Understanding Boolean Logic

-

 Understanding Boolean Logic

Before we can build logic gates or design digital systems, we need to understand the language that describes how they work, Boolean logic.

What Is Boolean Logic?

Boolean logic is a type of algebra that deals with true or false values. Represented as 1 (true/high) and 0 (false/low).
It’s the foundation of all digital electronics, from simple gates to full CPUs.
Every digital circuit you’ll ever build using 74LS chips, microcontrollers, or even a 6502 ultimately follows Boolean rules.

Think of Boolean logic as mathematics for truth, where instead of adding numbers, we combine logical conditions.

The Basic Boolean Operators

These three operators form the basis of Boolean algebra.

Every logic gate, circuit, and even program condition can be expressed using these symbols.

Truth Tables

A truth table lists all possible combinations of inputs and the resulting output.
They’re one of the most useful tools when designing or testing logic.

AND Gate Example

OR Gate Example

NOT Gate Example

Combining Boolean Expressions

You can combine logic operations just like you do in algebra:

  • (A·B) + C → Output is true if both A and B are true, or if C is true.

  • ¬(A + B) → Output is true only when both A and B are false.

  • (A + B)·C → Output is true if C is true and at least one of A or B is true.

When written in Boolean notation, brackets control operation order just like in normal algebra.

Boolean Laws and Rules

The following table shows Boolean identities which make simplifying expressions easier:


These rules let you reduce large expressions into simpler ones, which in turn means fewer logic gates in hardware.

Logic in Circuits

In real circuits:

  • Logic 1 (HIGH) → about +5 V for TTL logic.

  • Logic 0 (LOW) → around 0 V.

Each logic chip (like the 74LS08 AND, 74LS32 OR, or 74LS04 NOT) performs one of the Boolean operations above.
You can wire up inputs with switches or jumpers and display outputs with LEDs to see Boolean logic in action.

Simplifying Logic

When you design larger circuits, expressions can grow quickly.
Using Boolean algebra, De Morgan’s Theorems, or Karnaugh Maps (K-maps), you can simplify them down to the smallest set of gates that does the same job.

That’s how digital designers save cost, space, and power.

Try This Yourself

  1. Grab a 74LS08, 74LS32, and 74LS04.

  2. Build AND, OR, and NOT gates on a breadboard.

  3. Create simple truth tables and verify each one with LEDs.

  4. Then combine them to form small circuits such as half or full adders

Comments

Post a Comment

Popular posts from this blog

Math Behind Logic Gates

-
Image
Understanding the Real Electronics Behind Logic Gates (The Practical Math Behind Digital Circuits) When we build circuits with logic chips like the 74LS series, it’s easy to think only in 1s and 0s. But under the hood, every logic gate follows the same rules as any other circuit, Ohm’s Law, current limits, and voltage thresholds. So yes, there is real-world math behind those blinking LEDs and logic tables.  1. Logic Levels — The Digital View At the logic level, everything is simple: 0 V → LOW (logic 0) 5 V → HIGH (logic 1) We don’t care about exact voltages here just whether a signal crosses a defined threshold. For TTL (Transistor-Transistor Logic) chips like the 74LS86 XOR gate , these thresholds are: This means a logic gate “decides” based on whether the voltage is above or below that boundary, which is how binary logic becomes real voltage levels. 2. Inside the Circuit — Real Electrical Math Even though logic circuits process bits, they still obey Ohm’s Law ...
Read more

6502 - Part 2 Reset and Clock Circuit

-
Image
6502 Retro Console Part 2 Add a Reset and Clock Circuit In Part 1 we powered up the 6502 and watched it running a NOP test with just the CPU and LEDs on the address bus. Now we’ll give it a clock and a clean start every time you power on, rather than relying on a function generator. In this post we will cover a simple 555 Timer Clock circuit and reset circuit. 555 Timer The 6502 needs a steady pulse on its Φ0 (pin 37) to step through instructions. A NE555 timer is perfect for low-speed testing reliable, simple, and easy to wire. Parts List NE555 timer 1 kΩ resistor 10 kΩ resistor 100 nF capacitor (timing) 0.1 µF capacitor (decoupling) Breadboard + jumper wires Circuit Diagram How to Wire It Add a 0.1 µF decoupling cap across the +5V and GND rails stability. Typical speed: ≈ 7–10 kHz perfect for a visible NOP test. Use larger resistors or capacitors to slow it down. Tip: Swap the timing capacitor for 1 µF and the 10 kΩ for 100 kΩ to step slowly enough ...
Read more

Building a 6502 NOP Test

-
Image
 The Simplest Working 8-bit CPU Circuit Before we start building a full retro console, let’s do something beautifully simple get a 6502 CPU to run,  no RAM, no ROM, no glue logic, just the chip, a few LEDs, and a clock. This is called a NOP test (No Operation). It’s the digital equivalent of “does it breathe?” What You’ll Need Part Purpose MOS 6502 CPU The heart of our system Bench power supply (5 V) Power for the CPU Function generator Clock source 8 × LEDs To watch address lines change 8 × 470 Ω resistors Current-limiters for the LEDs 1 × 0.1 µF capacitor Decoupling between VCC and GND Breadboard and jumper wires For connections MOS 6502 Cpu Pin Out Check your IC and its datasheet to ensure that you have the correct pin out map as the steps below may vary slightly. Step 1 — Power and Decoupling Give the 6502 a clean 5 V supply: VCC (pin 8) → +5 V VSS (pins 1 and 21) → Ground Place the 0.1 µF capacitor between VCC and VSS close to the chip. That tiny capacitor...
Read more