If the universal set is \(U=\{x:x\in\mathbb{Z},\ 0\le x\le 10\}\) and \(A=\{x:x\in U\text{ and }x^2-5x+6=0\}\), which of the following is the complement \(A'\) of \(A\) with respect to \(U\)?
Answer and explanation
Correct answer: \(\{0,1,4,5,6,7,8,9,10\}\)
Factor the quadratic equation: \(x^2-5x+6=(x-2)(x-3)=0\). Its roots are 2 and 3, and both belong to the specified universe, so \(A=\{2,3\}\). The universal set is \(U=\{0,1,2,3,4,5,6,7,8,9,10\}\). Hence the complement \(A'=U\setminus A\) contains every element except 2 and 3, giving \(\{0,1,4,5,6,7,8,9,10\}\). Thus option A is correct; option B is the original set \(A\).
Frequently asked questions
What is the correct answer to this question?
\(\{0,1,4,5,6,7,8,9,10\}\)
Why is this the correct answer?
Factor the quadratic equation: \(x^2-5x+6=(x-2)(x-3)=0\). Its roots are 2 and 3, and both belong to the specified universe, so \(A=\{2,3\}\). The universal set is \(U=\{0,1,2,3,4,5,6,7,8,9,10\}\). Hence the complement \(A'=U\setminus A\) contains every element except 2 and 3, giving \(\{0,1,4,5,6,7,8,9,10\}\). Thus option A is correct; option B is the original set \(A\).
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.
