Question

Consider U = x|x is a positive integer greater than 1.

Which is an empty set?

a)x|x ∈ U and 1/2 x is prime

b)x|x ∈ U and 2x is prime

c)x|x ∈ U and 1/2 can be written as a fraction

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

Answer 1

Given the set U as positive integers greater than 1, the only empty set is option (a) which requires half of an element x to be prime, which is impossible because half of an integer greater than 1 is not an integer and thus not prime.

Step-by-step explanation:

A student asked which of the following sets is empty, given U = x | x is a positive integer greater than 1:

1. x | x ∈ U and 1/2 x is prime
2. x | x ∈ U and 2x is prime
3. x | x ∈ U and 1/2 can be written as a fraction
4. x | x ∈ U and 2x can be written as a fraction

Let's go through the options one by one:

• Option (a) suggests half of x must be prime. But since x is a positive integer greater than 1, half of x would not be an integer, so it cannot be prime. This means the set described in option (a) is indeed an empty set.
• Option (b) suggests that doubling x must yield a prime number. This is possible if x is 1 or a number which when doubled gives a prime number, but because x must be greater than 1 by the definition of U, if x is an integer that would give 2x as prime, then the set is not empty.
• Option (c) is not an empty set because 1/2 can indeed be written as a fraction, specifically as 1/2 itself.
• Option (d) is not an empty set either since 2x will always result in an integer which can certainly be expressed as a fraction.

Therefore, the only empty set is option (a).