If A and B are two sets such that `A xx B = phi`, then which of the following must be true ?

Options

A. `A = phi` and `B != phi`
B. `A = phi` and/or `B = phi`
C. `A != phi` and `B = phi`
D. `A = phi` and `B = phi`

Answer

To solve the problem, we need to analyze the condition given: \( A \times B = \phi \), where \( A \) and \( B \) are two sets, and \( \phi \) represents the empty set.

### Step-by-Step Solution:

1. **Understanding the Cartesian Product**: The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \). If \( A \times B = \phi \), it means there are no such pairs.

2. **Condition for the Empty Set**: The Cartesian product \( A \times B \) is empty if at least one of the sets \( A \) or \( B \) is empty. This is because:
- If \( A \) is empty (\( A = \phi \)), then there are no elements \( a \) to pair with any element \( b \) in \( B \).
- If \( B \) is empty (\( B = \phi \)), then there are no elements \( b \) to pair with any element \( a \) in \( A \).

3. **Conclusion**: Therefore, for \( A \times B \) to be empty, it must be true that either \( A \) is empty, \( B \) is empty, or both are empty. However, the strongest conclusion we can draw is that at least one of the sets must be empty.

4. **Final Statement**: Thus, we conclude that if \( A \times B = \phi \), then at least one of the sets \( A \) or \( B \) must be the empty set.

### Answer:
The correct conclusion is that either \( A = \phi \) or \( B = \phi \) (or both).

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