Boolean Algebra Theorems – Simplifying Logic Made Easy!

Boolean algebra uses a set of rules and theorems to manipulate logical expressions. These rules are the foundation of digital circuit design and simplification.


1. Identity Theorems

  • A + 0 = A

  • A · 1 = A

These mean that OR-ing with 0 or AND-ing with 1 doesn’t change the value.

2. Null Theorems

  • A + 1 = 1

  • A · 0 = 0

OR-ing anything with 1 always gives 1; AND-ing with 0 gives 0.

3. Idempotent Theorems

  • A + A = A

  • A · A = A

Repeated use of the same variable doesn’t change the result.

4. Complement Theorems

  • A + A’ = 1

  • A · A’ = 0

A variable OR-ed with its complement is always 1; AND-ed is always 0.

5. Involution Law

  • (A’)’ = A

Double negation brings back the original value.

6. Commutative Laws

  • A + B = B + A

  • A · B = B · A

Order doesn’t matter for OR and AND operations.

7. Associative Laws

  • A + (B + C) = (A + B) + C

  • A · (B · C) = (A · B) · C

Grouping doesn’t affect the result of OR or AND.

8. Distributive Laws

  • A · (B + C) = A·B + A·C

  • A + (B · C) = (A + B) · (A + C)

These help in expanding or factoring expressions.

9. Absorption Theorems

  • A + A·B = A

  • A · (A + B) = A

These remove redundant terms to simplify expressions.

10. DeMorgan’s Theorems

  • (A · B)’ = A’ + B’

  • (A + B)’ = A’ · B’

Important for converting between AND/OR and NAND/NOR.

Practical Use

These theorems are widely used in:

  • Simplifying logic circuits

  • Reducing gate count

  • Designing combinational logic (e.g., multiplexers, decoders)

  • Optimizing software logic (especially in embedded systems)

Stay tuned to hobitronics.blog

To know more about Logic Gates: Understanding Logic Gates

Comments

Post a Comment

Popular posts from this blog

How MOSFETs Work with Arduino: PWM Control, Applications, and Pro Tips

Image
If PWM is the brain signal, the MOSFET is the muscle that makes it move. From your Arduino's motor shield to your laptop charger and even your electric vehicle's inverter — MOSFETs silently power modern electronics with unmatched speed and precision. In this blog, we’ll break down how MOSFETs work, where they’re used, and how they bring digital control to life — especially when combined with PWM. In this blog, we’ll break down: What a MOSFET is How it works with PWM Real-world uses (motors, LEDs, inverters, etc.) Pro design tips for safe and effective usage 1. What is a MOSFET? MOSFET stands for Metal-Oxide-Semiconductor Field-Effect Transistor .   It’s a type of transistor widely used for switching and amplification , especially in high-speed, high-efficiency applications . Key Roles: Acts like a gate-controlled switch — voltage on the Gate decides whether current flows. Used to control large loads using small digital signals (e.g., from microcontr...
Read more

Controlling RGB LEDs with PWM Using Arduino

Image
P ulse Width Modulation (PWM) is not just a feature of microcontrollers — it’s a foundational technique in electronics. From dimming LEDs to simulating analog voltage levels, PWM is used in display systems , communication protocols , and motor drivers.  In this article, we’ll explore how PWM works technically and apply it to a fun and colorful project: controlling an RGB LED using an Arduino . Whether you’re an engineering student or a DIY hobbyist, this simple RGB LED demo will give you a hands-on feel for what PWM really means — both in concept and application. What is a Common Cathode RGB LED? An RGB LED combines three LEDs — Red, Green, and Blue — into a single package. In a common cathode RGB LED , all three internal LEDs share one ground (GND) pin. The remaining three pins individually control the red, green, and blue segments. By varying the amount of current (or rather, the voltage drop) across each color pin, you can create a wide spectrum of colors through additive...
Read more

Why Does My Old Phone Charge Slowly But Heat Up More?

Image
The Truth About Aging Batteries Welcome back to Hobitronics — where we decode the daily mysteries inside your gadgets. Today we’re getting into something everyone with a 3-year-old smartphone has asked themselves: "Why is my phone charging so slow? And why does it feel like it’s cooking itself while doing it?" You’re not imagining it. Aging phones really do charge slower and heat up more — and the reason lies deep inside the battery chemistry , internal resistance , and the invisible wear and tear of thousands of charging cycles. 1. Your Battery Ages — Even If It Looks Fine Smartphones use Li-ion or Li-Po batteries . These are fantastic when new, but over time: Electrochemical reactions degrade the anode/cathode SEI (Solid Electrolyte Interface) layer thickens Lithium plating reduces active ions Capacity drops with every charge-discharge cycle This degradation results in something very important: higher internal resistance . 2. Internal Resistance: Th...
Read more

ASK, FSK & PSK Explained

Image
In the world of modern communication—whether you're streaming videos, using Wi-Fi, or sending data over Bluetooth— digital modulation plays a critical role. It's the art of taking your binary data (0s and 1s) and efficiently converting it into analog waveforms so it can travel across airwaves or wires. Among the various digital modulation techniques, ASK (Amplitude Shift Keying), FSK (Frequency Shift Keying), and PSK (Phase Shift Keying) form the foundational trio. Understanding these will unlock your grasp of how digital signals are transmitted reliably and efficiently. Let’s explore each of these in detail: 🔷 1. Amplitude Shift Keying (ASK) ➤ What is ASK? Amplitude Shift Keying works by varying the amplitude of a carrier wave to represent binary data. Bit 1 is transmitted using a carrier wave of fixed amplitude. Bit 0 is transmitted by turning the carrier OFF (zero amplitude) or using a smaller amplitude. This is often called On-Off Keying (OOK) . ➤ Wavef...
Read more

Delta Modulation and Adaptive Delta Modulation: Simplifying Digital Voice Communication

Image
In the world of digital communication, Pulse Code Modulation (PCM) paved the way for digitizing analog signals. However, PCM isn't always efficient in terms of bandwidth and data rate. To address these limitations, two intelligent techniques were developed: Delta Modulation (DM) and its improved version, Adaptive Delta Modulation (ADM) . In this blog, we’ll take a deep dive into how Delta Modulation works, its advantages and limitations, and how Adaptive Delta Modulation overcomes those challenges. So, let’s decode this modulation magic! 🔁 Recap: What Came Before — PCM Before we explore DM and ADM, it’s essential to remember what PCM  (ref:  Pulse Code Modulation (PCM): The Digital Backbone of Modern Communication ) does: Samples the analog signal at regular intervals (Nyquist Rate). Quantizes the samples into discrete values. Encodes these values into binary for transmission. While PCM is powerful, it requires a high bit rate and complex quantization , makin...
Read more

Pulse Code Modulation (PCM): The Digital Backbone of Modern Communication

Image
Introduction: Why PCM Matters In the era of smartphones, digital streaming, and cloud-based communication, it's easy to forget that all digital information starts out analog. Whether it’s your voice, your favorite song, or a movie’s soundtrack, sound begins as a continuously varying signal. But to store, transmit, or process it digitally, we need to convert it into a digital form—and that's where Pulse Code Modulation (PCM) comes in. PCM is the foundation of digital audio , and it plays a crucial role in telecommunications, media storage, and even space communication. Let’s dive deep into how it works, why it’s important, and where it’s used. What is PCM? Pulse Code Modulation (PCM) is a method used to digitally represent analog signals. It involves sampling , quantizing , and encoding an analog input into a binary form suitable for transmission or storage. Think of PCM as translating a wave of sound into a digital language that machines can understand. The Three Core P...
Read more

Understanding Masking, Enabling, and Disabling of Logic Gates

Image
In digital electronics, logic gates are the basic building blocks of all digital systems. But did you know that we can control whether a gate actively participates in a circuit using masking , enabling , and disabling techniques? These are crucial concepts, especially when working with multiplexers , tri-state buffers , and control logic . Let’s break it down in simple terms. What is Masking in Logic Gates? Masking is a method of controlling the output of a logic gate by using a specific input value (often 0 or 1) to “mask” or block the effect of another input. It's widely used in data handling , interrupt systems , and bitwise operations . Example: AND Gate Masking Let A = input signal Let B = mask Output = A AND B If B = 1 → Output = A (no masking) If B = 0 → Output = 0 (signal is masked) Example: OR Gate Masking Let A = input signal Let B = mask Output = A OR B If B = 0 → Output = A (no masking) If B = 1 → Output = 1 (signal is masked) This is especia...
Read more

Sign Bit Extension Explained with Examples

Image
Understanding how numbers are stored and manipulated in binary is crucial in electronics, embedded systems, and computer architecture. One important concept in this area is Sign Bit Extension —especially when dealing with signed numbers and different bit widths. In this blog, we’ll break it down in a simple way with clear examples to help you grasp it fully. What is Sign Bit Extension? Sign Bit Extension (also called Sign Extension ) is the process of increasing the number of bits of a binary number while preserving its sign (positive or negative). This is commonly used when converting a signed number from a smaller bit size (like 8 bits) to a larger one (like 16 bits or 32 bits). Rule : If the number is positive (MSB is 0), pad with 0s on the left. If the number is negative (MSB is 1), pad with 1s on the left. This keeps the value the same even in the larger bit size. Why is Sign Bit Extension Needed? In embedded systems and low-level programming (like Arduino, microcontr...
Read more