Boolean Algebra, Formal Logic and Karnaugh Maps A formal, structural and epistemological exposition from the Theory of Symbolization and Pure Mathematical Teaching (TSEMP)

Boolean algebra constitutes one of the most refined formal constructions in modern discrete mathematics, since it represents propositional logic through a closed algebraic system whose validity depends exclusively on axioms and symbolic transformation rules. Therefore, its relevance goes far beyond the instrumental domain of digital engineering: Boolean algebra is, in the strict sense, a mathematical theory whose internal structure reflects the deductive essence of formal rationality. This article develops an extensive, rigorous and epistemologically grounded exposition of Boolean principles, including their axiomatization, derived laws, structural duality, formulation as a complemented distributive lattice, canonical forms SOP/POS, derived operators XOR/XNOR, and their interpretation as an algebra of functions over the Boolean hypercube. Furthermore, Karnaugh maps are examined as structural projections of the space Bn that preserve adjacency through Gray ordering and enable a formal reduction based on absorption and complementation laws. Finally, an epistemological and didactic interpretation is presented from the Theory of Symbolization and Pure Mathematical Teaching (TSEMP), arguing that Boolean logic constitutes a privileged scenario for advanced mathematical formation due to the primacy of symbols, proof and formal coherence as the foundations of knowledge. Keywords: Boolean algebra, propositional logic, formalism, duality, lattices, Karnaugh, Gray code, SOP, POS, XOR, XNOR, TSEMP, pure mathematics education.