Digital Electronics: Principles, Devices and Applications

202 Digital ElectronicsThe proof of theorem 16(a) is straightforward and is given as follows:fX XYZ= XfX XYZ+ XfX XYZ= Xf1 0YZ+ Xf0 1YZTheorem 15(a)AlsofX XYZ= X + fX XYZX + fX XYZ= X + f0 1YZX + f1 0YZTheorem 15(b)6.3.17 Theorem 17 (Involution Law)X = X (6.31)Involution law says that the complement of the complement of an expression leaves the expressionunchanged. Also, the dual of the dual of an expression is the original expression. This theorem formsthe basis of finding the equivalent product-of-sums expression for a given sum-of-products expression,and vice versa.Example 6.5Prove the following:1. LM + N+ LPQ = L + PQL + M + N2. AB + C + DD + E + FG = DAB + C+ DGE + FSolution1. Let us assume that L = X M + N= Y and PQ = Z.The LHS of the given Boolean equation then reduces to XY + XZ.Applying the transposition theorem,XY + XZ = X + ZX + Y = L + PQL + M + N= RHS2. Let us assume D = X AB + C = Y and E + FG = Z.The LHS of given the Boolean equation then reduces to X + YX + Z.Applying the transposition theorem,X + YX + Z = XZ + XY = DGE + F+ DAB + C = RHS