Question 7.12: Finding the Minimum POS Form for a Logic Function Find the m...
Finding the Minimum POS Form for a Logic Function
Find the minimum POS for the logic variable E of Example 7.11.
The Karnaugh map for E is shown in Figure 7.32. The map for \overline{E} is obtained by replacing 1s with 0s (blank squares) and vice versa. The result is shown in Figure 7.33.
Now, we look for the smallest number of the largest size cubes that cover the ones in the map. Clearly, there are no 8-cubes or 4-cubes contained in Figure 7.33.A total of eight 1s appear in the map. Thus, the best we can do is to cover the map with four 2-cubes. One option is the grouping shown in the figure, which yields
\overline{E}=ABC+A\overline{B}D+\overline{A}C\overline{D}+\overline{B}\overline{C}\overline{D}
Next, we apply De Morgan’s laws to obtain a minimum POS form:
E=(\overline{A}+\overline{B}+\overline{C})(\overline{A}+B+\overline{D} )(A+\overline{C}+D )(B+C+D)
Choosing a different grouping in Figure 7.33 produces another equally good form, which is
\overline{E}=A\overline{B}\overline{C}+\overline{A}\overline{B}\overline{D}+ACD +BC\overline{D}
Then, applying De Morgan’s laws gives another minimum POS form:
E=(\overline{A}+B+C)(A+B+D)(\overline{A}+\overline{C}+\overline{D})(\overline{B}+\overline{C}+D )
