BOOLEAN ALGEBRA
Keywords:
Boolean Algebra, Digital Logic Design, Logic Minimization, Karnaugh Maps, De Morgan’s Laws, Combinational Circuits, Truth Tables, Logic Gates, Quine-McCluskey Algorithm, Canonical Forms, Sum of Products (SOP), Product of Sums (POS), Sequential Logic, Duality Principle.Abstract
Boolean algebra serves as the indispensable mathematical framework underlying the analysis and design of digital logic systems. This comprehensive module, designed for higher education curricula in Computer Science and Electrical Engineering, systematically explores the transition from abstract logical postulates to physical circuit implementation. Beginning with the historical context established by George Boole, the text defines the fundamental algebraic structure, including the Huntington postulates, the principle of duality, and core theorems such as De Morgan’s laws. It further examines standard canonical forms—specifically Sum of Products (SOP) and Product of Sums (POS)—and details rigorous logic minimization techniques, ranging from visual Karnaugh maps to the algorithmic Quine-McCluskey method.. The discussion culminates in the practical application of these theoretical constructs to combinational and sequential logic design, demonstrating how Boolean concepts directly inform the architecture, efficiency, and reliability of modern computing systems.
References
1.Mano, M. M., & Ciletti, M. D. (2018). Digital Design: With an Introduction to the Verilog HDL, VHDL, and SystemVerilog (6th ed.). Pearson.
2.Rosen, K. H. (2019). Discrete Mathematics and Its Applications (8th ed.). McGraw-Hill Education
3.Roth, C. H., & Kinney, L. L. (2020). Fundamentals of Logic Design (7th ed.). Cengage Learning.
4.Whitesitt, J. E. (2010). Boolean Algebra and Its Applications. Dover Publications.
5.Boole, G. (1854). An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Macmillan.
6.Shannon, C. E. (1938). "A Symbolic Analysis of Relay and Switching Circuits". Transactions of the American Institute of Electrical Engineers, 57(12), 713–723.
7.Huntington, E. V. (1904). "Sets of Independent Postulates for the Algebra of Logic". Transactions of the American Mathematical Society, 5, 288-309.
8.References for the specific minimization algorithms discussed in the module.
9.Tojimamatov I., Ahmataliyeva S. Berilganlarni markazlashgan tarzda boshqarish tamoyillari //Академические исследования в современной науке. – 2025. – Т. 4. – №. 21. – С. 59-64. G‘aniev B. “Ta’lim jarayonida AKTdan foydalanish samaradorligi”. — Ta’lim va innovatsiyalar jurnali, 2022.(AKTning pedagogik samaradorligi ilmiy asoslangan.)
10. Ismoilova N. “Raqamli ta’lim: imkoniyatlar va muammolar”. — Pedagogika va psixologiya ilmiy jurnali, 2023.(Raqamli ta’limning ijobiy va salbiy jihatlari tahlil qilingan.)
11.Tojimamatov I., Adxamova C. AMALIY TIZIMLARDA BERILGANLAR BAZASINI BOSHQARISH TIZIMLARI O ‘RNI //Академические исследования в современной науке. – 2025. – Т. 4. – №. 21. – С. 77-
12.Quine, W. V. (1952). "The Problem of Simplifying Truth Functions". The American Mathematical Monthly, 59(8), 521–531.
13.McCluskey, E. J. (1956). "Minimization of Boolean Functions". Bell System Technical Journal, 35(6), 1417–1444.
