Question

re algebra i b-cr
consider u = {x|x is a positive integer greater than 1}.
which is an empty set?
○ {x|x ∈ u and $\frac{1}{2}x$ is prime}
○ {x|x ∈ u and 2x is prime}
○ {x|x ∈ u and $\frac{1}{2}x$ can be written as a fraction}
○ {x|x ∈ u and 2x can be written as a fraction}

Explanation:

Step1: Define universal set U

$U = \{2, 3, 4, 5, ...\}$ (all integers >1)

Step2: Analyze Option 1

Check if $\frac{1}{2}x$ is prime. For $x=4$, $\frac{1}{2}(4)=2$ (prime), so set is non-empty.

Step3: Analyze Option 2

Check if $2x$ is prime. For any $x\in U$, $2x$ is even and $\geq4$, so it cannot be prime (primes >2 are odd). This set has no elements.

Step4: Analyze Option 3

$\frac{1}{2}x$ is a fraction for any integer $x$, so all $U$ elements qualify; set is non-empty.

Step5: Analyze Option 4

$2x$ is an integer (a type of fraction), so all $U$ elements qualify; set is non-empty.

Answer:

B. {x|x ∈ U and 2x is prime}