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The image contains a mathematical statement that needs to be proven. The statement is:

"Prove that \(\overline{A \cup B} = \overline{A} \cap \overline{B}\)."

This is a well-known identity in set theory called **De Morgan's Law**.

### Proof:

We need to prove that:
\[
\overline{A \cup B} = \overline{A} \cap \overline{B}
\]
where:
- \( A \) and \( B \) are any two sets.
- \( \overline{A} \) denotes the complement of \( A \), i.e., the set of all elements that are not in \( A \).
- \( A \cup B \) is the union of sets \( A \) and \( B \), meaning all elements that are in \( A \) or \( B \) or both.

#### 1. Left-Hand Side (\(\overline{A \cup B}\)):

The complement of the union \( A \cup B \) consists of all elements that are **not** in \( A \cup B \). This means:
\[
x \in \overline{A \cup B} \implies x \notin A \cup B
\]
So, \( x \) is **not** in \( A \) and is **not** in \( B \). This can be written as:
\[
x \notin A \text{ and } x \notin B
\]
Thus:
\[
x \in \overline{A} \text{ and } x \in \overline{B}
\]
Therefore:
\[
x \in \overline{A} \cap \overline{B}
\]
So, we have shown that:
\[
\overline{A \cup B} \subseteq \overline{A} \cap \overline{B}
\]

#### 2. Right-Hand Side (\(\overline{A} \cap \overline{B}\)):

Now, consider the right-hand side, which is \( \overline{A} \cap \overline{B} \).

An element \( x \) that belongs to \( \overline{A} \cap \overline{B} \) must be in both \( \overline{A} \) and \( \overline{B} \). This means:
\[
x \in \overline{A} \text{ and } x \in \overline{B}
\]
Which implies:
\[
x \notin A \text{ and } x \notin B
\]
Therefore, \( x \notin A \cup B \), which means:
\[
x \in \overline{A \cup B}
\]
So, we have shown that:
\[
\overline{A} \cap \overline{B} \subseteq \overline{A \cup B}
\]

#### Conclusion:

Since both \( \overline{A \cup B} \subseteq \overline{A} \cap \overline{B} \) and \( \overline{A} \cap \overline{B} \subseteq \overline{A \cup B} \), we conclude that:
\[
\overline{A \cup B} = \overline{A} \cap \overline{B}
\]

This completes the proof.

---

Would you like any further details or have any questions?

Here are some related questions to explore:

1. How do De Morgan's laws extend to more than two sets?
2. Can De Morgan's laws be applied to logical statements, and if so, how?
3. What is the significance of set complements in probability theory?
4. How does the intersection operation differ from the union in set theory?
5. Can you provide an example where De Morgan's laws simplify a set expression?

**Tip:** De Morgan's laws are fundamental in both set theory and logic, helping simplify complex expressions involving unions, intersections, and complements.

Prove that \(\overline{A \cup B} = \overline{A} \cap \overline{B}\).

Explore the proof of De Morgan's Law in set theory, demonstrating the equality \(\overline{A \cup B} = \overline{A} \cap \overline{B}\). This step-by-step solution is designed for students in Grades 11-12, covering key concepts in set theory.

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