If \(U=\mathbb{N}\) and \(A=\{x:x\in\mathbb{N}\text{ and }x\text{ is odd}\}\), what is \(A'\) with respect to \(U\)?
Answer and explanation
Correct answer: \(\{x:x\in\mathbb{N}\text{ and }x\text{ is even}\}\)
A complement is determined by the universal set. Here the universal set is \(\mathbb{N}\), and \(A\) consists of all odd natural numbers. Every natural number is either odd or even, and no natural number is both. Therefore, the elements of \(\mathbb{N}\) that are not in \(A\) are precisely the even natural numbers: \(A'=\{x\in\mathbb{N}:x\text{ is even}\}\). Option B incorrectly changes the universe to \(\mathbb{Z}\), while C includes only primes.
Frequently asked questions
What is the correct answer to this question?
\(\{x:x\in\mathbb{N}\text{ and }x\text{ is even}\}\)
Why is this the correct answer?
A complement is determined by the universal set. Here the universal set is \(\mathbb{N}\), and \(A\) consists of all odd natural numbers. Every natural number is either odd or even, and no natural number is both. Therefore, the elements of \(\mathbb{N}\) that are not in \(A\) are precisely the even natural numbers: \(A'=\{x\in\mathbb{N}:x\text{ is even}\}\). Option B incorrectly changes the universe to \(\mathbb{Z}\), while C includes only primes.
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.
