Prove DeMorgan’s law: (𝐴 ∪ 𝐵)
̅̅̅̅̅̅̅̅̅̅ = 𝐴̅∩ 𝐵̅

DeMorgan's Law states that the complement of the union of two sets is equal to the intersection of their complements:

\[
\overline{A \cup B} = \overline{A} \cap \overline{B}.
\]

### Proof:

We will prove this using **set theory** and **logic equivalences**.

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#### Definitions:
1. **Set complement**: \(\overline{A} = \{x \mid x \notin A\}\).
2. **Set union**: \(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\).
3. **Set intersection**: \(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\).

To prove \(\overline{A \cup B} = \overline{A} \cap \overline{B}\), we show:
1. \(x \in \overline{A \cup B} \implies x \in \overline{A} \cap \overline{B}\),
2. \(x \in \overline{A} \cap \overline{B} \implies x \in \overline{A \cup B}\).

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### Part 1: Show \(\overline{A \cup B} \subseteq \overline{A} \cap \overline{B}\).

Assume \(x \in \overline{A \cup B}\).  
This means \(x \notin A \cup B\).  
By definition of union, \(x \notin A \cup B \implies x \notin A \text{ and } x \notin B\).

Thus:
\[
x \in \overline{A} \text{ and } x \in \overline{B}.
\]

Therefore, \(x \in \overline{A} \cap \overline{B}\).

---

### Part 2: Show \(\overline{A} \cap \overline{B} \subseteq \overline{A \cup B}\).

Assume \(x \in \overline{A} \cap \overline{B}\).  
This means \(x \in \overline{A}\) and \(x \in \overline{B}\), or equivalently:  
\(x \notin A\) and \(x \notin B\).

By definition of union:
\[
x \notin A \text{ and } x \notin B \implies x \notin A \cup B.
\]

Thus, \(x \in \overline{A \cup B}\).

---

### Conclusion:

Since we have shown both:
\[
\overline{A \cup B} \subseteq \overline{A} \cap \overline{B},
\]
and:
\[
\overline{A} \cap \overline{B} \subseteq \overline{A \cup B},
\]

we conclude:
\[
\overline{A \cup B} = \overline{A} \cap \overline{B}.
\]

---

Would you like details on a related example or further explanation?

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### Follow-up Questions:
1. How does DeMorgan's law apply to Boolean algebra?
2. What is the analogous form of DeMorgan's law for intersections?
3. Can you prove DeMorgan's law using a Venn diagram?
4. How does DeMorgan's law generalize for more than two sets?
5. How is DeMorgan's law used in computer science for logical negation?

### Tip:
DeMorgan's laws are essential for simplifying logical expressions, particularly in programming and circuit design!

Prove DeMorgan’s law: (𝐴 ∪ 𝐵)̅ = 𝐴̅ ∩ 𝐵̅

Explore the proof of DeMorgan's Law, which demonstrates that the complement of the union of two sets equals the intersection of their complements. This detailed explanation includes key set theory concepts, logical equivalences, and step-by-step reasoning, suitable for advanced high school and undergraduate students.

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