If the universal set is \(U=\mathbb{R}\) and \(A=\{x:x^2-1\ge 0\}\), what is \(A'\)?
Answer and explanation
Correct answer: \((-1,1)\)
Factor the inequality as \(x^2-1=(x-1)(x+1)\ge0\). The product is non-negative outside the roots, so \(A=(-\infty,-1]\cup[1,\infty)\). Since the universal set is all real numbers, the complement consists of the real numbers strictly between the roots. Thus \(A'=(-1,1)\), and the endpoints are excluded because they belong to \(A\).
Frequently asked questions
What is the correct answer to this question?
\((-1,1)\)
Why is this the correct answer?
Factor the inequality as \(x^2-1=(x-1)(x+1)\ge0\). The product is non-negative outside the roots, so \(A=(-\infty,-1]\cup[1,\infty)\). Since the universal set is all real numbers, the complement consists of the real numbers strictly between the roots. Thus \(A'=(-1,1)\), and the endpoints are excluded because they belong 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.
