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The image contains an exercise about set theory involving Cartesian products of sets. The problem is from *Exercise 2.5.10*, which asks to show that the following identities hold for three sets \(A\), \(B\), and \(C\):

1. \( A \times (B \cup C) = (A \times B) \cup (A \times C) \)
2. \( A \times (B \cap C) = (A \times B) \cap (A \times C) \)
3. \( A \times (B \setminus C) = (A \times B) \setminus (A \times C) \)

### Explanation of the approach:

1. **First identity**: 
   - The left-hand side represents the Cartesian product of set \(A\) with the union of sets \(B\) and \(C\). The union operation means taking all elements from both \(B\) and \(C\).
   - The right-hand side is the union of two Cartesian products: \(A \times B\) and \(A \times C\), which means taking pairs from both.

   **Proof sketch**: 
   An element of \(A \times (B \cup C)\) is of the form \((a, x)\), where \(a \in A\) and \(x \in B \cup C\). Thus, \(x\) must be in either \(B\) or \(C\), so \((a, x)\) belongs to either \(A \times B\) or \(A \times C\), proving equality.

2. **Second identity**: 
   - The left-hand side represents the Cartesian product of set \(A\) with the intersection of \(B\) and \(C\).
   - The right-hand side is the intersection of two Cartesian products: \(A \times B\) and \(A \times C\), meaning pairs that belong to both products.

   **Proof sketch**: 
   An element of \(A \times (B \cap C)\) is of the form \((a, x)\), where \(x \in B \cap C\). Therefore, \((a, x)\) must be in both \(A \times B\) and \(A \times C\), proving equality.

3. **Third identity**: 
   - The left-hand side is the Cartesian product of set \(A\) with the difference \(B \setminus C\).
   - The right-hand side is the difference between \(A \times B\) and \(A \times C\).

   **Proof sketch**: 
   An element of \(A \times (B \setminus C)\) is of the form \((a, x)\), where \(x \in B\) and \(x \notin C\). Thus, \((a, x)\) belongs to \(A \times B\) but not \(A \times C\), proving equality.

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Do you need more details or have any questions about these proofs?

Here are 5 related questions to think about:
1. How does the Cartesian product interact with other set operations like symmetric difference?
2. Can these properties be generalized to more than three sets?
3. What happens when the sets \(A\), \(B\), or \(C\) are empty?
4. How would these identities change if the operation were the power set instead of Cartesian product?
5. Can you think of real-world examples where Cartesian products are useful?

**Tip**: When working with Cartesian products, it's useful to think of them as forming "ordered pairs" that combine elements from each set, and visualize how the operations affect the pairing.

Let A, B, and C be three sets. Show that the following hold:
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
(iii) A × (B \ C) = (A × B) \ (A × C)

Explore the distributive properties of Cartesian products over set operations in this detailed problem. Learn how to prove that A × (B ∪ C) = (A × B) ∪ (A × C), A × (B ∩ C) = (A × B) ∩ (A × C), and A × (B \ C) = (A × B) \ (A × C). Suitable for advanced high school and undergraduate students studying set theory.

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