Question 3.5: Derive the standard SOP expression and the standard POS expr......
Derive the standard SOP expression and the standard POS expression according to the truth table shown in Table 3.4.4.
| Table 3.4.4: Truth table. | |||
| Input | Output | ||
| A | B | C | F |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
There are five 1s in the output column and the corresponding binary values of the input variables are 010, 011, 101, 110, 111. These binary values are converted to the product terms as follows:
010→\bar{A}B\bar{C}~(m_{2});~011→\bar{A}B C~(m_{3});101→ A \bar B{C}~(m_{5});~110→ A B\bar{C}~(m_{6});~111→ A B C~(m_{7})The resulting standard SOP expression for the output F is
F=\overline{{{A}}}B\overline{{{C}}}+\overline{{{A}}}B C+{{{A}}}\overline{B} C+A B\overline{{{C}}}+A B C=\sum m(2,3,5,6,7)For POS expression, there are three 0s in the output column and the corresponding binary values of the input variables are 000, 001, 100. These binary values are converted to the sum terms as follows:
000 → A+B+C\;(M_{0});\;001 → A+B+\bar{C}\;(M_{1});\;100 → \bar{A}+B+C\;(M_{4})The resulting standard POS expression for the output F is
F=(A+B+C)(A+B+\overline{{{C}}})(\overline{{{A}}}+B+C)=\prod M(0,1,4)Comparing the standard SOP expression with the standard POS expression, you can find that the labels in minterm expression and maxterm expression are complemented for the same truth table. That is, the labels occurring in minterm expression will not appear in maxterm expression.