Boolean Algebra
Introduction to Boolean Algebra
Boolean algebra, developed by George Boole, is a branch of algebra in which the values of the variables are the truth values true (1) and false (0). It is fundamental to the design and analysis of digital logic circuits found in computers and other electronic devices. Basic operations include AND (conjunction, $\cdot$), OR (disjunction, $+$), and NOT (negation, $\overline{A}$ or $A'$).
Basic Boolean Functions & Switching Circuits
OR Function
Boolean Expression (2-input): $Y = A + B$
Boolean Expression (3-input): $Y = A + B + C$
Switching Circuits (Parallel):
2-Input OR
3-Input OR
Truth Table (2-input OR):
AND Function
Boolean Expression (2-input): $Y = A \cdot B$
Boolean Expression (3-input): $Y = A \cdot B \cdot C$
Switching Circuits (Series):
2-Input AND
3-Input AND
Truth Table (2-input AND):
NOT Function (Inverter)
Boolean Expression: $Y = \overline{A}$
Truth Table (1-input NOT):
Logic Gate Symbols & Truth Tables
AND Gate (3-input)
Expression: $Y = A \cdot B \cdot C$
OR Gate (3-input)
Expression: $Y = A + B + C$
NOT Gate
Expression: $Y = \overline{A}$
NAND Gate (3-input)
Expression: $Y = \overline{A \cdot B \cdot C}$
NOR Gate (3-input)
Expression: $Y = \overline{A + B + C}$
XOR Gate (2-input)
Expression: $Y = A \oplus B$
Laws and Rules of Boolean Algebra
Used to manipulate and simplify Boolean expressions:
- Commutative: $A+B = B+A$; $A \cdot B = B \cdot A$
- Associative: $(A+B)+C = A+(B+C)$; $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
- Distributive: $A \cdot (B+C) = A \cdot B + A \cdot C$; $A + (B \cdot C) = (A+B) \cdot (A+C)$
- Identity: $A+0 = A$; $A \cdot 1 = A$
- Annihilation (Dominance): $A+1 = 1$; $A \cdot 0 = 0$
- Idempotent: $A+A = A$; $A \cdot A = A$
- Complement: $A+\overline{A} = 1$; $A \cdot \overline{A} = 0$
- Involution (Double Negation): $\overline{\overline{A}} = A$
- Absorption: $A+(A \cdot B) = A$; $A \cdot (A+B) = A$
De Morgan's Laws:
- $\overline{A+B} = \overline{A} \cdot \overline{B}$
- $\overline{A \cdot B} = \overline{A} + \overline{B}$
Interactive Boolean Expression Simplification
This example demonstrates step-by-step simplification of the expression $Y = A + \overline{A}B$. Other common expressions for practice include $A(A+B)$, $AB + A\overline{B}$, $(A+B)(A+\overline{B})$.
Current Expression:
Simplify with Karnaugh Maps (K-maps)
A graphical method for simplifying Boolean expressions for 2, 3, or 4 variables.
Alternatively, click cells in the K-map below to set values (0, 1, X).
Enter minterms or click K-map cells, then click "Simplify K-map".
Universal Gates (NAND and NOR)
NAND and NOR gates are called universal gates because any other logic gate (AND, OR, NOT) can be implemented using only NAND gates or only NOR gates.
NAND as Universal Gate
- NOT from NAND: $ \overline{A} = \overline{A \cdot A} $
- AND from NAND: $ A \cdot B = \overline{\overline{A \cdot B}} $
- OR from NAND: $ A+B = \overline{\overline{A} \cdot \overline{B}} $
Logic Circuit Problem Examples
Select a predefined Boolean expression to see its standard logic circuit and an equivalent implementation using only NAND gates.
Select a problem to view details.