Logic Gates with Boolean Functions

Signals & Logic Circuits

• Digital systems use two discrete signal levels: $0$ (LOW / OFF) and $1$ (HIGH / ON)
• Logic circuit = electronic circuit that makes decisions based on binary inputs
• Two classes:
  • Basic gates (AND, OR, NOT)
  • Combinational gates (built from basics, e.g., NOR, NAND)

Basic Logic Gates

AND Gate

• Boolean: $Q = A \cdot B$ (read “A AND B”)
• Output $1$ only when all inputs $1$; otherwise $0$
• With $n$ inputs: $Q = 1$ only if every input $=1$

OR Gate

• Boolean: $Q = A + B$ (read “A OR B”)
• Output $1$ when at least one input $1$; $0$ only if all inputs $0$

NOT Gate

• Boolean: $Q = \overline{A}$
• Inverts the input: $0 \rightarrow 1$, $1 \rightarrow 0$

Combinational Gates

NOR Gate (NOT-OR)

• Structure: OR gate → NOT gate
• Boolean: $Q = \overline{A + B}$
• Output $1$ only when all inputs $0$

NAND Gate (NOT-AND)

• Structure: AND gate → NOT gate
• Boolean: $Q = \overline{A \cdot B}$
• Output $0$ only when all inputs $1$

Designing Circuits & Boolean Expressions

• Convert algebra → gates by reading operators left-to-right
  • Example: $Q = A \cdot (A + B) + B$
  1. OR: $A + B$
  2. AND: $A \cdot (A + B)$
  3. OR with $B$ for final $Q$
• Number of rows in a truth table = $2^n$ ( $n$ = number of inputs )

Truth Tables (Quick Reference)

• AND: only $1$ row = all $1$ inputs
• OR: only $0$ row = all $0$ inputs
• NOT: single input inversion
• NAND: inverse of AND table
• NOR: inverse of OR table

Integrated Circuits (ICs)

• IC = miniaturised network of transistors / resistors / diodes performing a defined logic function
• Logic gates packaged in ICs (e.g., 7408 → 4 × AND gates, 7432 → 4 × OR gates)
• Pin groups: inputs on outer pins, outputs on dedicated pins

Practical Applications

• Home alarm: multiple window/door sensors → OR gate → siren (alarm = $1$ if any sensor $1$)
• Street-light controller: Manual switch OR (Timer AND Dark-sensor)
• Vehicle security: NOT(Engine) AND (Movement OR Shutter-damage) → alarm

Quick Facts

• $\text{AND}$ → $\cdot$
• $\text{OR}$ → $+$
• $\text{NOT}$ → overbar / prime
• $\text{NAND}$, $\text{NOR}$ are universal: any logic circuit can be built solely from NAND or NOR gates