Consider u = {x|x is a positive integer greater than 1}. which is an empty set? {x|x ∈ u and 1/2 x is prime} {x|x ∈ u and 2x is prime} {x|x ∈ u and 1/2 can be written as a fraction} {x|x ∈ u and 2x can be written as a fraction}

{x|x ∈ U and 2x is prime} is an empty set.

Justification:

U = {x|x is a positive integer greater than 1} = {2, 3, 4, 5, 6, ...}

1) {x|x ∈ U and (1/2) x is prime}

Take x = 4 => (1/2) 4 = 4 / 2 = 2 = prime, so this is not empty.

{x|x ∈ U and 2x is prime}

2) 2x = {4, 6, 8, 10, 12, } so none of them is prime, hence this is an empty set.

3) {x|x ∈ U and (1/2) x can be written as a fraction}

Take any x that is odd. For example x = 3

=> (1/2)x = (1/2)3 = 3/2 which is a fraction, so this is not an empty set.

2) {x|x ∈ U and 2x can be written as a fraction}

Take any value of x, for example x = 7 => 2 = 14, and 14 = 28 / 2 so this is not an empty set.