Lecture Notes: Boolean Algebra & Logic Circuits

1. Introduction to Logic Circuits

Circuit Representation

• Block Diagram: A logic circuit is conceptually represented as a block where inputs enter on one side and outputs emerge from the other.
• Signal Nature: The input and output signals are discrete or digital. Unlike analog signals that vary continuously, digital signals are always represented by two distinct voltage levels:
  • High Voltage: Representing Logic 1.
  • Low Voltage: Representing Logic 0.

Digital vs. Analog Circuits

While analog circuits handle continuous signals, digital circuits offer specific advantages and trade-offs:

Feature	Digital Circuit Advantage	Notes
Reliability	More Reliable	Digital circuits are simpler and less susceptible to noise interference.
Accuracy	High Accuracy	States are determinate (0 or 1), making results precise.
Response Time	Slower	A disadvantage compared to analog circuits is often a slower response time.
Modeling	Mathematical Model	The main advantage is the ability to use Boolean Algebra for design, analysis, and simplification.

---

2. Boolean Algebra Fundamentals

Origins and Concept

• History: Boolean Algebra (BA) is named after George Boole, who developed it to study human logical reasoning and the calculus of propositions.
• Switch Algebra: Shannon later introduced "Switch Algebra," applying two-value Boolean Algebra to represent stable switching circuits (e.g., switches in series for AND, parallel for OR).

The Two-Value System

• Elements: The set consists of exactly two elements: .
• Physical Mapping: In digital electronics, these elements map to voltage levels:
  • High Level
  • Low Level .
• Operations: The system uses three primary binary operators:
  1. AND ()
  2. OR ()
  3. NOT ( or )

Fundamental Postulates

For a set with operations and , the following postulates are true:

1. Closure:
   • If you take any two valid binary values (0 or 1) and combine them with an operation (+ or ⋅), the result is still a valid binary value (0 or 1).
   • For every , the result of is also in .
   • For every , the result of is also in .
2. Commutative Law:
   • The order in which you apply variables to an OR or AND gate does not matter.
3. Associative Law:
   • How you group variables together does not matter.
4. Identity Element:
   • This defines the "neutral" value for an operation.
   • 0 is the identity for +:
   • 1 is the identity for :
5. Distributive Law:
   • This allows you to "multiply out" or "factor" expressions, similar to standard algebra, but with a unique twist in Boolean Algebra.
   • (Note: This second form is unique to Boolean algebra)
6. Complement:
   • Every element has an inverse.
   • For every , there exists a unique element such that:

Operator Priority

To reduce the need for brackets, operations are performed in the following strict order:

1. NOT () - Highest priority
2. AND ()
3. OR () - Lowest priority

Example: The expression is interpreted as , not . Brackets can be used to overwrite this priority.

Duality Principle

The duality principle implies a "buy one, get one free" relationship for theorems. A Boolean identity remains valid if you interchange the operators and identity elements:

• Swap
• Swap

Example:

• Original:
• Dual:

3. Boolean Functions

A Boolean function is an expression formed by binary variables, logic operators (OR, AND, NOT), and an equal sign.

• Order of Precedence: NOT (), AND (), OR ().
  • Example: means .

Complement Function

The complement of a function is . It is obtained by exchanging 1s with 0s in the output, or algebraically using DeMorgan's Theorem repeatedly.

• Generalized DeMorgan:

4. Standard and Canonical Forms

Standard Forms

• Literals: A variable in normal () or complement () form.
• Product Term: Literals connected by AND (e.g., , ).
• Sum Term: Literals connected by OR (e.g., , ).

1. Sum-of-Products (SOP): Several product terms summed together (OR).
   • Example:
2. Product-of-Sums (POS): Several sum terms multiplied together (AND).
   • Example:

Minterms and Maxterms

For variables, there are possible combinations.

• Minterm (): A product term containing all variables. Represents a logic 1 in the truth table.
• Maxterm (): A sum term containing all variables. Represents a logic 0 in the truth table.

Relationship:
A minterm is the complement of its corresponding maxterm: .

Canonical Forms

A function expressed as a unique sum of minterms or product of maxterms.

1. Sum of Minterms (SOM):
   • Collect all minterms where the function output is 1.
2. Product of Maxterms (POM):
   • Collect all maxterms where the function output is 0.

Conversions

Between SOM and POM for function :

• Indices missing from the SOM list are present in the POM list.
• Equivalent POM:

Between and (Complement):

• SOM of SOM of : Use the minterms missing from .
• POM of POM of : Use the maxterms missing from .
• SOM of POM of : Just change to with the same indices (Inverse relationship).

---

Note: This material is based on "Boolean Algebra (1).pptx" and "Boolean Algebra (2).pptx".