Lecture on Binary Numbers and Logic Gates

Introduction to Binary Numbers

• Binary Numbers: Comprises two possibilities, 0 or 1.
• Circuit Representation:
  • 0 = Circuit Off
  • 1 = Circuit On
• True/False Statements:
  • Off = False
  • On = True
• Voltage Representation:
  • Off = 0 volts
  • On = Typically 5 volts

The Buffer Gate

• Symbol: Triangle pointing right.
• Truth Table:
  • Input 1 -> Output 1
  • Input 0 -> Output 0
• Circuit Representation: Utilizes NPN transistor, power source, and LED.
  • NPN Transistor: Base, collector, emitter.
  • LED represents the output:
    • Input On = LED On
    • Input Off = LED Off

The NOT Gate

• Symbol: Similar to buffer gate with a circle.
• Truth Table:
  • Input 0 -> Output 1
  • Input 1 -> Output 0
• Circuit Representation:
  • Similar to buffer but location of LED differs.
  • LED connected across collector and emitter.
  • Complementary relationship between input and output.

The AND Gate

• Symbol: Two inputs A, B.
• Truth Table:
  • Both A and B On = Output On
  • Any other state = Output Off
• Circuit Representation:
  • Requires two transistors in series.
  • Current flows only if both transistors are On.

The OR Gate

• Symbol: Inputs A, B.
• Truth Table:
  • Either A or B On = Output On
  • Both A and B Off = Output Off
• Circuit Representation:
  • Requires two transistors in parallel.
  • Current can flow if either transistor is On.

The NAND Gate

• Complement of AND Gate: Symbol like AND with a circle.
• Truth Table:
  • Both A and B On = Output Off
  • Any other state = Output On
• Constructing a NAND Gate:
  • Combination of AND and NOT gates.
  • Input A = A × A complement using Boolean algebra.

The NOR Gate

• Complement of OR Gate: Symbol like OR with a circle.
• Truth Table:
  • Either A or B On = Output Off
  • Both A and B Off = Output On

Writing Functions from Block Diagrams

• Identify logic gates: AND = Multiplication, OR = Addition.
• Construct functions step-by-step from diagrams.

Drawing Block Diagrams from Functions

• SOP (Sum of Products):
  • Use AND gates for products, OR gates for sums.
• POS (Product of Sums):
  • Use OR gates for sums, AND gates for products.

Boolean Algebra Rules

• Commutative Property: A + B = B + A, A × B = B × A
• Associative Property: A + (B + C) = (A + B) + C
• Identity Rule: A + 0 = A, A × 1 = A
• Null Property: A + 1 = 1, A × 0 = 0
• Complement: A + A' = 1, A × A' = 0

Additional Concepts

• Literal: Single variable in an expression.
• Sum/Product Terms: Combinations of literals under addition/multiplication.
• Min/Max Terms: Standard terms that include all variables.

By understanding these fundamental concepts, students can design and interpret logic circuits and their operations effectively.