If \(U=\{1,2,\ldots,32\}\), \(A=\{x:x\text{ is a multiple of }4\}\), and \(B=\{x:x\text{ is a multiple of }16\}\), how many elements are in \(A'\cup B\)?
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Answer and explanation
Correct answer: 26
There are \(\lfloor32/4\rfloor=8\) multiples of 4 in \(U\), so \(|A'|=32-8=24\). Every multiple of 16 is also a multiple of 4, hence \(B\subseteq A\), which means \(A'\cap B=\varnothing\). The multiples of 16 in the universe are 16 and 32, so \(|B|=2\). Therefore \(|A'\cup B|=24+2=26\).
Frequently asked questions
What is the correct answer to this question?
26
Why is this the correct answer?
There are \(\lfloor32/4\rfloor=8\) multiples of 4 in \(U\), so \(|A'|=32-8=24\). Every multiple of 16 is also a multiple of 4, hence \(B\subseteq A\), which means \(A'\cap B=\varnothing\). The multiples of 16 in the universe are 16 and 32, so \(|B|=2\). Therefore \(|A'\cup B|=24+2=26\).
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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