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The new image contains the same set theory problem as before:

\[
A - B = A \cap B^c = B^c - A^c
\]

Let’s go through the proof step by step once again.

### 1. **Proving \( A - B = A \cap B^c \)**

- **Definition of \( A - B \)**: The set difference \( A - B \) is defined as the set of elements that are in \( A \) but not in \( B \). Symbolically:
  \[
  A - B = \{x \mid x \in A \text{ and } x \notin B\}.
  \]
  
- **Definition of \( A \cap B^c \)**: The complement of \( B \), denoted \( B^c \), consists of all elements not in \( B \). The intersection \( A \cap B^c \) represents the set of elements that are both in \( A \) and not in \( B \):
  \[
  A \cap B^c = \{x \mid x \in A \text{ and } x \notin B\}.
  \]

Both expressions define the same set of elements, meaning:
\[
A - B = A \cap B^c.
\]

### 2. **Proving \( A \cap B^c = B^c - A^c \)**

- **Definition of \( B^c - A^c \)**: The set \( B^c - A^c \) consists of elements in \( B^c \) that are not in \( A^c \). Since \( A^c \) is the complement of \( A \), this translates to elements that are in \( B^c \) and in \( A \) (because \( x \notin A^c \) implies \( x \in A \)). Symbolically:
  \[
  B^c - A^c = \{x \mid x \in B^c \text{ and } x \notin A^c\} = \{x \mid x \in B^c \text{ and } x \in A\}.
  \]

This is precisely the definition of \( A \cap B^c \). Hence:
\[
A \cap B^c = B^c - A^c.
\]

### Conclusion:

The full equality is proven as follows:
\[
A - B = A \cap B^c = B^c - A^c.
\]

This completes the proof.

Would you like any further clarification on any step?

---

Here are five related questions to explore:
1. How does the proof change if we consider the union of sets instead of intersections and differences?
2. Can you prove the identity \( (A \cup B)^c = A^c \cap B^c \)?
3. What would happen if the sets \( A \) and \( B \) were disjoint? Does the equality still hold?
4. Can you represent the equality \( A - B = A \cap B^c = B^c - A^c \) using a Venn diagram?
5. How do De Morgan’s laws assist in simplifying complex set expressions?

**Tip**: When working with set identities, try applying specific examples (like finite sets of numbers) to see the behavior and verify results step by step!

Prove that A - B = A ∩ B^c = B^c - A^c.

Learn how to prove the set identity A - B = A ∩ B^c = B^c - A^c using set difference, intersection, and complement rules. This detailed solution is suitable for advanced high school students or undergraduates studying set theory.

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