Let A= {{1,2,3},{4,5},{6,7,8}}. Determine which of the following is true or false :
(i) `1inA`
`{1,2,3}subA`
(iii) `{6,7,8}inA`
(iv) `{{4,5}}subA`
(v) `phiinA`
(vi) `phisubA`.

Answer

To solve the problem, we need to analyze each statement regarding the set \( A = \{ \{1, 2, 3\}, \{4, 5\}, \{6, 7, 8\} \} \) and determine whether each statement is true or false.

### Step-by-step Solution:

1. **Statement (i): \( 1 \in A \)**
- **Analysis:** The statement claims that the element \( 1 \) is a member of the set \( A \). However, the elements of \( A \) are the sets \( \{1, 2, 3\} \), \( \{4, 5\} \), and \( \{6, 7, 8\} \). The number \( 1 \) is not an element of \( A \) itself; it is part of the subset \( \{1, 2, 3\} \).
- **Conclusion:** This statement is **False**.

2. **Statement (ii): \( \{1, 2, 3\} \subset A \)**
- **Analysis:** This statement claims that the set \( \{1, 2, 3\} \) is a subset of \( A \). A subset means that all elements of the subset must be elements of the larger set. However, \( \{1, 2, 3\} \) is an element of \( A \), not a subset. For it to be a subset, it would need to be written as \( \{\{1, 2, 3\}\} \).
- **Conclusion:** This statement is **False**.

3. **Statement (iii): \( \{6, 7, 8\} \in A \)**
- **Analysis:** The statement claims that the set \( \{6, 7, 8\} \) is an element of \( A \). Looking at the elements of \( A \), we see that \( \{6, 7, 8\} \) is indeed one of the elements.
- **Conclusion:** This statement is **True**.

4. **Statement (iv): \( \{\{4, 5\}\} \subset A \)**
- **Analysis:** This statement claims that the set containing the set \( \{4, 5\} \) is a subset of \( A \). Since \( \{4, 5\} \) is an element of \( A \), the set \( \{\{4, 5\}\} \) is not a subset of \( A \) because it is not included as a single element.
- **Conclusion:** This statement is **False**.

5. **Statement (v): \( \emptyset \in A \)**
- **Analysis:** This statement claims that the empty set \( \emptyset \) is an element of \( A \). Since \( A \) contains the sets \( \{1, 2, 3\} \), \( \{4, 5\} \), and \( \{6, 7, 8\} \), and does not contain \( \emptyset \), this statement is false.
- **Conclusion:** This statement is **False**.

6. **Statement (vi): \( \emptyset \subset A \)**
- **Analysis:** This statement claims that the empty set \( \emptyset \) is a subset of \( A \). By definition, the empty set is a subset of every set, including \( A \).
- **Conclusion:** This statement is **True**.

### Final Answers:
- (i) False
- (ii) False
- (iii) True
- (iv) False
- (v) False
- (vi) True