To solve the problem, we need to determine whether the given set \( P \times P \times P = \{(1,1,1), (2,2,2), (1,2,2), (2,1,1)\} \) is correct or not.
### Step-by-step Solution:
1. **Identify the Set \( P \)**:
- We are given \( P = \{1, 2\} \).
2. **Calculate \( P \times P \)**:
- The Cartesian product \( P \times P \) consists of all ordered pairs formed by taking one element from \( P \) and pairing it with every element in \( P \).
- Therefore, \( P \times P = \{(1,1), (1,2), (2,1), (2,2)\} \).
3. **Calculate \( P \times P \times P \)**:
- Now, we need to compute \( P \times P \times P \), which is the Cartesian product of \( P \times P \) with \( P \).
- This means we will take each element from \( P \times P \) and pair it with every element in \( P \):
- From \( (1,1) \): \( (1,1,1), (1,1,2) \)
- From \( (1,2) \): \( (1,2,1), (1,2,2) \)
- From \( (2,1) \): \( (2,1,1), (2,1,2) \)
- From \( (2,2) \): \( (2,2,1), (2,2,2) \)
4. **List All Elements of \( P \times P \times P \)**:
- Combining all these, we get:
- \( (1,1,1) \)
- \( (1,1,2) \)
- \( (1,2,1) \)
- \( (1,2,2) \)
- \( (2,1,1) \)
- \( (2,1,2) \)
- \( (2,2,1) \)
- \( (2,2,2) \)
5. **Count the Total Elements**:
- There are a total of 8 elements in \( P \times P \times P \).
6. **Compare with the Given Set**:
- The provided set \( \{(1,1,1), (2,2,2), (1,2,2), (2,1,1)\} \) contains only 4 elements.
- Since \( P \times P \times P \) should have 8 elements, the statement is false.
### Conclusion:
The statement that \( P \times P \times P = \{(1,1,1), (2,2,2), (1,2,2), (2,1,1)\} \) is **false**.
