d) \( (A \cup(B \cup C))^{\prime} \cap\left(A^{\prime} \cup(B \cap C)\right)^{\prime} \)

To simplify the expression \( (A \cup (B \cup C))^{\prime} \cap \left(A^{\prime} \cup (B \cap C)\right)^{\prime} \), we will use De Morgan's laws and properties of set operations. 1. **Apply De Morgan's Law**: - The first part is \( (A \cup (B \cup C))^{\prime} \): \[ (A \cup (B \cup C))^{\prime} = A^{\prime} \cap (B \cup C)^{\prime} = A^{\prime} \cap (B^{\prime} \cap C^{\prime}) = A^{\prime} \cap B^{\prime} \cap C^{\prime} \] - The second part is \( \left(A^{\prime} \cup (B \cap C)\right)^{\prime} \): \[ \left(A^{\prime} \cup (B \cap C)\right)^{\prime} = (A^{\prime})^{\prime} \cap (B \cap C)^{\prime} = A \cap (B^{\prime} \cup C^{\prime}) \] 2. **Combine the results**: Now we have: \[ (A \cup (B \cup C))^{\prime} \cap \left(A^{\prime} \cup (B \cap C)\right)^{\prime} = (A^{\prime} \cap B^{\prime} \cap C^{\prime}) \cap (A \cap (B^{\prime} \cup C^{\prime})) \] 3. **Distribute the intersection**: We can distribute the intersection: \[ = (A^{\prime} \cap B^{\prime} \cap C^{\prime}) \cap A \cap (B^{\prime} \cup C^{\prime}) \] Since \( A^{\prime} \cap A = \emptyset \), the entire expression simplifies to: \[ = \emptyset \] Thus, the final result is: \[ \boxed{\emptyset} \]