If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,2,3,4\}\), and \(B=\{3,4,5,6\}\), what is \(A'\cup B'\)?
Answer and explanation
Correct answer: \(\{1,2,5,6,7,8\}\)
All complements are taken relative to \(U\). Thus \(A'=U\setminus A=\{5,6,7,8\}\) and \(B'=U\setminus B=\{1,2,7,8\}\). Their union contains every element appearing in either complement: \(A'\cup B'=\{1,2,5,6,7,8\}\). This also agrees with De Morgan’s law, \(A'\cup B'=(A\cap B)'\), since \(A\cap B=\{3,4\}\). Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
\(\{1,2,5,6,7,8\}\)
Why is this the correct answer?
All complements are taken relative to \(U\). Thus \(A'=U\setminus A=\{5,6,7,8\}\) and \(B'=U\setminus B=\{1,2,7,8\}\). Their union contains every element appearing in either complement: \(A'\cup B'=\{1,2,5,6,7,8\}\). This also agrees with De Morgan’s law, \(A'\cup B'=(A\cap B)'\), since \(A\cap B=\{3,4\}\). 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.
