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The image presents a mathematical proposition and proof about set theory. Here's a transcription and explanation:

### Proposition:
**"If \( A \), \( B \), and \( C \) are sets, then:**
\[
A \setminus (B \setminus C) = (A \setminus B) \cup (A \cap C)
\]**

### Proof:
**"Note that:**
\[
x \in A \setminus (B \setminus C) \iff x \in (A \setminus B) \cup (A \cap C)
\]

**Detailed steps:**
1. \( x \in A \setminus (B \setminus C) \)
2. \( \iff (x \in A \text{ and } x \notin (B \setminus C)) \)
3. \( \iff (x \in A \text{ and } (x \notin B \text{ or } x \in C)) \)
4. \( \iff (x \in A \text{ and } x \notin B) \text{ or } (x \in A \text{ and } x \in C) \)
5. \( \iff (x \in (A \setminus B)) \text{ or } (x \in (A \cap C)) \)
6. \( \iff x \in (A \setminus B) \cup (A \cap C) \)

Thus, we conclude:
\[
A \setminus (B \setminus C) = (A \setminus B) \cup (A \cap C)
\]**

This proof uses basic set-theoretic logic, breaking down the difference between two sets \( B \setminus C \) and applying union and intersection rules.

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Do you need more details on the proof or a clarification on any steps? 

Here are five related questions to consider:
1. What are the formal definitions of set difference and intersection?
2. How does De Morgan’s Law relate to this type of set operation?
3. Can we generalize this identity to more than three sets?
4. How do we visualize set operations using Venn diagrams?
5. What role does distributivity play in set theory?

**Tip**: In set theory, always carefully analyze each component of set operations like \( \setminus \), \( \cup \), and \( \cap \) to avoid mistakes in proofs or operations.

If A, B, and C are sets, prove that A\(B\C) = (A\B) ∪ (A ∩ C).

Learn how to prove that A\(B\C) = (A\B) ∪ (A ∩ C) using basic set theory operations like difference, union, and intersection. This proof breaks down key set-theoretic logic with step-by-step details, suitable for university-level mathematics.

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