Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Ask your questions and receive precise answers from experienced professionals across different disciplines. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Consider [tex]$U=\{x \mid x$ is a negative real number [tex]$\}$[/tex][/tex].

Which is an empty set?

A. [tex]$\{x \mid x \in U$ and [tex]$x$[/tex] has a negative cube root $\}$[/tex]

B. [tex][tex]$\{x \mid x \in U$[/tex] and [tex]$x$[/tex] has a negative square root $\}$[/tex]

C. [tex][tex]$\{x \mid x \in U$[/tex] and [tex]$x$[/tex] is equal to the product of a positive number and -1$\}$[/tex]

D. [tex][tex]$\{x \mid x \in U$[/tex] and [tex]$x$[/tex] is equal to the sum of one negative and one positive number $\}$[/tex]

Sagot :

GinnyAnswer
Let's analyze each set description and determine whether it is empty or not.

1. [tex]\(\{x \mid x \in U \text{ and } x \text{ has a negative cube root}\}\)[/tex]:

- [tex]\(U\)[/tex] is the set of negative real numbers.
- A negative number [tex]\(x\)[/tex] can have a negative cube root because the cube root of a negative number is also negative (e.g., [tex]\(\sqrt[3]{-8} = -2\)[/tex]).
- Therefore, this set contains all negative real numbers because every negative real number in [tex]\(U\)[/tex] has a negative cube root. This set is not empty.

2. [tex]\(\{x \mid x \in U \text{ and } x \text{ has a negative square root}\}\)[/tex]:

- [tex]\(U\)[/tex] is the set of negative real numbers.
- The square root of a negative number is not a real number; it is an imaginary number (e.g., [tex]\(\sqrt{-4} = 2i\)[/tex]).
- Since [tex]\(x\)[/tex] must be a real number and we are considering real numbers only, there are no real numbers in [tex]\(U\)[/tex] that have a real square root, let alone a negative square root.
- Therefore, this set is empty.

3. [tex]\(\{x \mid x \in U \text{ and } x \text{ is equal to the product of a positive number and -1}\}\)[/tex]:

- [tex]\(U\)[/tex] is the set of negative real numbers.
- A number [tex]\(x\)[/tex] that is the product of a positive number and [tex]\(-1\)[/tex] is negative (e.g., [tex]\(x = -1 \cdot 3 = -3\)[/tex]).
- All numbers in [tex]\(U\)[/tex] are negative by definition, and they can be expressed as the product of a positive number and [tex]\(-1\)[/tex].
- Therefore, this set contains all negative real numbers in [tex]\(U\)[/tex] and is not empty.

4. [tex]\(\{x \mid x \in U \text{ and } x \text{ is equal to the sum of one negative and one positive number}\}\)[/tex]:

- [tex]\(U\)[/tex] is the set of negative real numbers.
- A negative number can be expressed as the sum of a negative number and a (larger in magnitude) negative number or a smaller positive number (e.g., [tex]\(-5 = -8 + 3\)[/tex]).
- All numbers in [tex]\(U\)[/tex] can be expressed as the sum of a smaller positive number and a larger negative number.
- Therefore, this set is not empty.

So, the only set that is empty is:
[tex]\[ \{x \mid x \in U \text{ and } x \text{ has a negative square root}\} \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.