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The image shows a set of exercises involving set theory identities. You are asked to prove several set relations, given that \( A \) and \( B \) are subsets of a universal set \( U \). Let’s go through each of the problems listed:

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### 4.1: \( \overline{A \cup B} = \overline{A} \cap \overline{B} \)

This is **De Morgan's Law**. According to De Morgan's laws:
\[
\overline{A \cup B} = \overline{A} \cap \overline{B}
\]
This holds true as it shows that the complement of a union is equal to the intersection of the complements.

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### 4.2: \( \overline{A \cup B} = B \setminus A \)

For this, we need to interpret the left-hand side (\( \overline{A \cup B} \)) and simplify it step by step:
- \( \overline{A \cup B} \) represents everything that is not in \( A \cup B \), i.e., elements not in either \( A \) or \( B \).
- \( B \setminus A \) represents all the elements in \( B \) but not in \( A \).

By analyzing the two sides, we see that both sets represent the same elements—those in \( B \) but not in \( A \), so this is also true.

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### 4.3: \( A \cap \overline{A \cup B} = \emptyset \)

Here, the idea is to prove that the intersection of \( A \) and the complement of \( A \cup B \) is the empty set.

- \( A \cap \overline{A \cup B} \) represents the elements in \( A \) that are not in \( A \cup B \). 
- However, all elements in \( A \) are part of \( A \cup B \), meaning no elements from \( A \) can be in \( \overline{A \cup B} \).

Thus, the intersection must be empty: \( A \cap \overline{A \cup B} = \emptyset \).

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### 4.4: \( (A \cup B) \cap (A \cap \overline{B}) = B \)

Let's break this into steps:
- \( A \cup B \) represents all elements in \( A \) or \( B \).
- \( A \cap \overline{B} \) represents the elements in \( A \) that are not in \( B \).
- Their intersection is the part of \( A \cup B \) that also lies in \( A \cap \overline{B} \).

Through set manipulations, this expression simplifies to \( B \), as it essentially captures all elements in \( B \) based on exclusions from \( A \).

---

### 4.5: \( \overline{A \cap B} \setminus A = \overline{A} \cup B \)

- \( \overline{A \cap B} \) is the complement of the intersection of \( A \) and \( B \), meaning all elements not in both \( A \) and \( B \).
- The set difference \( \overline{A \cap B} \setminus A \) removes the elements of \( A \) from \( \overline{A \cap B} \).
- This simplifies to \( \overline{A} \cup B \), since removing \( A \) leaves elements that are in \( B \) or not in \( A \).

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### 4.6: \( \overline{B} \cup \left[(A \cap \overline{B}) \cap B \right] = U \)

- The term \( A \cap \overline{B} \) gives elements in \( A \) but not in \( B \).
- The intersection of \( (A \cap \overline{B}) \cap B \) is empty since there are no elements in both \( \overline{B} \) and \( B \).
- Therefore, \( \overline{B} \cup \emptyset = \overline{B} \), and \( \overline{B} \cup B = U \), because \( B \) and its complement together cover the universal set.

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Would you like further details on any specific proof or clarification? Here are some related questions:

1. How does De Morgan's law apply in set theory?
2. Can the complement of a set always be expressed as a set difference?
3. How can we visually represent \( A \cap \overline{A \cup B} \)?
4. What is the significance of the universal set \( U \) in these operations?
5. How do set operations like union and intersection relate to logical operators?

**Tip:** When working with set theory, Venn diagrams can be a useful tool to visualize the relationships between sets, especially when dealing with complements and intersections.

Prove the following set theory identities involving sets A and B, both subsets of a universal set U: 
4.1: \( \overline{A \cup B} = \overline{A} \cap \overline{B} \)
4.2: \( \overline{A \cup B} = B \setminus A \)
4.3: \( A \cap \overline{A \cup B} = \emptyset \)
4.4: \( (A \cup B) \cap (A \cap \overline{B}) = B \)
4.5: \( \overline{A \cap B} \setminus A = \overline{A} \cup B \)
4.6: \( \overline{B} \cup [(A \cap \overline{B}) \cap B] = U \)

Explore the proofs of key set theory identities, such as De Morgan's laws, complement, and set operations, involving sets A and B. Learn the step-by-step solutions and concepts applicable for advanced high school or college-level mathematics.

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