If the universal set is \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{x\mid x\in U\text{ and }x^2-5x+6=0\}\), what is the complement \(A^c\) of \(A\) with respect to \(U\)?
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Answer and explanation
Correct answer: \(\{1,4,5,6,7,8,9,10\}\)
Factor the condition: \(x^2-5x+6=(x-2)(x-3)=0\). Thus the solutions that lie in U are \(x=2\) and \(x=3\), so \(A=\{2,3\}\). The complement relative to U is \(U\setminus A\), obtained by removing 2 and 3 from U. Hence \(A^c=\{1,4,5,6,7,8,9,10\}\), which is option A.
Frequently asked questions
What is the correct answer to this question?
\(\{1,4,5,6,7,8,9,10\}\)
Why is this the correct answer?
Factor the condition: \(x^2-5x+6=(x-2)(x-3)=0\). Thus the solutions that lie in U are \(x=2\) and \(x=3\), so \(A=\{2,3\}\). The complement relative to U is \(U\setminus A\), obtained by removing 2 and 3 from U. Hence \(A^c=\{1,4,5,6,7,8,9,10\}\), which is option 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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