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

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 they are true or false.

### Step-by-Step Solution:

1. **Statement (i): \( \phi \in A \)**
- **Analysis**: The symbol \( \phi \) represents the empty set. The set \( A \) contains three elements: \( \{1, 2, 3\} \), \( \{4, 5\} \), and \( \{6, 7, 8\} \). None of these elements is the empty set.
- **Conclusion**: This statement is **False**.

2. **Statement (ii): \( \phi \subset A \)**
- **Analysis**: The empty set \( \phi \) is a subset of every set, including \( A \).
- **Conclusion**: This statement is **True**.

3. **Statement (iii): \( 1 \in A \)**
- **Analysis**: The number \( 1 \) is not an element of \( A \). The elements of \( A \) are the sets \( \{1, 2, 3\} \), \( \{4, 5\} \), and \( \{6, 7, 8\} \), not the individual numbers.
- **Conclusion**: This statement is **False**.

4. **Statement (iv): \( \{1, 2, 3\} \subset A \)**
- **Analysis**: The set \( \{1, 2, 3\} \) is an element of \( A \), but it is not a subset of \( A \). A subset would require every element of \( \{1, 2, 3\} \) to be an element of \( A \), which is not the case.
- **Conclusion**: This statement is **False**.

5. **Statement (v): \( \{6, 7, 8\} \in A \)**
- **Analysis**: The set \( \{6, 7, 8\} \) is indeed one of the elements of \( A \).
- **Conclusion**: This statement is **True**.

6. **Statement (vi): \( \{\{4, 5\}\} \subset A \)**
- **Analysis**: The set \( \{4, 5\} \) is an element of \( A \), and therefore, the set containing \( \{4, 5\} \) as an element, i.e., \( \{\{4, 5\}\} \), is a subset of \( A \).
- **Conclusion**: This statement is **True**.

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