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Boolean Algebra Cheat Sheet (DRAFT) by

This cheat sheet aims to help students understand the basic aspects of Boole's Algebra applied to digital electronic circuits.

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Introd­uction

The Boolean algebra, named after the mathem­atician George Boole, is a branch of mathem­atics that deals with binary variables and logical operat­ions. Developed in the mid-19th century, Boolean algebra laid the foundation for digital logic and computer science. It was primarily introduced as a way to express logical statements in a symbolic form, offering a method to manipulate logical propos­itions algebr­aic­ally.

The main idea behind Boolean algebra is to operate on binary values, typically repres­ented as 0 (false) and 1 (true). The algebraic structure follows a set of basic operat­ions: AND, OR, and NOT, which correspond to logical conjun­ction, disjun­ction, and negation. These operations are the backbone of digital circuits, making Boolean algebra crucial in the design of electronic systems such as computers, teleco­mmu­nic­ation devices, and automation systems.

Boolean algebra has found extensive applic­ations in various fields, most notably in the develo­pment of computer hardware and software. It is used to optimize circuits, simplify complex logical expres­sions, and create algorithms for decisi­on-­making processes. Over time, its influence has expanded, and it continues to be a vital component of modern comput­ational techno­logies.

Historic

1847
George Boole published a pamphlet titled The Mathem­atical Analysis of Logic in response to a contro­versy between De Morgan and Sir Hamilton.
1854
Boole published the book The Laws of Thought with a different formul­ation from his previous work.
1860s
Boolean algebra emerged in the works of William Jevons and Charles Peirce.
1890
Schröder published his best work, in three volumes, called Vorles­ungen über die Algebra der Logik. This work was the first systematic presen­tation of Boolean algebra and distri­butive matrices.
1904
The Boolean algebra is seen as an axiomatic algebraic structure thanks to the works of Huntington.
1927
Zhegalkin demons­trated that tradit­ional algebra using integer numerical values modulo 2 (instead of the tradit­ional modulo 10) behaves exactly like Boolean algebra. This fact has led to some ambiguity about the true nature of Boolean algebra: it can be understood as logical algebra or as numerical algebra.
1930s
Boolean algebra became mathem­ati­cally rigorous thanks to the works of Marshall Stone (1930s), and to Birkhoff's matrix theory (1940).
1938
Shannon demons­trated that electronic circuits with relays could be modeled through Boolean algebra.

The Basics

Boolean Algebra consists of a set of values, operations, and axioms, from which identities and theorems are derived.

Venn Diagrams

John Venn was a British mathem­atician and logician known for introd­ucing Venn diagrams, which visually represent relati­onships between sets. These diagrams simplify the unders­tanding of operations like union, inters­ection, and comple­ment, making them valuable in logic, statis­tics, and computer science. In Boolean Algebra, they help visualize logical operat­ions, aiding in the analysis of Boolean expres­sions, circuit simpli­fic­ation, and unders­tanding fundam­ental principles of mathem­atical logic.