##### 17.1 INTRODUCTION

Digital systems mainly handle binary data, i.e., 0s and 1s. The truth table represents the input–output relationship of a logic circuit. The output of a logic circuit in terms of its input variables is given by the Boolean function. The desired logical output, represented by a Boolean function, may be realized by using either physical switches or basic digital building blocks such as AND, OR and NOT gates and their combinations. Boolean algebra is the basis of the design of digital systems. Digital systems essentially consist of logic circuits. A combinational circuit consists of logic gates whose outputs at a given time are determined directly from the present combination of inputs without regard to the previous inputs. Some logic gates such as AND, OR, NAND and NOR, using discrete components, have been considered earlier. Also, the merits and demerits of each of the logic families have been discussed. However, for better stability and reliability, these gates are best designed as integrated circuits called digital integrated circuits. NAND and NOR gates are called universal gates as it is possible to derive the other gates using them. When a logic circuit is complex, then, to implement a desired logic requirement with minimum cost, it becomes necessary to reduce the complexity of the circuit by simplifying the Boolean function. Thus, primarily, Boolean theorems, properties and identities along with De Morgan’s laws are the tools that help us in simplifying a Boolean function. However, sometimes the Boolean simplification may not necessarily yield the optimum circuit configuration. Also the procedure for simplification is a bit involved. Karnaugh maps, on the other hand, help in realizing the optimum circuit configuration, with minimal effort. If the Boolean function is expressed either as the sum of products (SOP) or as the product of sums (POS) mapped on to a Karnaugh map and the rules for simplification followed, it is a much easier to arrive at the optimal form of the Boolean function. The method of simplification of Boolean functions using Boolean rules and Karnaugh maps are discussed.