The expression $(A \cup B)'$ represents the complement of the union of sets $A$ and $B$. In set theory, this means it includes all the elements that are not in the union of $A$ and $B$.

Using De Morgan's laws, we can rewrite $(A \cup B)'$ as:

$(A \cup B)' = A' \cap B'$

Explanation:

• $A \cup B$: The union of sets $A$ and $B$ contains all elements that are in either $A$, $B$, or both.
• $(A \cup B)'$: The complement of $A \cup B$ includes all elements that are not in $A \cup B$.
• De Morgan's Law: This law helps to simplify the expression by distributing the complement over the union, turning it into an intersection: $(A \cup B)' = A' \cap B'$.

Thus, the result is the intersection of the complements of $A$ and $B$, which contains all elements that are neither in $A$ nor in $B$.

Would you like further details or have any questions about this explanation?

Related Questions:

1. What is the complement of a set in set theory?
2. How do De Morgan's laws apply to other set operations like intersection?
3. Can you provide an example with specific sets to illustrate $(A \cup B)'$?
4. What is the meaning of $(A \cap B)'$ and how is it different from $(A \cup B)'$?
5. How do you visualize set operations using Venn diagrams?

Tip:

When using De Morgan's laws, remember that the complement of a union becomes an intersection of complements, and vice versa.