Question 3.3: Derive a truth table for the SOP expression F = A + BC...
Derive a truth table for the SOP expression F=A+{\bar{B}}C
Step 1 There are three variables in the domain, so there are eight possible combinations of binary values of the variables as listed in the left input columns of Table 3.4.2
Step 2 Substitute each binary value into the SOP expression: F=A+{\bar{B}}C. When ABC = 000, F = 0; when ABC = 001, F = 1; when ABC = 010, F = 0; when ABC = 010, F = 0; when ABC = 011, F = 0; and so on. The resulting output values are listed in the output column of Table 3.4.2.
Alternatively, you can convert the given SOP expression into the standard form. From example 3.1, you already convert the expression F=A+{\bar{B}}C into a standard form and the resulting expression is
F=A+\overline{{{B}}}C=A B C+A B\overline{{{C}}}+A\overline{{{B}}}C+A\overline{{{BC}}}+\overline{{{AB}}}{C}= m_{1}+m_{4}+m_{5}+m_{6}+m_{7}=\sum m(1,4,5,6,7)
Therefore, the binary values that make the standard SOP expression equal to 1 are \bar A\bar{B}{C}\,(m_{1}):001,\,A\bar{B}\bar{C}\,(m_{4}):100,\,A\bar{B}C\,(m_{5}):101,\,A B\bar{C}\,(m_{6}):110,\,A B{C}\,(m_{7}):111. For each of these binary values, a 1 is placed in the output column as shown in Table 3.4.2. For every remaining binary combination, a 0 is placed in the output column. In other words, one row of the truth table corresponds to a minterm. If a minterm exists in the SOP expression, the corresponding output is a 1. If a minterm does not exist in the SOP expression, the corresponding output is a 0.
| Table 3.4.2: Truth table. | |||
| Input | Output | ||
| A | B | C | F |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |