Digital Electronics: Principles, Devices and Applications

200 Digital ElectronicsX1X2XnX1X2Xn(a)X1X2X3X1X2Xn(b)Figure 6.3DeMorgan’s theorem.LHS = X 1 + X 2 + X 3 + ···X n = 1 + 0 + 0 +···+0 = 1 = 0RHS = X 1 X 2 X 3 X n = 100 0 = 011 1 = 0Therefore, again LHS = RHS.The same holds good when more than one or all variables are in the logic ‘1’ state. Therefore,theorem 13(a) stands proved. Since theorem 13(b) is the dual of theorem 13(a), the same also standsproved. Theorem 13(b), though, can be proved on similar lines.6.3.14 Theorem 14 (Transposition Theorem)and(a) XY + XZ = X + ZX + Y(b) X + YX + Z = XZ + XY (6.24)This theorem can be applied to any sum-of-products or product-of-sums expression having two terms,provided that a given variable in one term has its complement in the other. Table 6.4 gives the proofof theorem 14(a) using the method of perfect induction. Theorem 14(b) is the dual of theorem 14(a)and hence stands proved.As an example,AB + AB = A + BA + B and AB + AB = A + BA + BIncidentally, the first expression is the representation of a two-input EX-OR gate, while the secondexpression gives two forms of representation of a two-input EX-NOR gate.