Digital Logic🖥️

Did you know that nearly every device in the modern world operates using digital logic? From a simple calculator to complex artificial intelligence (AI) systems, it all starts here✨

The contents of this chapters are:

• Boolean algebra
• Gates
• Combinational circuits

Boolean Algebra

Mathematical discipline used to design and analyze the behavior of the digital cicuitry in digital computers and other digital systems.

Claude Shannon suggested boolean algebra could be used to solve problems in relay-switching circuit design

The functions of boolean algebra is to do analysis and design.

Analysis:

• its an economical way of describing the function of digital circiuitry Design:
• it can be applied to develop a simplified implementation of that function

Boolean variables and operations

Boolean Operation & Vector Expression.

In boolean algebra, variables and operations must be logical

• variable may take value 1 (TRUE) or 0 (FALSE)
• basic logical operations are AND, OR and NOT

Identities of Boolean Algebra

There are basic identities and other identities of boolean algebra in thi chapter.

In the basic identities, there are:

Commutative Laws Distributive Laws Identify Elements Inverse Elements

On the other hand, other identities consist of:

Associative Laws DeMorgan’s Theorem

Gates

The gate is the fundamental building block of all digital logic circuits, enabling the implementation of logical functions through their interconnection.

Digital Logic Gates

A gate is an electronic circuit that produces an output signal that is a simple Boolean operation on its input signals. The basic gates used in digital logic are AND, OR, NOT, NAND and NOR.

Each gate is defined in three ways:

• graphical symbol
• algebraic notation
• truth table

Digital Logic Gates

Combinational Circuits

refers to the type of digital circuit where the output depends solely on the current inputs, without any memory or feedback from past inputs.

Combinational circuits can be defined in 3 ways:

• Truth table — For each of the 2^n possible combinations of input signals, the binary value of each of the m output signals is listed.
• Graphical symbols — The interconnected layout of gates is depicted.
• Boolean equations — Each input signal is expressed as a boolean function of its input signals.

Boolean equations has 2 types which are:

1. SOP (Sum of Product) — output 1

2. POS (Product of Sum) — output 0

Simplification Method

• Karnaugh maps (K-maps)

Example of simplification using K-maps.

• Algebraic Simplification
• Quine-McCluskey Tables