Boolean Algebra and Number Systems

This syllabus is designed as a 10-week foundational "Bridge" course. It moves from pure numerical theory into the abstract logic that governs both hardware (NAND gates) and software (Assembly/C bitwise operations).

All resources listed are Creative Commons (CC) or Open-Source.

Course Syllabus: Foundations of Digital Logic & Boolean Algebra

Primary Resources:

1. [Kuphaldt] Lessons In Electric Circuits, Vol IV (Digital) – Tony R. Kuphaldt (CC BY).

2. [Kahn] Digital Circuit Projects – Stephen Kahn/Charles Kann, LibreTexts (CC BY).

3. [MIT] 6.004 Computation Structures – MIT OpenCourseWare (CC BY-NC-SA).

4. [OpenStax] Introduction to Computer Science – (CC BY 4.0).

Phase 1: The Language of Bits (Weeks 1–2)

Goal: Mastery of number systems and data representation.

• Week 1: Binary, Octal, and Hexadecimal.

  • Topics: Positional notation, base conversion (Dec $\leftrightarrow$ Bin $\leftrightarrow$ Hex), and "Why Hex?" (visualizing 4-bit nibbles).

  • Reading: [Kuphaldt] Chapter 1: Numeration Systems.

  • Hook: Discuss how Hex is used in memory addresses and C-style 0x notation.

• Week 2: Binary Arithmetic.

  • Topics: Addition, subtraction (Two's Complement), and overflow.

  • Reading: [Kuphaldt] Chapter 2: Binary Arithmetic.

  • Reading: [OpenStax CS] Section 3.1: Data Representation.

Phase 2: The Building Blocks (Weeks 3–4)

Goal: Moving from numbers to logical "decisions."

• Week 3: Logic Gates (The Physical "Why").

  • Topics: NOT, AND, OR, NAND, NOR, XOR, and XNOR.

  • Reading: [Kahn] Chapter 4: Gates.

  • Reading: [Kuphaldt] Chapter 3: Logic Gates.

• Week 4: Truth Tables & Universal Gates.

  • Topics: Building complex tables; proving that NAND/NOR can build any other gate.

  • Reading: [MIT 6.004] Tutorial 4: Gates and Boolean Equations.

  • Hook: Explain that modern CPUs are essentially billions of NAND gates "glued" together.

Phase 3: Boolean Algebra & Simplification (Weeks 5–7)

Goal: Mathematical optimization of logic.

• Week 5: Laws of Boolean Algebra.

  • Topics: Identities, Commutative/Associative/Distributive laws, and De Morgan’s Theorems.

  • Reading: [Kuphaldt] Chapter 7: Boolean Algebra.

• Week 6: Algebraic Simplification.

  • Topics: Reducing complex expressions to Sum of Products (SOP) and Product of Sums (POS).

  • Exercise: [MIT 6.004] Problem Set 1 (Logic Section).

• Week 7: Karnaugh Mapping (K-Maps).

  • Topics: Visual simplification, 2/3/4-variable maps, and "Don't Care" conditions.

  • Reading: [Kuphaldt] Chapter 8: Karnaugh Mapping.

Phase 4: Combinational Foundations (Weeks 8–9)

Goal: Using logic to perform specific functions.

• Week 8: Adders & Multiplexers.

  • Topics: Half-adders, Full-adders, and using MUXs to implement Boolean functions.

  • Reading: [Kahn] Chapter 6: Adders & Chapter 8: Multiplexers.

• Week 9: Decoders and Encoders.

  • Topics: 7-segment displays and address decoding.

  • Reading: [Kahn] Chapter 7: Decoders.

  • Hook: How the CPU "chooses" which memory address to read.

Phase 5: The Software Bridge (Week 10)

Goal: Explicitly connecting logic to the upcoming Assembly/C courses.

• Week 10: Bitwise Operations & Masking.

  • Topics: Bitwise AND (&), OR (|), XOR (^), and NOT (~) in C. Setting, clearing, and toggling bits.

  • Resource: [OpenStax CS] Section 4.3: Boolean Operations.

  • Assembly Hook: Look at the FLAGS register in x86/ARM and how Boolean logic determines "Jump if Zero" (JZ) or "Jump if Carry" (JC).

Recommended "Theory" Mapping Table

Course Concept and a Practical Assembly/C "Hook"

Two's Complement

• Understanding why signed integers wrap around to negative numbers.

Bitwise AND

• Masking: Extracting specific bits from a status register (e.g., if (reg & 0x01)).

Bitwise OR

• Setting Bits: Turning on a specific hardware peripheral without affecting others.

Bitwise XOR

• Toggling Bits: Switching an LED state or simple encryption/obfuscation.

De Morgan's Laws

• Optimizing if statements in C (e.g., !(A && B) is the same as `!A

Instructional Tip: Since students will later learn about Assembly, emphasize Week 10 heavily. Showing them how an AND gate in hardware is identical to a & operation in C is the "lightbulb moment" that connects the two worlds.