Applications of Boolean Algebra: Claude Shannon and Circuit Design

Abstract

This Primary Source Project (PSP) is based on the award-winning paper “A Symbolic Analysis of Relay and Switching Circuits” by American mathematician and electrical engineer Claude Shannon. After introducing the use of Boolean expressions to represent parallel and series circuits, the project then develops the standard properties of a Boolean algebra within the concrete context of the two-valued Boolean algebra associated with such circuits. It then provides students with practice in using those basic identities to simplify and manipulate Boolean expressions and introduces the concept of a ‘disjunctive normal form’ for Boolean expressions. It continues by exploring an additional, more sophisticated, method for applying Boolean algebra to the problem of simplifying complicated circuits. While the project does assume some (minimal) familiarity with the set operations of union and intersection, that pre-requisite material may be introduced by completing the companion Boole-Venn-Peirce project listed below, through reading a standard textbook treatment thereof, or via a short class discussion/lecture. Although no other specific pre-requisite knowledge is necessary for any part of the project, Sections 3 and 4 do assume slightly higher levels of mathematical maturity on the part of the students, roughly commensurate with that of a student who has completed Calculus I (for Section 3) and Calculus II (for Section 4).

This PSP is designed for use in an introductory or intermediate course in discrete or finite mathematics that considers Boolean algebra from either a mathematical or computer science perspective. The following two PSPs are further available as possible companions for the current project in such a course:

• “Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce”
• “Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization”

Either or both of these companion projects could be used independently of the current (Shannon) project, or as a preliminary to or as a follow-up to the other companion project.