9. Boolean Identities and Basic Theorems
Just like standard algebra, Boolean algebra has fundamental laws that allow us to simplify complex expressions, which directly translates to reducing the number of gates required in a circuit.
Commutative Laws
$$A + B = B + A$$ $$A \cdot B = B \cdot A$$
Associative Laws
$$(A + B) + C = A + (B + C)$$ $$(A \cdot B) \cdot C = A \cdot (B \cdot C)$$
Distributive Law
$$A \cdot (B + C) = (A \cdot B) + (A \cdot C)$$ $$A + (B \cdot C) = (A + B) \cdot (A + C)$$
Basic Identity Rules
| OR Rules | AND Rules | Complement Rules |
|---|---|---|
| $A + 0 = A$ | $A \cdot 1 = A$ | $\overline{\overline{A}} = A$ |
| $A + 1 = 1$ | $A \cdot 0 = 0$ | $A + \overline{A} = 1$ |
| $A + A = A$ | $A \cdot A = A$ | $A \cdot \overline{A} = 0$ |
Absorption Law
These are powerful simplification tools:
- $$A + (A \cdot B) = A$$
- $$A \cdot (A + B) = A$$
Knowing these theorems by heart significantly accelerates circuit simplification.