To solve the question, we need to analyze each statement regarding the set \( A = \{1, 2, \{3, 4\}, 5\} \) and determine whether each statement is true or false based on the definitions of "belongs to" (∈) and "subset" (⊆).
### Step-by-Step Solution:
1. **Statement (i): \( \{3, 4\} \subset A \)**
- **Analysis**: The statement claims that the set \( \{3, 4\} \) is a subset of \( A \). For \( \{3, 4\} \) to be a subset of \( A \), both 3 and 4 must be elements of \( A \). However, \( A \) contains \( \{3, 4\} \) as a single element, not as individual elements.
- **Conclusion**: **False**
2. **Statement (ii): \( \{3, 4\} \in A \)**
- **Analysis**: This statement claims that \( \{3, 4\} \) is an element of \( A \). Since \( A \) contains \( \{3, 4\} \) as one of its elements, this statement is true.
- **Conclusion**: **True**
3. **Statement (iii): \( \{\{3, 4\}\} \subset A \)**
- **Analysis**: The statement claims that the set containing the single element \( \{3, 4\} \) is a subset of \( A \). Since \( \{3, 4\} \) is an element of \( A \), this statement is true.
- **Conclusion**: **True**
4. **Statement (iv): \( 1 \in A \)**
- **Analysis**: This statement claims that 1 is an element of \( A \). Since 1 is explicitly listed in \( A \), this statement is true.
- **Conclusion**: **True**
5. **Statement (v): \( 1 \subset A \)**
- **Analysis**: This statement claims that 1 is a subset of \( A \). However, 1 is not a set; it is an individual element. Therefore, it cannot be a subset.
- **Conclusion**: **False**
6. **Statement (vi): \( \{1, 2\} \subset A \)**
- **Analysis**: This statement claims that the set \( \{1, 2\} \) is a subset of \( A \). Since both 1 and 2 are elements of \( A \), this statement is true.
- **Conclusion**: **True**
7. **Statement (vii): \( \{1, 2, 5\} \in A \)**
- **Analysis**: This statement claims that the set \( \{1, 2, 5\} \) is an element of \( A \). However, \( A \) does not contain this set as a single element. Therefore, this statement is false.
- **Conclusion**: **False**
8. **Statement (viii): \( \{1, 2, 3\} \subset A \)**
- **Analysis**: This statement claims that \( \{1, 2, 3\} \) is a subset of \( A \). Since 3 is not an element of \( A \) (it is part of the set \( \{3, 4\} \)), this statement is false.
- **Conclusion**: **False**
9. **Statement (ix): \( \emptyset \in A \)**
- **Analysis**: This statement claims that the empty set is an element of \( A \). Since \( A \) does not contain the empty set as an element, this statement is false.
- **Conclusion**: **False**
10. **Statement (x): \( \emptyset \subset A \)**
- **Analysis**: This statement claims that the empty set is a subset of \( A \). The empty set is a subset of every set, so this statement is true.
- **Conclusion**: **True**
11. **Statement (xi): \( \{\emptyset\} \subset A \)**
- **Analysis**: This statement claims that the set containing the empty set is a subset of \( A \). Since \( \emptyset \) is not an element of \( A \), this statement is false.
- **Conclusion**: **False**
### Summary of Conclusions:
- (i) False
- (ii) True
- (iii) True
- (iv) True
- (v) False
- (vi) True
- (vii) False
- (viii) False
- (ix) False
- (x) True
- (xi) False
