Boolean Operations and Laws

Discrete Math · Axiom Academy

LESSON Boolean Algebra: Operations and Laws Master the fundamental operations, axioms, and theorems of Boolean algebra, including the elegant duality principle that reveals hidden symmetries. Boolean algebra has four fundamental operations that combine binary values: Boolean algebra is built on a foundation of axioms (self-evident truths) that define the behavior of the operations: From the axioms, we can derive important theorems that simplify Boolean expressions: One of the most elegant properties of Boolean algebra is the duality principle : every valid Boolean equation remains valid if you swap AND with OR and swap 0 with 1. Replace every AND (·) with OR (+) Replace every OR (+) with AND (·) Keep all variables and complements unchanged We can prove Boolean laws using truth tables by showing both sides of an equation produce identical outputs for all possible input combinations. Example: Proving De Morgan's Law ¬(x · y) = ¬x + ¬y Since columns 4 and 7 are identical for all input combinations, we've proven that ¬(x · y) = ¬x + ¬y.

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