A\) in the universal set: \(C=\{2,4,6,8\}\). Taking the complement of \(C\) again gives all elements of \(U\) that are not in \(C\), namely \(\{1,3,5,7\}\). Thus \(C^c=(A^c)^c=A\), so option A is correct. This illustrates the double-complement identity: the complement of a complement is the original set.
Frequently asked questions
What is the correct answer to this question?
\(\{1,3,5,7\}\)
Why is this the correct answer?
Since \(C=A^c\), first find the complement of \(A\) in the universal set: \(C=\{2,4,6,8\}\). Taking the complement of \(C\) again gives all elements of \(U\) that are not in \(C\), namely \(\{1,3,5,7\}\). Thus \(C^c=(A^c)^c=A\), so option A is correct. This illustrates the double-complement identity: the complement of a complement is the original set.
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.
