State De Morgan's law: (A ∪ B)^c = ?

• A. A ∩ B
• B. A^c ∪ B^c
• C. A^c ∩ B^c
• D. A ∪ B

Answer: (C)

De Morgan's law tells us how the complement distributes over a union. The set A ∪ B contains everything that is in A or in B. Its complement, everything not in A and not in B, is exactly the intersection of the complements: A^c ∩ B^c. So (A ∪ B)^c = A^c ∩ B^c.

This makes sense with a Venn-diagram view: shading outside A and outside B is the common area outside both—precisely the intersection of the two complements. The other forms don’t match: A^c ∪ B^c would include elements outside A or outside B (not necessarily outside both), which is the complement of A ∩ B. A ∪ B is just the uncomplemented union, and A ∩ B is the overlap, neither of which equals the complement of the union.