If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6\}\), and \(B=\{4,6,8\}\), what is \(A'\cap B'\)?
Answer and explanation
Correct answer: \(\{1,3,5,7\}\)
The complement of a set is taken with respect to the universal set \(U\). Thus, \(A'=U\setminus A=\{1,3,5,7,8\}\), because these elements are in \(U\) but not in \(A\). Similarly, \(B'=U\setminus B=\{1,2,3,5,7\}\). The intersection contains only elements common to both complements: \(A'\cap B'=\{1,3,5,7\}\). Therefore, option A is correct. This also agrees with De Morgan’s law: \(A'\cap B'=(A\cup B)'\).
Frequently asked questions
What is the correct answer to this question?
\(\{1,3,5,7\}\)
Why is this the correct answer?
The complement of a set is taken with respect to the universal set \(U\). Thus, \(A'=U\setminus A=\{1,3,5,7,8\}\), because these elements are in \(U\) but not in \(A\). Similarly, \(B'=U\setminus B=\{1,2,3,5,7\}\). The intersection contains only elements common to both complements: \(A'\cap B'=\{1,3,5,7\}\). Therefore, option A is correct. This also agrees with De Morgan’s law: \(A'\cap B'=(A\cup B)'\).
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.
