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The exercise you're asking about appears to be a problem on set theory or Cartesian products. It asks:

**Exercise 2.5.12**:
"Under what conditions does \( A \times B = C \times D \) imply that \( A = C \) and \( B = D \)?"

### Explanation:
This question is about the equality of Cartesian products of two sets. The Cartesian product of sets \( A \times B \) is the set of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

To determine when \( A \times B = C \times D \), we need to analyze the structure of Cartesian products. The main condition required for this equality to imply \( A = C \) and \( B = D \) is when the sets are non-empty. If \( A \times B = C \times D \) holds and both products are non-empty, it can be shown that:

- \( A = C \)
- \( B = D \)

This is because the elements of the product are ordered pairs, and the first components of the pairs must match in order for the products to be equal. Similarly, the second components must match as well.

However, if one or both of the sets are empty, the situation becomes more complicated. For instance:
- \( A \times B = \emptyset \) when either \( A = \emptyset \) or \( B = \emptyset \).
- In this case, it's possible for \( A \times B = C \times D \) without \( A = C \) or \( B = D \).

Thus, the equality \( A \times B = C \times D \) **implies** \( A = C \) and \( B = D \) **if and only if neither of the sets involved is empty**.

Would you like more details on this explanation or related concepts? Here are a few questions to think about:

1. How does the Cartesian product change when one or both sets are empty?
2. Can you give an example where \( A \times B = C \times D \) but \( A \neq C \) or \( B \neq D \)?
3. How would the solution differ if the sets involved are finite vs infinite?
4. What role does the order of elements in ordered pairs play in Cartesian product equality?
5. How can we formally prove that \( A = C \) and \( B = D \) if \( A \times B = C \times D \)?

**Tip**: When working with Cartesian products, always check whether sets are empty as this can influence the equality of products significantly.

Under what conditions does A × B = C × D imply that A = C and B = D?

Explore the conditions under which A × B = C × D implies that A = C and B = D. This problem covers key concepts in set theory and Cartesian products, including the role of non-empty sets and ordered pairs. Suitable for undergraduate students studying set theory or discrete mathematics.

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