If \(U=\{1,2,\ldots,64\}\) and \(A=\{x:x=2^n,\ n\in\mathbb N,\ 1\le n\le6\}\), what is the cardinality \(|A'|\) of the complement of \(A\) in \(U\)?
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Answer and explanation
Correct answer: 58
For \(n=1,2,3,4,5,6\), the distinct values of \(2^n\) are \(2,4,8,16,32,64\). Thus \(|A|=6\). The universal set \(U=\{1,2,\ldots,64\}\) contains 64 elements. Since \(A'\) contains all elements of \(U\) not in \(A\), \(|A'|=|U|-|A|=64-6=58\). Neither 0 nor 1 belongs to \(A\).
Frequently asked questions
What is the correct answer to this question?
58
Why is this the correct answer?
For \(n=1,2,3,4,5,6\), the distinct values of \(2^n\) are \(2,4,8,16,32,64\). Thus \(|A|=6\). The universal set \(U=\{1,2,\ldots,64\}\) contains 64 elements. Since \(A'\) contains all elements of \(U\) not in \(A\), \(|A'|=|U|-|A|=64-6=58\). Neither 0 nor 1 belongs to \(A\).
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.
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