If `P={m,n)` and `Q={n,m}`, then `PxxQ={(m,n),(n,m)}`

Answer

To determine whether the statement "If \( P = \{m, n\} \) and \( Q = \{n, m\} \), then \( P \times Q = \{(m, n), (n, m)\} \)" is true or false, we need to find the Cartesian product \( P \times Q \).

### Step-by-step Solution:

1. **Identify the Sets**:
- We have \( P = \{m, n\} \) and \( Q = \{n, m\} \).

2. **Definition of Cartesian Product**:
- The Cartesian product \( P \times Q \) is defined as the set of all ordered pairs \( (a, b) \) where \( a \) is an element of \( P \) and \( b \) is an element of \( Q \).

3. **List the Ordered Pairs**:
- We will create ordered pairs by taking each element from \( P \) and pairing it with each element from \( Q \):
- Pair \( (m, n) \): where \( m \) is from \( P \) and \( n \) is from \( Q \).
- Pair \( (m, m) \): where \( m \) is from \( P \) and \( m \) is from \( Q \).
- Pair \( (n, n) \): where \( n \) is from \( P \) and \( n \) is from \( Q \).
- Pair \( (n, m) \): where \( n \) is from \( P \) and \( m \) is from \( Q \).

4. **Write the Cartesian Product**:
- Therefore, the Cartesian product \( P \times Q \) is:
\[ P \times Q = \{(m, n), (m, m), (n, n), (n, m)\} \]

5. **Compare with Given Statement**:
- The statement claims that \( P \times Q = \{(m, n), (n, m)\} \).
- However, we found \( P \times Q = \{(m, n), (m, m), (n, n), (n, m)\} \), which includes additional pairs.

6. **Conclusion**:
- Since \( P \times Q \) contains more elements than stated in the question, the statement is **false**.