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Back to Basics - Boolean Algebra

One of the things I have been wanting to do for a while is create a series of blogs which cover the basics of digital logic design, starting from the beginning.

In this first instalment I will be focusing on the history of Boolean algebra and how it relates to our digital world.

At the heart of every digital circuit lies a simple yet profound mathematical system: Boolean Algebra. This cornerstone of digital logic design, first formulated by George Boole in his seminal 1854 work, "An investigation of the laws of thought," provides the essential framework for understanding the operations of modern logic design and synthesis engines.

George Boole's exploration of logic and probability paved the way for Claude Shannon's 1937 master's thesis at MIT, "A Symbolic Analysis of Relay and Switching Circuits". Shannon's work demonstrated that Boolean algebra could be applied to switching circuits, establishing a fundamental relationship that underpins all digital technology.

The Core Operations: Addition and Multiplication in Boolean Context

In Boolean algebra, we deal with two primary operations:

  • Boolean Addition, where the SUM is 1 if one or more operands are 1

0 + 0 = 0 1 + 0 = 1

0 + 1 = 1 1 + 1 = 1

0 = 0 1 + 0 = 0 + 1 = 1 1 + 1 = 1

  • Boolean Multiplication, yields PRODUCT of 1 when both operands are 1

0 * 0 = 0 0 * 1 = 0

1 * 0 = 0 1 * 1 = 1

Understanding these operations is critical when we move to the practical application of designing logic gates and complex digital circuits, as we will see in later instalments.

Key Laws of Boolean Algebra

To streamline our digital designs and minimize the use of logic gates, it's crucial to master several laws of Boolean algebra:

  • Commutative Law: Order doesn't matter

A + B = B + A

  • Associative Law: Grouping doesn't change the outcome

(A + (B + C) = (A + B) + C)

  • Distributive Law: Distributes multiplication over addition

A(B + C) = AB + AC

Each of these laws offers advantages in optimizing circuit designs, reducing complexity, and enhancing performance.

Along with these laws there are also twelve rules of Boolean Logic we must also be familiar with.

A + 0 = A

A * A = A

A + 1 = 1

A * !A = 0

A * 0 = 0

!!A = A

A * 1 = A

A + A*B = A

A + A = A

A + !A*!B = A + B

A + !A = 1

(A + B)*(A+C) = A + B*C

Deep Dive into DeMorgan’s Theorem

A pivotal element in Boolean logic is DeMorgan's Theorem, which provides a method for simplifying expressions involving complements:

This principle, often remembered by the mnemonic "break the bar, change the sign," is indispensable for anyone involved in circuit design or logic optimization.

From Theory to Application

With a solid grasp of Boolean algebra, we can effectively map these principles into logic gates that form the building blocks of all digital devices as we will see in our next article.

This understanding not only enhances our ability to innovate but also deepens our appreciation of the intricate link between mathematical theory and practical application.

Workshops and Webinars

If you enjoyed the blog why not take a look at the free webinars, workshops and training courses we have created over the years. Highlights include

Embedded System Book   

Do you want to know more about designing embedded systems from scratch? Check out our book on creating embedded systems. This book will walk you through all the stages of requirements, architecture, component selection, schematics, layout, and FPGA / software design. We designed and manufactured the board at the heart of the book! The schematics and layout are available in Altium here   Learn more about the board (see previous blogs on Bring up, DDR validation, USB, Sensors) and view the schematics here.


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