3 - Boolean Algebra

Abstract

This chapter reviews Boolean algebra, certain fundamentals, including a comparison of ordinary algebra and Boolean algebra, procedures for simplifying Boolean algebraic equations, and the basis for representing switching functions in binary computers by Boolean algebraic equations. The subject of Boolean algebra has important application in the design of systems composed of storage elements capable of assuming a discrete number of stable states and switching devices that trigger these elements from one stable state to another. An electronic digital computer is such a system. If the two assigned values of a Boolean algebraic variable are interpreted as representing the numerical values of the binary number system, namely 0 and 1, then the three operators are analogous to addition, multiplication, and complementation, respectively, of binary elements. There is an exception to the analogy in that the result of the OR (Boolean addition) operation on two or more 1s produces 1 as a result. The inclusive or operation of Boolean algebra is designated literally by OR and in equations by the symbol +. The and operation of Boolean algebra is designated literally by AND and in equations by placing the variables so to be operated upon adjacent to one another. The complementation or negation operation of Boolean algebra is designated literally by NOT and in equations by placing a bar over the variable or variables so to be operated upon. The commutative, associative, and distributive laws of number algebra apply to Boolean algebra.