The expression $A^c \cap B^c$ represents the intersection of the complements of sets $A$ and $B$.

This can be understood through De Morgan's Laws in set theory, which state:

$A^c \cap B^c = (A \cup B)^c$

Explanation:

• $A^c$ is the complement of set $A$, meaning it includes all elements not in $A$.
• $B^c$ is the complement of set $B$, meaning it includes all elements not in $B$.
• $A^c \cap B^c$ is the set of elements that are not in $A$ and also not in $B$. This is equivalent to the complement of the union of $A$ and $B$.

Summary:

$A^c \cap B^c = (A \cup B)^c$

This is a key result in set theory and is useful in various applications, including probability and logic.

Would you like further details or have any questions?

Relative Questions:

1. What are the implications of De Morgan's Laws in probability?
2. How can $A^c \cap B^c$ be visualized using a Venn diagram?
3. How does the intersection of complements relate to logical AND and OR operations?
4. Can you provide an example with specific sets for $A$ and $B$?
5. What is the difference between $A^c \cap B$ and $A \cap B^c$?

Tip: Understanding set operations and their complements is crucial in probability, logic, and various branches of mathematics.