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We are given that the set \( A = \{1, \{\emptyset\}, \{1, \emptyset\}\} \). Now, let's evaluate each of the statements:

1. **Statement (a)**: \( \{\emptyset\}, \{1, \emptyset\} \in A \)
   - \( A \) contains the element \( \{\emptyset\} \) and \( \{1, \emptyset\} \). 
   - Hence, both \( \{\emptyset\} \) and \( \{1, \emptyset\} \) are elements of \( A \), making statement (a) **true**.

2. **Statement (b)**: \( \{\emptyset\}, \{1, \emptyset\} \subseteq A \)
   - For this to be true, each element of the set \( \{\emptyset\}, \{1, \emptyset\} \) must be an element of \( A \).
   - Both \( \{\emptyset\} \) and \( \{1, \emptyset\} \) are elements of \( A \), so this is **true**.

3. **Statement (c)**: \( \{\{\emptyset\}, \{1, \emptyset\}\} \subseteq A \)
   - This statement claims that the set \( \{\{\emptyset\}, \{1, \emptyset\}\} \) is a subset of \( A \).
   - For it to be true, each element of \( \{\{\emptyset\}, \{1, \emptyset\}\} \) must be an element of \( A \).
   - Both \( \{\emptyset\} \) and \( \{1, \emptyset\} \) are elements of \( A \), so this is **true**.

4. **Statement (d)**: \( \{1, \emptyset\} \subseteq A \)
   - The set \( \{1, \emptyset\} \) is a subset of \( A \) if both 1 and \( \emptyset \) are elements of \( A \).
   - The element 1 is in \( A \), but \( \emptyset \) is not an element of \( A \); hence, this statement is **false**.

5. **Statement (e)**: \( \{\{1\}, \{\emptyset\}\} \subseteq A \)
   - This statement claims that the set \( \{\{1\}, \{\emptyset\}\} \) is a subset of \( A \).
   - The set \( \{1\} \) is not an element of \( A \), but \( \{\emptyset\} \) is. Therefore, this statement is **false**.

### Conclusion:
The correct statements are (a), (b), and (c), so the correct answer is:

**D) a and c**

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Would you like more details on this solution or any further clarification?

Here are 5 related questions for further practice:
1. What is the difference between an element of a set and a subset?
2. How would the answer change if \( \emptyset \) were an element of \( A \)?
3. What does it mean for a set to be a subset of another set?
4. How would you represent the power set of \( A \)?
5. Can a set contain both elements and subsets simultaneously?

**Tip:** When evaluating membership (\( \in \)) and subset (\( \subseteq \)) relationships, always carefully distinguish between sets and their elements.

Let A = {1, {∅}, {1, ∅}}. Consider the following statements: a) {∅}, {1, ∅} ∈ A b) {∅}, {1, ∅} ⊆ A c) {{∅}, {1, ∅}} ⊆ A d) {1, ∅} ⊆ A e) {{1}, {∅}} ⊆ A Which of the following are correct?

Explore the concepts of set membership and subset relations with the problem A = {1, {∅}, {1, ∅}}. Learn which of the given statements about set membership and subsets are correct with step-by-step reasoning. Suitable for advanced high school students.

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