If the universal set is \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{1,2,3,4\}\), and \(B=\{3,4,5,6\}\), what is the value of \((A^c\cap B)^c\)?
Answer and explanation
Correct answer: \(\{1,2,3,4,7,8,9,10\}\)
First find the complement of \(A\) in \(U\): \(A^c=U\setminus A=\{5,6,7,8,9,10\}\). Intersecting this with \(B=\{3,4,5,6\}\) gives \(A^c\cap B=\{5,6\}\), because 5 and 6 are the only elements common to both sets. Finally, take the complement of \(\{5,6\}\) in \(U\): \(U\setminus\{5,6\}=\{1,2,3,4,7,8,9,10\}\). Therefore, option A is correct.
Frequently asked questions
What is the correct answer to this question?
\(\{1,2,3,4,7,8,9,10\}\)
Why is this the correct answer?
First find the complement of \(A\) in \(U\): \(A^c=U\setminus A=\{5,6,7,8,9,10\}\). Intersecting this with \(B=\{3,4,5,6\}\) gives \(A^c\cap B=\{5,6\}\), because 5 and 6 are the only elements common to both sets. Finally, take the complement of \(\{5,6\}\) in \(U\): \(U\setminus\{5,6\}=\{1,2,3,4,7,8,9,10\}\). Therefore, option A is correct.
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.
