Y x z the representation of these two functions in

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Unformatted text preview: words, is it possible to find two algebraic expressions that specify the same Boolean function? The answer to this question is yes. As a matter of fact, the manipulation of Boolean algebra is applied mostly to the problem of finding simpler expressions for a given expression. For example, let us consider the following two functions: F1= x-y-z+x-y-z+x-y and F2 = x . y+ x-z The representation of these two functions in the form of truth table is shown in Figure 6.14. From the table we find that the function F 2 is the same as the function Fi since both have identical 0s and Is for each combination of values of the three binary variables x, y and z. In general, two functions of n binary variables are said to be equal if they have the same value for all possible 2n combinations of the n literals. X y z F1 F2 0 0 0 0 1 1 1 1 0 0 l l 0 0 l l 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 .1 1 0 0 Figure 6.14. Truth table for the Boolean functions: F1 = x -y z +x -y-z+x-y and F2 = xy + xz Minimization of Boolean Functions When a Boolean function is implemented with logic gates (discussed later in this chapter), each literal in the function designates an input to a gate and each term is implemented with a gate. Thus for a given Boolean function, the minimization of the number of literals and the number of terms will result in a circuit with less components. For example, since functions Fi and F2 of Figure 6.14 are equal Boolean functions, it is more constitutes the best form of Boolean function depends on the particular application. However, we will give consideration only to the criterion of component minimization, which is achieved by literal minimization. There are several methods used for minimizing the number of literals in a Boolean function. However, a discussion of all these methods is beyond the scope of this book. Hence here we will consider only the method of algebraic manipulations. Unfortunately, in this method, there are no specific rules or guidelines to be followed that will guarantee the final answer. The only method available is cut-and-try procedure employing the postulates, the basic theorems, and any other manipulation method, which becomes familiar with use. The following examples illustrate this procedure. Example 6.1. Simplify the following Boolean functions to a minimum number of literals, (a) x + x • y (b)x-(x + y) (c...
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This document was uploaded on 04/07/2014.

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