Digital Electronics: Principles, Devices and Applications

6

Boolean Algebra and Simplification Techniques

Boolean algebra is mathematics of logic. It is one of the most basic tools available to the logic designer and thus can be effectively used for simplification of complex logic expressions. Other useful and widely used techniques based on Boolean theorems include the use of Karnaugh maps in what is known as the mapping method of logic simplification and the tabular method given by Quine–McCluskey. In this chapter, we will have a closer look at the different postulates and theorems of Boolean algebra and their applications in minimizing Boolean expressions. We will also discuss at length the mapping and tabular methods of minimizing fairly complex and large logic expressions.

6.1 Introduction to Boolean Algebra

Boolean algebra, quite interestingly, is simpler than ordinary algebra. It is also composed of a set of symbols and a set of rules to manipulate these symbols. However, this is the only similarity between the two. The differences are many. These include the following:

1. In ordinary algebra, the letter symbols can take on any number of values including infinity. In Boolean algebra, they can take on either of two values, that is, 0 and 1.
2. The values assigned to a variable have a numerical significance in ordinary algebra, whereas in its Boolean counterpart they have a logical significance.
3. While ‘.’ and ‘+’ are respectively the signs of multiplication and addition in ordinary algebra, in Boolean algebra ‘.’ means an AND operation ...