3.3 Simplification of Boolean Functions

3.3 Simplification of Boolean Functions

Simplification of Boolean functions is a particularly important process in designing an efficient digital circuit. Such simplified functions require fewer gates making them compact in size, energy efficient and faster than the complicated ones. Simplification means applying of some Boolean algebra rules to make the functions less complicated.

Basic Boolean Algebra Rules:

Here are some fundamental Boolean algebra rules used for simplification:

1. Identity Laws
   • $A + 0 = A$
   • $A \cdot 1 = A$
2. Null Laws
   • $A + 1 = 1$
   • $A \cdot 0 = 0$
3. Idempotent Laws
   • $A + A = A$
   • $A \cdot A = A$
4. Complement Laws
   • $A + \overline{A} = 1$
   • $A \cdot \overline{A} = 0$
5. Commutative Laws
   • $A + B = B + A$
   • $A \cdot B = B \cdot A$
6. Associative Laws
   • $(A + B) + C = A + (B + C)$
   • $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
7. Distributive Laws
   • $A \cdot (B + C) = (A \cdot B) + (A \cdot C)$
   • $A + (B \cdot C) = (A + B) \cdot (A + C)$
8. Absorption Laws
   • $A + (A \cdot B) = A$
   • $A \cdot (A + B) = A$
9. De Morgan's Theorems
   • $\overline{A + B} = \overline{A} \cdot \overline{B}$
   • $\overline{A \cdot B} = \overline{A} + \overline{B}$
10. Double Negation Law
    • $\overline{\overline{A}} = A$