Question

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

Which is an empty set?

{x|x ∈ U andOne-half is prime}
{x|x ∈ U and 2x is prime}
{x|x ∈ U andOne-half can be written as a fraction}
{x|x ∈ U and 2x can be written as a fraction}

Answer 1

To determine which of the sets is empty, let's analyze each option:

1. **{x | x ∈ U and One-half is prime}**:
- One-half (or \( \frac{1}{2} \)) is not a prime number. In fact, it is not an integer at all. Therefore, there are no positive integers greater than 1 for which One-half is prime. This set is indeed empty.

2. **{x | x ∈ U and 2x is prime}**:
- This set consists of positive integers \( x \) such that \( 2x \) is a prime number. For any integer \( x \geq 2 \), \( 2x \) is an even number. The only even prime number is 2, which can only happen if \( x = 1 \), but \( x \) must be greater than 1 according to the defined universe \( U \). Therefore, this set is also empty.

3. **{x | x ∈ U and One-half can be written as a fraction}**:
- One-half can be expressed as \( \frac{1}{2} \), which is a fraction. However, the condition does not empty the set of positive integers greater than 1. This set will not be empty because it does not impose any restriction related to \( x \); thus, it will contain all positive integers greater than 1 because of the universal quantifier.

4. **{x | x ∈ U and 2x can be written as a fraction}**:
- The expression \( 2x \) will produce integer values when \( x \) is a positive integer. Every positive integer can be expressed as a fraction (e.g., \( 2x = \frac{2x}{1} \)). Therefore, this set is not empty and includes all integers greater than 1.

### Conclusion:
The empty sets are the first and second options:

- **{x | x ∈ U and One-half is prime}** (Empty)
- **{x | x ∈ U and 2x is prime}** (Empty)

However, since only one response can be selected, the answer would be the first option:

**{x | x ∈ U and One-half is prime}** is an empty set.