$$ \left(a\cup b\right)^{\prime}=a^{\prime}\cap b^{\prime} $$

The expression you provided represents **De Morgan's Law** in set theory, which relates the complement of a union of two sets to the intersection of their complements.

### De Morgan's Law
For any two sets $ A $ and $ B $, De Morgan's laws state:

1. **Complement of a Union:**
   \[
   (A \cup B)' = A' \cap B'
   \]

2. **Complement of an Intersection:**
   \[
   (A \cap B)' = A' \cup B'
   \]

### Explanation of Your Expression

The expression you provided is:

\[
(a \cup b)' = a' \cap b'
\]

Here:
- $ a \cup b $ denotes the **union** of sets $ a $ and $ b $. This is the set containing all elements that are in $ a $, in $ b $, or in both.
- $ (a \cup b)' $ represents the **complement** of the union of sets $ a $ and $ b $. This is the set of all elements not in $ a \cup b $.
- $ a' $ represents the **complement** of set $ a $, which is the set of all elements not in $ a $.
- $ b' $ represents the **complement** of set $ b $, which is the set of all elements not in $ b $.
- $ a' \cap b' $ represents the **intersection** of the complements of sets $ a $ and $ b $. This is the set containing all elements that are in both $ a' $ and $ b' $, or equivalently, all elements that are not in either $ a $ or $ b $.

### Intuitive Proof

To see why this law holds, consider an element $ x $:

- $ x \in (a \cup b)' $ means that $ x $ is **not** in $ a \cup b $. Thus, $ x $ is neither in $ a $ nor in $ b $.
- Therefore, $ x $ must be in both $ a' $ and $ b' $. This means $ x \in a' \cap b' $.

Conversely:

- If $ x \in a' \cap b' $, then $ x $ is in both $ a' $ and $ b' $. This implies that $ x $ is not in $ a $ and not in $ b $.
- Therefore, $ x $ is not in $ a \cup b $, which means $ x \in (a \cup b)' $.

Since both sides of the equation represent the same set of elements, we conclude:

\[
(a \cup b)' = a' \cap b'
\]

### Conclusion
De Morgan's laws are foundational in set theory, logic, and various fields of mathematics and computer science. They are essential for understanding how operations on sets relate to each other, particularly in the context of complements, unions, and intersections.

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Would you like further details on this topic, or do you have any questions?

### Related Questions:

1. How does De Morgan's law apply to logical expressions?
2. What is the proof for De Morgan's law using Venn diagrams?
3. Can De Morgan's laws be extended to more than two sets?
4. How are De Morgan's laws used in digital circuit design?
5. What are some real-life applications of De Morgan's laws?

### Tip:
To visualize set operations, try using Venn diagrams to see how the elements are grouped, intersected, or complemented. This helps to build intuition about these concepts.

$$ (a \cup b)^{\prime}=a^{\prime}\cap b^{\prime} $$

Explore De Morgan's Law with the set operation (a \cup b)' = a' \cap b'. This solution provides an intuitive explanation of set complements, unions, and intersections, focusing on De Morgan's Laws in set theory. Ideal for high school students studying set theory in Grades 9-12.

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