To determine whether the statement \( P \times P \times P = \{(1,1,1), (2,2,2), (1,2,2), (2,1,1)\} \) is true or false, we will follow these steps:
### Step 1: Identify the Set \( P \)
The set \( P \) is given as:
\[
P = \{1, 2\}
\]
### Step 2: Calculate the Cartesian Product \( P \times P \times P \)
The Cartesian product \( P \times P \times P \) consists of all possible ordered triples where each element of the triple is taken from the set \( P \).
Since \( P \) has 2 elements, the number of elements in \( P \times P \times P \) is calculated as:
\[
|P| \times |P| \times |P| = 2 \times 2 \times 2 = 8
\]
### Step 3: List All Possible Ordered Triples
Now, we will list all the ordered triples formed by taking each element from \( P \) for each position in the triple:
1. \( (1, 1, 1) \)
2. \( (1, 1, 2) \)
3. \( (1, 2, 1) \)
4. \( (1, 2, 2) \)
5. \( (2, 1, 1) \)
6. \( (2, 1, 2) \)
7. \( (2, 2, 1) \)
8. \( (2, 2, 2) \)
Thus, the complete set \( P \times P \times P \) is:
\[
P \times P \times P = \{(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2)\}
\]
### Step 4: Compare with the Given Set
The given set is:
\[
\{(1,1,1), (2,2,2), (1,2,2), (2,1,1)\}
\]
### Step 5: Conclusion
Since the complete set \( P \times P \times P \) contains 8 elements and is different from the given set, we conclude that the statement is false.
### Final Answer
The statement \( P \times P \times P = \{(1,1,1), (2,2,2), (1,2,2), (2,1,1)\} \) is **false**.
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