If the universal set is \(U=\{1,2,\ldots,32\}\) and \(A=\{x:x=2^n,\ n\in\mathbb{N},\ 0\le n\le 5\}\), what is the value of \(|A'|\)?
Answer and explanation
Correct answer: 26
The allowed values of \(n\) are 0, 1, 2, 3, 4, and 5. Hence, \(A=\{2^0,2^1,2^2,2^3,2^4,2^5\}=\{1,2,4,8,16,32\}\), which contains six distinct elements. Since the complement is taken relative to \(U\), every element of U not in A belongs to \(A'\). Therefore, \(|A'|=|U|-|A|=32-6=26\). Thus, option B is correct. The value 32 is included in A because the condition permits \(n=5\).
Frequently asked questions
What is the correct answer to this question?
26
Why is this the correct answer?
The allowed values of \(n\) are 0, 1, 2, 3, 4, and 5. Hence, \(A=\{2^0,2^1,2^2,2^3,2^4,2^5\}=\{1,2,4,8,16,32\}\), which contains six distinct elements. Since the complement is taken relative to \(U\), every element of U not in A belongs to \(A'\). Therefore, \(|A'|=|U|-|A|=32-6=26\). Thus, option B is correct. The value 32 is included in A because the condition permits \(n=5\).
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.
