Digital vs Analog

In digital systems, we work with only two values:

• 0 (LOW, False, Ground, 0V)
• 1 (HIGH, True, VDD, 1.8V/3.3V/5V)

This simplicity makes digital circuits reliable and easy to design.

Basic Logic Gates

NOT Gate (Inverter)

Flips the input. 0 becomes 1, 1 becomes 0.

AND Gate

Output is 1 only when ALL inputs are 1.

OR Gate

Output is 1 when ANY input is 1.

XOR Gate (Exclusive OR)

Output is 1 when inputs are DIFFERENT.

NAND and NOR Gates

These are "universal gates" - you can build any circuit using only NAND (or only NOR).

Gate	Function	Output
NAND	NOT + AND	Y = (A · B)'
NOR	NOT + OR	Y = (A + B)'

Boolean Algebra

Rules for simplifying logic expressions:

Basic Laws

Identity:     A + 0 = A       A · 1 = A
Null:         A + 1 = 1       A · 0 = 0
Idempotent:   A + A = A       A · A = A
Complement:   A + A' = 1      A · A' = 0
Involution:   (A')' = A

Commutative, Associative, Distributive

Commutative:  A + B = B + A           A · B = B · A
Associative:  (A+B)+C = A+(B+C)       (A·B)·C = A·(B·C)
Distributive: A·(B+C) = A·B + A·C     A+(B·C) = (A+B)·(A+C)

De Morgan's Theorem

Very important for logic simplification!

(A · B)' = A' + B'    (Break the bar, change the sign)
(A + B)' = A' · B'

Karnaugh Maps (K-Maps)

Visual method to simplify Boolean expressions.

2-Variable K-Map

4-Variable K-Map

Number Systems

System	Base	Digits	Example
Binary	2	0, 1	1010₂ = 10
Octal	8	0-7	12₈ = 10
Decimal	10	0-9	10₁₀ = 10
Hex	16	0-9, A-F	A₁₆ = 10

Conversions

Binary to Decimal:
  1010₂ = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10

Decimal to Binary (divide by 2):
  10 ÷ 2 = 5 remainder 0
  5  ÷ 2 = 2 remainder 1
  2  ÷ 2 = 1 remainder 0
  1  ÷ 2 = 0 remainder 1
  Read remainders bottom-up: 1010₂

Hex to Binary (each hex digit = 4 bits):
  A₁₆ = 1010₂
  F₁₆ = 1111₂
  3₁₆ = 0011₂

Binary Arithmetic

Addition

  0 + 0 = 0
  0 + 1 = 1
  1 + 0 = 1
  1 + 1 = 10 (0 with carry 1)

Example:
    1011  (11)
  + 0110  (6)
  ------
   10001  (17)

2's Complement (for Negative Numbers)

To get -N from N:
1. Flip all bits (1's complement)
2. Add 1

Example: -5 in 4-bit
  5 = 0101
  Flip: 1010
  Add 1: 1011 = -5

Why? So that 5 + (-5) = 0:
  0101 + 1011 = 10000 (overflow ignored) = 0000 ✓