Boolean Algebra And Logic Circuits (Copy)

15.2.1 Boolean Algebra

• Boolean algebra, named after George Boole, is a mathematical approach to dealing with logic operations.
• It operates on two fundamental values:
  • TRUE (1)
  • FALSE (0)
• Boolean algebra is crucial in representing and simplifying logic circuits.
• Used extensively in computer science, digital electronics, and circuit design.

Basic Boolean Operations

• AND (⋅) Operation
  • Symbolized as A ⋅ B or AB.
  • Output is 1 only if both inputs are 1.
  • Truth table:

    A | B | Output (A ⋅ B)
    --|--|---------
    0 | 0 | 0
    0 | 1 | 0
    1 | 0 | 0
    1 | 1 | 1
    

• OR (+) Operation
  • Symbolized as A + B.
  • Output is 1 if at least one input is 1.
  • Truth table:

    A | B | Output (A + B)
    --|--|---------
    0 | 0 | 0
    0 | 1 | 1
    1 | 0 | 1
    1 | 1 | 1
    

• NOT (¬) Operation
  • Inverts the input.
  • Represented as A' or ¬A.
  • Truth table:

    A | Output (¬A)
    --|---------
    0 | 1
    1 | 0
    

Boolean Laws and Rules

Commutative Laws

• A + B = B + A (OR is commutative)
• A ⋅ B = B ⋅ A (AND is commutative)

Associative Laws

• A + (B + C) = (A + B) + C
• A ⋅ (B ⋅ C) = (A ⋅ B) ⋅ C

Distributive Laws

• A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C)
• (A + B) ⋅ (A + C) = A + (B ⋅ C)

Identity Laws

• A + 0 = A, A ⋅ 1 = A
• A + 1 = 1, A ⋅ 0 = 0

Idempotent Laws

• A + A = A
• A ⋅ A = A

Inverse Laws

• A + A’ = 1
• A ⋅ A’ = 0

Absorption Laws

• A + (A ⋅ B) = A
• A ⋅ (A + B) = A

De Morgan’s Theorems

• (A ⋅ B)’ = A’ + B’
• (A + B)’ = A’ ⋅ B’

15.2.2 Logic Circuits

• Logic circuits are composed of logic gates to process digital signals.
• Half-Adder Circuit:
  • Adds two binary digits producing sum and carry.
  • Uses XOR for sum and AND for carry.
• Full-Adder Circuit:
  • Extends the half-adder to three inputs (A, B, Carry-in).
  • Uses multiple XOR, AND, and OR gates.

15.2.3 Flip-Flop Circuits

• Flip-flops are sequential circuits that store binary states.
• Types of Flip-Flops:
  • SR (Set-Reset) Flip-Flop
  • JK Flip-Flop (resolves invalid states of SR Flip-Flop)
  • D (Data) Flip-Flop
  • T (Toggle) Flip-Flop (used in counters)

15.2.4 Boolean Algebra and Logic Circuits

• Boolean expressions represent logic circuits.
• Example:
  • Given logic circuit:
    • Stage 1: A AND B
    • Stage 2: B OR C
    • Stage 3: (A AND B) OR (B OR C)
    • Stage 4: A OR (NOT C)
    • Stage 5: ((A AND B) OR (B OR C)) AND (A OR (NOT C))
  • Boolean expression: ((A ⋅ B) + (B + C)) ⋅ (A + C’)

15.2.5 Karnaugh Maps (K-Maps)

• A graphical method to simplify Boolean expressions.
• Uses Gray codes to ensure only one bit changes between adjacent values.
• Rules for K-Maps:
  • Only 1s in truth tables are considered.
  • Adjacent 1s are grouped in powers of 2 (1, 2, 4, 8, …).
  • Groups must be as large as possible.

Examples

1. Simplify (A ⋅ B) + (A ⋅ C) + (B ⋅ C) using K-Map
   • Solution: A + B + C
2. Find simplified Boolean expression for a given circuit
   • Given: A ⋅ B + A ⋅ C + B ⋅ C
   • Using K-Map, simplified to A + B + C.