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.

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.