Question

Consider ( U = {x \mid x ) is a positive integer greater than 1 }. Which is an empty set?A. ( {XIX \in U \text{ and is prime} } )B. ( {x \in U \text{ and } 2x \text{ is prime} } )C. ( {x \\otin U \text{ and } x \text{ can be written as a fraction} } )D. ( {x \mid x \in U \text{ and } 2x \text{ can be written as a fraction} } )

Answer 1

The set is empty because, for positive integers greater than 1, doubling them results in even numbers. Since the only even prime number is 2, the set requires x values equal to 1, which contradicts the given condition. Therefore, there are no elements in this set. So, the correct option is B.
`( {x \in U \text{ and } 2x \text{ is prime} } )`.

Explanation:

The correct option is B.
`( {x \in U \text{ and } 2x \text{ is prime} } )`. This set is empty because, for any positive integer greater than 1, when you double it (2x), the result will always be an even number. The only even prime number is 2, and this occurs when x is equal to 1.

However, the given set explicitly excludes x values less than or equal to 1, leaving no valid elements. Therefore, there are no positive integers greater than 1 such that 2x is prime, making this set empty.

Understanding the nature of prime numbers is crucial for this conclusion. Prime numbers greater than 2 are always odd, as any even number greater than 2 is divisible by 2. The doubling operation on positive integers greater than 1 ensures an even result, making it impossible for 2x to be prime in this context. Hence, the set is devoid of elements.