Draw the circuits that will perform the functions described by both sides of the first of De Morgan’s theorems (Equation 1.16: \overline{A+B} =\overline{A} \cdot \overline{B} \Rightarrow \overline{A+B+C+...} =\overline{A} \cdot \overline{B} \cdot \overline{C} ...) given in Table 1.4, and also demonstrate the theorem is true using a truth table.
| Table 1.4 Multivariable Boolean theorems | |
| Commutative laws: Show that the order of operation under AND and OR is unimportant: | |
| A·B=B·A | (1.10) |
| A+B=B+A | (1.11) |
| Associative laws: Show how variables are grouped together: | |
| (A·B)·C=A·B·C=A·(B·C) | (1.12) |
| (A+B)+C=A+B+C=A+(B+C) | (1.13) |
| Distributive laws: Show how to expand equations out: | |
| A·(B+C)=A·B+A·C | (1.14) |
| A+(B·C)=(A+B)·(A+C) | (1.15) |
| De Morgan’s theorem: | |
| \overline{A+B} =\overline{A} · \overline{B} \Rightarrow \overline{A+B+C+...} =\overline{A} · \overline{B} · \overline{C} ... | (1.16) |
| \overline{A·B} =\overline{A} + \overline{B} \Rightarrow \overline{A·B·C...} =\overline{A} + \overline{B} + \overline{C} ... | (1.17) |
| Other laws which can be proved from the above are the: | |
| Absorption laws: | |
| A·(A+B)=A | (1.18) |
| A+(A·B)=A | (1.19) |
| and ‘other identities’: | |
| A·(\overline{A}+B )=A·B | (1.20) |
| A+(\overline{A}·B )=A+B | (1.21) |
