Digital Basics

• Computers use digital electronics with only two voltage levels: high (1 / true / asserted) and low (0 / false / deasserted).
• 0 and 1 are complements/inverses.
• Binary numbers naturally match digital electronics.
• Pitfall: sampling a signal while it’s transitioning between high/low can cause errors.

Combinational vs. Sequential Logic

• Combinational logic: no memory; output depends only on current inputs.
• Sequential logic: has memory (state); output depends on inputs + stored state.

Truth Tables

• Fully describe any combinational logic function by listing outputs for every possible input combination.
• For n inputs → 2ⁿ rows.
• Example:
  • D = true if at least one input true.
  • E = true if exactly two inputs true.
  • F = true only if all three inputs true.
  • Eight-row truth table (for A, B, C) lists outputs D, E, F.
• Often only the “1” (true) output rows are listed for shorthand.

Boolean Algebra

• Variables take only 0 or 1.
• Operators:
  • OR (+): 1 if either input is 1.
  • AND (·): 1 only if both inputs are 1.
  • NOT (overbar or ‘): inverts input (0 becomes 1 and 1 becomes 0)
• Useful laws (to simplify equations):
  • Identity:
    • A + 0 = A
    • A·1 = A
  • Zero/One:
    • A + 1 = 1
    • A·0 = 0
  • Inverse:
    • A + A' = 1
    • A·A' = 0
  • Commutative:
    • A + B = B + A
    • A·B = B·A
  • Associative:
    • (A + B) + C = A + (B + C)
  • Distributive:
    • A·(B + C) = A·B + A·C
• DeMorgan’s laws (important):
  • (A·B)' = A' + B'
  • (A + B)' = A'·B'

Logic Equations Example (D, E, F):

• D = A + B + C (OR of inputs)