Question

Consider U = {x|x is a negative real number}.

Which is an empty set?

{x|x ∈ U and x has a negative cube root}
{x|x ∈ U and x has a negative square root}
{x|x ∈ U and x is equal to the product of a positive number and –1}
{x|x ∈ U and x is equal to the sum of one negative and one positive number}

Answer 1

Let's analyze each of the provided sets to determine whether they are empty or not.

1. **Set 1: {x | x ∈ U and x has a negative cube root}**
- A negative number always has a negative cube root. For example, if \( x = -8 \), the cube root is \( \sqrt[3]{-8} = -2 \), which is negative. Therefore, this set is not empty.

2. **Set 2: {x | x ∈ U and x has a negative square root}**
- A negative number does not have a real square root (because the square root of a negative number is imaginary). Therefore, this set is empty.

3. **Set 3: {x | x ∈ U and x is equal to the product of a positive number and –1}**
- A negative real number can indeed be expressed as the product of a positive number (let's say \( a > 0 \)) and -1 (i.e., \( x = -a \)). For example, \( -3 \) can be expressed as \( -1 \times 3 \). Therefore, this set is not empty.

4. **Set 4: {x | x ∈ U and x is equal to the sum of one negative and one positive number}**
- A negative number can be represented as the sum of a negative number and a positive number. For example, \( -2 \) can be represented as \( -3 + 1 \). Hence, this set is also not empty.

In summary:
- Set 1: Not empty
- Set 2: **Empty set**
- Set 3: Not empty
- Set 4: Not empty

Therefore, the only empty set among the options given is **Set 2**: {x|x ∈ U and x has a negative square root}.