Find whether the following statements are true or false. If the statement is false, then write its correct statement:
(i) If P={m,n} and Q={n,m},then `PxxQ={(m,n),(n,m)}`.
(ii) If A and B are non-empty sets, then `AxxB` is a non-empty set of the ordered pairs (x,y) such that `x in A and y in B`.

Explanation

To solve the question, we will analyze each statement one by one.

### Statement (i):
**Given:** If \( P = \{m, n\} \) and \( Q = \{n, m\} \), then \( P \times Q = \{(m, n), (n, m)\} \).

**Step 1:** Determine the Cartesian product \( P \times Q \).

The Cartesian product \( P \times Q \) consists of all ordered pairs \( (x, y) \) where \( x \) is from set \( P \) and \( y \) is from set \( Q \).

- The elements of \( P \) are \( m \) and \( n \).
- The elements of \( Q \) are \( n \) and \( m \).

**Step 2:** List all possible ordered pairs.

1. Pair \( (m, n) \) from \( m \in P \) and \( n \in Q \).
2. Pair \( (m, m) \) from \( m \in P \) and \( m \in Q \).
3. Pair \( (n, n) \) from \( n \in P \) and \( n \in Q \).
4. Pair \( (n, m) \) from \( n \in P \) and \( m \in Q \).

Thus, \( P \times Q = \{(m, n), (m, m), (n, n), (n, m)\} \).

**Step 3:** Compare with the given statement.

The statement claims that \( P \times Q = \{(m, n), (n, m)\} \), but we found that \( P \times Q = \{(m, n), (m, m), (n, n), (n, m)\} \).

**Conclusion for Statement (i):** The statement is **False**.

**Correct Statement:** \( P \times Q = \{(m, n), (m, m), (n, n), (n, m)\} \).

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### Statement (ii):
**Given:** If \( A \) and \( B \) are non-empty sets, then \( A \times B \) is a non-empty set of the ordered pairs \( (x, y) \) such that \( x \in A \) and \( y \in B \).

**Step 1:** Understand what it means for sets \( A \) and \( B \) to be non-empty.

If \( A \) and \( B \) are non-empty, it means there exists at least one element in each set.

**Step 2:** Formulate the Cartesian product \( A \times B \).

The Cartesian product \( A \times B \) will consist of all ordered pairs \( (x, y) \) where \( x \) is from set \( A \) and \( y \) is from set \( B \).

**Step 3:** Since both sets are non-empty, we can conclude:

- Let \( a \in A \) and \( b \in B \) (since both sets are non-empty).
- Then the pair \( (a, b) \) will be in \( A \times B \).

Thus, \( A \times B \) contains at least one ordered pair, which means \( A \times B \) is non-empty.

**Conclusion for Statement (ii):** The statement is **True**.

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### Summary of Results:
1. Statement (i) is **False**. Correct statement: \( P \times Q = \{(m, n), (m, m), (n, n), (n, m)\} \).
2. Statement (ii) is **True**.

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