Let A be a set and let B, C, D, and E be subsets of A. Show and explain how the set of minsets of B, C, D, and E form a partition of A. Provide a concrete example.

2. What happens if you mix the concepts of the Duality Principle and minsets? You get maxsets. Do maxsets also form a partition? Prove or disprove. see: https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/04%3A_More_on_Sets/4.04%3A_The_Duality_Principlefor helpful examples.

3. Use the Duality Principle to derive one of DeMorgan's Laws for Sets from the other. Prove both of the laws by use of set membership tables. Now prove both again by the use of Venn diagrams.

4. What is true for set identiesis equally true, mutatis mutandis, for logical equivalencies. Therefore, you can use the Duality Principle to derive one of the forms of the distributive laws for logic from the other. Prove both forms of the distrinutive laws of logic by the use of truth tables.

5. Use the Duality Principle to form the dual (S*) of the following identity S: (A intersect B) intersect (A intersect B complement) = empty set

Prove the truth of S* by any means you choose.

SHOW ALL WORK and doyour own work