consider u = {x|x is a positive integer greater than 1}. which is an empty set?\no {x|x ∈ u and…

consider u = {x|x is a positive integer greater than 1}. which is an empty set?\no {x|x ∈ u and $\frac{1}{2}$x is prime}\no {x|x ∈ u and 2x is prime}\no {x|x ∈ u and $\frac{1}{2}$x can be written as a fraction}\no {x|x ∈ u and 2x can be written as a fraction}

consider u = {x|x is a positive integer greater than 1}. which is an empty set?\no {x|x ∈ u and $\frac{1}{2}$x is prime}\no {x|x ∈ u and 2x is prime}\no {x|x ∈ u and $\frac{1}{2}$x can be written as a fraction}\no {x|x ∈ u and 2x can be written as a fraction}

Answer

Explanation:

Step1: Analyze the first option

Let (x\in U), if (\frac{1}{2}x) is prime. When (x = 2), (\frac{1}{2}x=1) which is not prime. When (x = 4), (\frac{1}{2}x = 2) (prime), so this set is not empty.

Step2: Analyze the second option

Let (x\in U). If (x) is a positive - integer greater than 1, then (2x) is an even number greater than 2. Since all even numbers greater than 2 are not prime (by the definition of prime numbers: a prime number has only two distinct positive divisors: 1 and itself, and even numbers greater than 2 are divisible by 2), the set ({x|x\in U\text{ and }2x\text{ is prime}}) has no elements, so it is an empty set.

Step3: Analyze the third option

Let (x\in U). (\frac{1}{2}x) is a rational number for all (x\in U) (since (x) is an integer, (\frac{1}{2}x) can be written as a fraction), so this set is not empty.

Step4: Analyze the fourth option

Let (x\in U). (2x) is an integer for all (x\in U), and all integers can be written as a fraction (e.g., (n=\frac{n}{1})), so this set is not empty.

Answer:

({x|x\in U\text{ and }2x\text{ is prime}})