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