If the universal set is \(U=\mathbb{R}\), \(A=(-\infty,2]\), and \(B=[-1,\infty)\), what is \(A'\cap B'\)?
Answer and explanation
Correct answer: \(\varnothing\)
Since \(A=(-\infty,2]\), its complement in \(\mathbb{R}\) is \(A'=(2,\infty)\); the endpoint 2 is excluded because it belongs to A. Since \(B=[-1,\infty)\), its complement is \(B'=(-\infty,-1)\); the endpoint −1 is excluded because it belongs to B. No real number can be both greater than 2 and less than −1, so \(A'\cap B'=\varnothing\).
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
Since \(A=(-\infty,2]\), its complement in \(\mathbb{R}\) is \(A'=(2,\infty)\); the endpoint 2 is excluded because it belongs to A. Since \(B=[-1,\infty)\), its complement is \(B'=(-\infty,-1)\); the endpoint −1 is excluded because it belongs to B. No real number can be both greater than 2 and less than −1, so \(A'\cap B'=\varnothing\).
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.
