∃ an item I such that ∀ student S, S did not choose I.True. Every student chose every available item on the menu.True. Although some students didn't choose a certain items from the menu, there existed an option for them to choose from. False. Uta did not have a beverage.False. Tim only had one dessert.False. Every item was chosen by at least one student.(f)∃ a station Z such that ∀ S, ∃ an item I such that S chose I from Z. True. Every student chose a main course, every student chose a dessert, and every student chose a beverage.True. Although some students didn't choose a food from some all of the stations, there existed an option for them to choose from. False. Uta did not have a beverage.False. Tim only had one dessert.False. None of the items were chosen from any station by any of the students. Below is the transcription of the given image which can be used for an Educational website.---**Propositional Logic Exercise: Answer the Questions**In the following questions, please determine the truth value for each given statement.**(b) ∀ student S, ∃ a salad T such that S chose T.**- O True. Every student chose at least one salad: Uta chose green salad, Tim chose fruit salad, and Yuen chose green salad.- O True. Although some students didn’t choose salad, there existed an option for them to choose from.- O False. Yuen did not have a salad.- O False. Tim had more than one salad.- O False. None of the students had a salad.**(c) ∃ a dessert D such that ∀ student S, S chose D.**- O True. Every student chose pie.- O True. Although some students didn’t choose dessert, there existed an option for them to choose from.- O False. Uta did not have dessert.- O False. Tim had more than one dessert.- O False. None of the students chose dessert.**(d) ∃ a beverage B such that ∀ student D, D chose B.**- O True. Every student chose soda.- O True. Although some students didn’t choose a beverage, there existed an option for them to choose from.- O False. Uta did not have a beverage.- O False. Tim only had one beverage.- O False. There was no particular beverage chosen by every student.---Explanation of the problems and choice selection:1. **Problem (b)** discusses the concept of universal and existential quantifiers applied to salads chosen by students.2. **Problem (c)** involves determining the distribution of a dessert among students.3. **Problem (d)** touches upon the distribution of a beverage among students.Students are asked to evaluate each statement and choose the correct one based on the logical structure provided. This exercise helps in understanding logical propositions and quantifiers. ### Analysis of Students' Food ChoicesTo determine the truthfulness of various statements related to students' food choices, we analyze the information provided in the figure.**Figure Description**The figure illustrates a list of students and their selections from various food categories: Salads, Main courses, Desserts, and Beverages.- **Salads:** green salad, fruit salad- **Main courses:** spaghetti, fish- **Desserts:** pie, cake- **Beverages:** milk, soda, coffee**Student Selections:**- **Uta:** green salad, fish, pie, soda- **Tim:** fish, fruit salad, pie, cake- **Yuen:** spaghetti, pie, coffee**Questions Analysis**Using the information, we address the statement:**(a) $\forall$ student $S$, $\exists$ a dessert $D$ such that $S$ chose $D$.** **Options:**1. **True. Every student chose at least one dessert: Uta chose pie, Tim chose both pie and cake, and Yuen chose pie.**2. **True. Although some students didn’t choose dessert, there existed an option for them to choose from.**3. **False. Uta did not have dessert.**4. **False. Tim had more than one dessert.**5. **False. None of the students chose dessert.****Determining the Correct Answer:**Based on the figure:- **Uta** chose pie.- **Tim** chose both pie and cake.- **Yuen** chose pie.Thus, each student selected at least one dessert.**Correct Answer:**1. **True. Every student chose at least one dessert: Uta chose pie, Tim chose both pie and cake, and Yuen chose pie.**This content demonstrates how to analyze graphical data to verify the accuracy of different statements regarding the selections of individuals.

Name: ∃ an item I such that ∀ student S, S did not choose I.True. Every stud
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Description: ∃ an item I such that ∀ student S, S did not choose I.True. Every student chose every available item on the menu.True. Although some students didn't choose a certain items from the menu, there existed an option for them to choose from. False. Uta

We are entitled to solve first three subparts of the questions, I am providing the answer of the…

Answered: Use the figure below to determine…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(f)

∃ a station Z such that ∀ S, ∃ an item I such that S chose I from Z.

True. Every student chose a main course, every student chose a dessert, and every student chose a beverage.

True. Although some students didn't choose a food from some all of the stations, there existed an option for them to choose from.

False. Uta did not have a beverage.

False. Tim only had one dessert.

False. None of the items were chosen from any station by any of the students.

Below is the transcription of the given image which can be used for an Educational website.

---

**Propositional Logic Exercise: Answer the Questions**

In the following questions, please determine the truth value for each given statement.

**(b) ∀ student S, ∃ a salad T such that S chose T.**

- O True. Every student chose at least one salad: Uta chose green salad, Tim chose fruit salad, and Yuen chose green salad.
- O True. Although some students didn’t choose salad, there existed an option for them to choose from.
- O False. Yuen did not have a salad.
- O False. Tim had more than one salad.
- O False. None of the students had a salad.

**(c) ∃ a dessert D such that ∀ student S, S chose D.**

- O True. Every student chose pie.
- O True. Although some students didn’t choose dessert, there existed an option for them to choose from.
- O False. Uta did not have dessert.
- O False. Tim had more than one dessert.
- O False. None of the students chose dessert.

**(d) ∃ a beverage B such that ∀ student D, D chose B.**

- O True. Every student chose soda.
- O True. Although some students didn’t choose a beverage, there existed an option for them to choose from.
- O False. Uta did not have a beverage.
- O False. Tim only had one beverage.
- O False. There was no particular beverage chosen by every student.

---

Explanation of the problems and choice selection:

1. **Problem (b)** discusses the concept of universal and existential quantifiers applied to salads chosen by students.
2. **Problem (c)** involves determining the distribution of a dessert among students.
3. **Problem (d)** touches upon the distribution of a beverage among students.

Students are asked to evaluate each statement and choose the correct one based on the logical structure provided. This exercise helps in understanding logical propositions and quantifiers.

### Analysis of Students' Food Choices

To determine the truthfulness of various statements related to students' food choices, we analyze the information provided in the figure.

**Figure Description**
The figure illustrates a list of students and their selections from various food categories: Salads, Main courses, Desserts, and Beverages.

- **Salads:** green salad, fruit salad
- **Main courses:** spaghetti, fish
- **Desserts:** pie, cake
- **Beverages:** milk, soda, coffee

**Student Selections:**
- **Uta:** green salad, fish, pie, soda
- **Tim:** fish, fruit salad, pie, cake
- **Yuen:** spaghetti, pie, coffee

**Questions Analysis**
Using the information, we address the statement:

**(a) $\forall$ student $S$, $\exists$ a dessert $D$ such that $S$ chose $D$.**

**Options:**
1. **True. Every student chose at least one dessert: Uta chose pie, Tim chose both pie and cake, and Yuen chose pie.**
2. **True. Although some students didn’t choose dessert, there existed an option for them to choose from.**
3. **False. Uta did not have dessert.**
4. **False. Tim had more than one dessert.**
5. **False. None of the students chose dessert.**

**Determining the Correct Answer:**

Based on the figure:
- **Uta** chose pie.
- **Tim** chose both pie and cake.
- **Yuen** chose pie.

Thus, each student selected at least one dessert.

**Correct Answer:**
1. **True. Every student chose at least one dessert: Uta chose pie, Tim chose both pie and cake, and Yuen chose pie.**

This content demonstrates how to analyze graphical data to verify the accuracy of different statements regarding the selections of individuals.

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