# Question Explain the significance of De Morgan's Laws in set theory. What fundamental relationship do they reveal? ## Answer Choices (A) They show that complement operations are distributive over union and intersection (B) They establish that union and intersection are inverse operations (C) They demonstrate the duality between union and intersection under complementation (D) They prove that all sets have complements (E) They define the relationship between subsets and supersets # Solution De Morgan's Laws state: 1. $(A \cup B)^c = A^c \cap B^c$ 2. $(A \cap B)^c = A^c \cup B^c$ These laws reveal a fundamental duality in set theory: union and intersection are dual operations that transform into each other when we take complements. Specifically: - The complement of a union becomes an intersection of complements - The complement of an intersection becomes a union of complements This duality is significant because it shows a deep structural symmetry in set theory. It means that any theorem involving unions can be converted to a corresponding theorem involving intersections (and vice versa) by taking complements. This principle extends beyond just two sets and applies to arbitrary unions and intersections. The laws are not about distributivity (which is a separate property), inverse operations, or defining complements—they specifically reveal how the union and intersection operations are interchanged under complementation. **Answer: (C)**