Boolean Algebra<br > and Boolean Function<br > Switching theory deals primarily with the analysis (characterization， minimization，<br > etc，) and synthesis (realization) of a special type of function， defined on a special type<br > of algebra known as switching algebra. Switching algebra is， in turn， a special type of<br > Boolean algebra，" and the special type of function， known as the switching function， is<br > a mapping defined on switching algebra. Switching algebra that contains two elements，<br > 0 and 1， is the two-element Boolean algebra (the simplest nondegenerate Boolean<br > algebra). To understand how switching algebra is derived， one must first learn Boolean<br > algebra， its mathematical foundation. In fact， Boolean algebra is the mathematical<br > :foundation of the entire field of switching theory.<br > The algebraic structure of Boolean algebra is derived from the ordered set. We<br > i begin the chapter by introducing ordered sets and the one-to-one relationship between<br > elements in set theory and elements in algebra. Before introducing Boolean algebra，<br > we first define lattice， which is a special subclass of the class of ordered sets. Boolean<br > ，algebra is a special class of a subclass of lattices known as the complcmented distributive<br > .lattice， or the Boolean lattice. Important properties of Boolean algebra are discussed<br > in detail. Finally， the formal definition of Boolean function and its canonical forms<br > are presented. The existence of the canonical forms for every Boolean hmction pro-<br > vides us with a convenient means of determining the equivalence between two Boolean<br > functions and with a basis for deriving switching-function minimization methods，<br > which will be discussed in Chapter 2.<br > 1.1 Sets， Ordered Sets， and Algebras<br > Set theory is often referred to as the "root" ofmathelnatics. We can consider every<br > branch of mathematics to be a study of sets of objects of one kind or another. For<br > instance， roughly speaking， geometry is a study of sets of points. Algebra is concerned<br > with sets of numbers and operations on those sets. AnaIysis deals mainly with sets of<br > functions. The study of sets and their use in the foundations of mathematics was begun<br > in the latter part of the nineteenth century by the German mathematician Georg<br >