How can De Morgan's theorem be derived using the concept of duality?

make one easy question and answer about the Knuth-Morris-Pratt algorithm with complete analysis

Simplifying Boolean Equations: Simplify the following Boolean equations. Show your work and list which axiom or theorem you used in each step. Your final equation should be in minimized sum-of-products(SOP) form.   M = (A + B'C)(A'D + AB + AB’C + AC')B Please list each therom used line by line during the solving process.

Answer the following questions: a. Apply DeMorgan's Theorems to the following expression F = (ABC + DEF)' b. Apply DeMorgan's Theorems to the following expression F = ((A+B+C). D)' c. Use DeMorgan's Theorem in algebraic form to Prove F = A' + B' (A. B)' d. Use DeMorgan's Theorem in algebraic form to Prove F = A'. B' = (A + B)' e. Minimize the following Logic Expressions F, Y Algebraically and PROVE the function F to be equal to A'.C' + B, and the function Y to be equal to A+BC. A'.B'.C + A'.B.C'+A'.B.C+A.B.C'+A.B.C F (A, B, C) Y (A, B, C) (A+B) (A+C)