https://file.aiquickdraw.com/mathbot/file/1726527035746j6l6lvz3.jpg

Let's analyze the question step by step, focusing on the relationships between the sets \( A \), \( B \), \( C \), \( D \), and \( E \), and then verify which of the statements (I, II, III) are true.

### Given Relations:
1. **All elements of \( A \) are in \( B \)**: \( A \subseteq B \).
2. **All elements of \( B \) are in \( C \)**: \( B \subseteq C \), meaning \( A \subseteq B \subseteq C \).
3. **Some, but not all, elements of \( D \) are in \( A \)**: \( D \cap A \neq \emptyset \), but \( D \nsubseteq A \).
4. **All elements of \( D \) are in \( B \)**: \( D \subseteq B \).
5. **No elements of \( E \) are in \( C \)**: \( E \cap C = \emptyset \).

### Now, let's analyze the statements:

#### Statement I: 
**All elements of \( A \) are in \( C \).**
- From the given, \( A \subseteq B \subseteq C \). This means \( A \subseteq C \) is true.
- **Conclusion: Statement I is true**.

#### Statement II:
**Some elements of \( E \) are in \( B \).**
- We know that \( E \cap C = \emptyset \), and since \( B \subseteq C \), this implies that \( E \cap B = \emptyset \) (no elements of \( E \) are in \( B \)).
- **Conclusion: Statement II is false**.

#### Statement III:
**No elements of \( B \) are in \( E \).**
- Since \( E \cap C = \emptyset \) and \( B \subseteq C \), this implies \( B \cap E = \emptyset \). Therefore, no elements of \( B \) are in \( E \).
- **Conclusion: Statement III is true**.

### Correct Answer:
- The true statements are I and III.
- Thus, the correct alternative is **c. I and III**.

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Do you have any questions or need further clarification?

Here are 5 related questions to explore further:
1. If \( A \subseteq B \) and \( B \subseteq C \), can we infer \( A \subseteq C \) in all cases?
2. What is the difference between \( A \subseteq B \) and \( A \cap B \neq \emptyset \)?
3. Can two sets have no common elements but still both be subsets of a third set?
4. How do set operations like union and intersection change when we know \( A \subseteq B \)?
5. What happens to the intersection of sets when one set is completely disjoint from another?

**Tip**: When solving set theory problems, drawing a Venn diagram can be a helpful way to visualize relationships between the sets.

Given the following relations between nonempty sets: All elements of A are in B. All elements of B are in C. Some, but not all, elements of D are in A. All elements of D are in B. No elements of E are in C. Which of the following statements are TRUE? I = All elements of A are in C. II = Some elements of E are in B. III = No elements of B are in E.

Explore the relationships between sets A, B, C, D, and E, focusing on subset transitivity and element memberships. This analysis helps clarify which statements about these sets are true, enhancing comprehension of fundamental set theory concepts suitable for college-level students.

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