Simplification in both SOP and POS forms

Example: Simplify the function S = Σ(1, 4, 9, 12, 13) and obtain the solutions. Simplify in both SOP and POS forms by means of K-Map.

Solution: The entries are as shown in Fig. 2.36 for the SOP form and Fig. 2.37 for the POS form. From Fig. 2.36, we obtain the SOP output as:

                                                            S = BCD + ACD + BCD                         (2.36)

From Fig. 2.37, we obtain the POS output as:

                                                            P = C′(B + D)(A + B′ + D)                              (2.37)

We now draw the respective implementations of Eqs. (2.36) and (2.37) using appropriate logic gates.  Figure 2.38 shows the implementation of the SOP Eq. (2.38) and Fig. 2.39 shows the implementation of the POS Eq. (2.37).

From Fig. 2.38 we find that we require a total of seven gates (three NOTs, three ANDs and one OR) to implement the function given by Eq. (2.36). From Fig. 2.39, we find that the POS implementation requires only six gates (three NOTs, two ORs and one NAND).  Since the POS form has lesser number of gates than the corresponding SOP form, implementation using the POS form is more economical in this case

In many cases, we find that the number of gates will be equal in both the SOP and POS forms.  In such cases, as explained above, we generally count the total number of input pins of the gates required for implementing the POS and SOP forms and choose the more economical form accordingly.

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