⚡ Module 3: Boolean Algebra
Duration: Weeks 8-10
📚 Core Topics
Boolean Functions (Week 8)
- Truth Tables: Complete function specification
- Basic Operations: AND (∧), OR (∨), NOT (¬), XOR (⊕), NAND (↑), NOR (↓)
- Boolean Laws: Commutative, associative, distributive, De Morgan’s
- Duality Principle: Swapping ∧↔∨ and 0↔1
- Normal Forms:
- DNF (Disjunctive): Sum of products
- CNF (Conjunctive): Product of sums
- Perfect/Canonical forms
Digital Circuits (Week 9)
- Logic Gates: Physical implementation of Boolean operations
- Circuit Design: From truth table to circuit
- Analysis: From circuit to Boolean expression
- Multi-level Circuits: Optimization and complexity
- Functional Completeness: Minimal operation sets
- Universal Gates: NAND and NOR alone suffice
Minimization (Week 10)
- Karnaugh Maps (K-maps): Visual minimization (2-4 variables)
- Don’t Care Conditions: Flexible outputs for optimization
- Quine-McCluskey: Algorithmic minimization (any number of variables)
- Prime Implicants: Essential and non-essential
🔑 Key Concepts
| Concept | Definition | Example |
|---|---|---|
| Boolean Function | f: {0,1}ⁿ → {0,1} | f(x,y) = x ∧ ¬y |
| DNF | Sum of products (OR of ANDs) | (x∧y) ∨ (¬x∧z) |
| CNF | Product of sums (AND of ORs) | (x∨y) ∧ (¬x∨z) |
| Functionally Complete | Can express any Boolean function | {∧,∨,¬}, {NAND}, {NOR} |
💡 De Morgan’s Laws: ¬(x ∧ y) = ¬x ∨ ¬y ¬(x ∨ y) = ¬x ∧ ¬y
💡 Applications
Where you’ll use this:
- 💻 Digital circuit design and hardware
- 🖥️ Computer architecture (ALU, CPU design)
- 🧩 SAT solvers and automated reasoning
✅ Learning Outcomes
By the end of this module, you will be able to:
- Construct and analyze truth tables for Boolean expressions
- Convert between DNF, CNF, and arbitrary Boolean forms
- Simplify expressions using Boolean laws and duality
- Design and analyze digital circuits using logic gates
- Minimize Boolean functions with Karnaugh maps
- Prove sets of operations are functionally complete