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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), OR (disjunction), and NOT (negation).

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
ABY
3-Input OR
ABCY

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
ABY
3-Input AND
ABCY

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)

Boolean Expression: $Y = A \cdot B \cdot C$

ABCY

OR Gate (3-input)

Boolean Expression: $Y = A + B + C$

ABCY

NOT Gate (Inverter)

Boolean Expression: $Y = \overline{A}$

AY

NAND Gate (3-input)

Boolean Expression: $Y = \overline{A \cdot B \cdot C}$

ABCY

NOR Gate (3-input)

Boolean Expression: $Y = \overline{A + B + C}$

ABCY

XOR Gate (2-input)

Boolean Expression: $Y = A \oplus B = A\overline{B} + \overline{A}B$

ABY

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: $A+1 = 1$; $A \cdot 0 = 0$
  • Idempotent: $A+A = A$; $A \cdot A = A$
  • Complement: $A+\overline{A} = 1$; $A \cdot \overline{A} = 0$
  • Involution: $\overline{\overline{A}} = A$
  • Absorption: $A+(A \cdot B) = A$; $A \cdot (A+B) = A$

De Morgan's Laws:

  1. $\overline{A+B} = \overline{A} \cdot \overline{B}$
  2. $\overline{A \cdot B} = \overline{A} + \overline{B}$

Interactive Simplification Example:

Original Expression:

Simplified Expression:

Simplify with Karnaugh Maps (K-maps)

A graphical method for simplifying Boolean expressions.

Click cells to cycle through 0, 1, X (don't care).

Simplified Expression (Sum of Products):

Enter values and click "Simplify K-map".

Y = ?

Universal Gates (NAND and NOR)

NAND and NOR gates can implement any other logic gate.

NAND as Universal Gate

  • NOT: $ \overline{A} = \overline{A \cdot A} $
    NOT from NAND
  • AND: $ A \cdot B = \overline{\overline{A \cdot B}} $
    AND from NAND
  • OR: $ A+B = \overline{\overline{A} \cdot \overline{B}} $
    OR from NAND

NOR as Universal Gate

  • NOT: $ \overline{A} = \overline{A+A} $
    NOT from NOR
  • OR: $ A+B = \overline{\overline{A+B}} $
    OR from NOR
  • AND: $ A \cdot B = \overline{\overline{A} + \overline{B}} $
    AND from NOR

Devising Logic Systems from Boolean Expressions

Translate Boolean expressions into circuit diagrams.

Example: $Y = (A \cdot B) + \overline{C}$

Logic circuit for Y = (A.B) + C'

Using only NANDs for $Y = (A \cdot B) + \overline{C}$ (which is $\overline{ (\overline{A \cdot B}) \cdot C }$ ):

NAND Logic circuit for Y = (A.B) + C'

Logic Circuit Problem Solver

Select a problem to see its expression and corresponding logic circuit diagrams.

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