Digital Electronics: Principles, Devices and Applications

Boolean Algebra and Simplification Techniques 195YZXX+(Y+Z)XYZZ+(X+Y)(a)YZXX.(Y.Z)XYZ(X.Y).Z(b)Figure 6.1Associative laws.6.3.7 Theorem 7 (Distributive Laws)(a) XY + Z = XY + XZ and (b) X + YZ = X + YX + Z (6.17)Theorem 7(b) is the dual of theorem 7(a). The distribution law implies that a Boolean expression canalways be expanded term by term. Also, in the case of the expression being the sum of two or morethan two terms having a common variable, the common variable can be taken as common as in the caseof ordinary algebra. Table 6.1 gives the proof of theorem 7(a) using the method of perfect induction.Theorem 7(b) is the dual of theorem 7(a) and therefore its proof is implied. Theorems 7(a) and (b) arefurther illustrated by the logic diagrams in Figs 6.2(a) and (b). As an illustration, theorem 7(a) can beused to simplify AB + AB + AB + AB as follows:AB + AB + AB + AB = AB + B + AB + B = A1 + A1 = A + A = 1Table 6.1Proof of distributive law.X Y Z Y+Z XY XZ X(Y+Z) XY+XZ0 0 0 0 0 0 0 00 0 1 1 0 0 0 00 1 0 1 0 0 0 00 1 1 1 0 0 0 01 0 0 0 0 0 0 01 0 1 1 0 1 1 11 1 0 1 1 0 1 11 1 1 1 1 1 1 1