Elements of Boolean Algebra for the Study of Information-Handling Systems

Abstract

Boolean algebra is the algebra of classes or sets of objects. It consists essentially of systematic rules for the use of the fundamental connectives "or," "and," "not." The relevance of Boolean algebra to the information-handling field stems from the fact that, in digital systems, information can be represented only by means of sets of discrete physical states of the machinery involved. This paper has been prepared principally to present an adequate mathematical basis for the application of Boolean algebra to the study of information-handling systems. An important purpose of this application is the minimization of the physical elements required in information-handling (or computing) circuits. Consequently, some fundamental methods of simplifying Boolean functions are explained in detail. These methods make possible the unconditional minimization of what are called here "elementary" functions, but of these functions only. Because much of the symbolism and many of the methods in use at the present time are far from uniform, another purpose of this paper has been to put into use an adequate and consistent symbolism, to provide needed fundamental definitions and to derive all theorems and propositions required from the basic postulates. The treatment is essentially an algebraic one. The methods are those of abstract algebra rather than of mathematical logic. The dualization laws have been systematically used to derive fundamental principles which cannot conveniently be obtained in any other way. The application of Boolean algebra to the design of computer circuits is illustrated by actual examples.