PPT: Boolean Algebra & Minimization Techniques | Digital Logic - Computer Science Engineering (CSE) PDF Download

 Page 1


Boolean 
Algebra
Page 2


Boolean 
Algebra
Introduction
Boolean algebra forms the mathematical framework for analyzing and designing digital 
circuits using binary variables (0, 1). It serves as the foundation for all digital systems 
including computers and microprocessors.
The key concepts include variables that represent binary values, operations such as AND 
(·), OR (+), and NOT (¬), and truth values of True (1) and False (0). These elements work 
together to enable the analysis and design of digital logic systems.
Boolean algebra finds applications in simplifying logic circuits, designing combinational 
and sequential circuits, and minimizing hardware resources - all essential aspects of 
modern digital electronics.
Page 3


Boolean 
Algebra
Introduction
Boolean algebra forms the mathematical framework for analyzing and designing digital 
circuits using binary variables (0, 1). It serves as the foundation for all digital systems 
including computers and microprocessors.
The key concepts include variables that represent binary values, operations such as AND 
(·), OR (+), and NOT (¬), and truth values of True (1) and False (0). These elements work 
together to enable the analysis and design of digital logic systems.
Boolean algebra finds applications in simplifying logic circuits, designing combinational 
and sequential circuits, and minimizing hardware resources - all essential aspects of 
modern digital electronics.
Logic Gates & Truth Tables
Basic Gates
AND: Output = 1 if all inputs are 1
OR: Output = 1 if any input is 1
NOT: Inverts input (0³1, 1³0)
Derived Gates
NAND: NOT of AND operation
NOR: NOT of OR operation
XOR: Output = 1 if odd number of 
inputs are 1
XNOR: Output = 1 if even number 
of inputs are 1
Truth Tables
Lists all possible input combinations and corresponding outputs
Example (AND): A=0, B=0 ³ 0; A=1, B=1 ³ 1
Logic gates are the basic building blocks of digital circuits that perform 
Boolean operations. They implement Boolean functions in hardware, allowing 
for the creation of complex digital systems from simple components.
Page 4


Boolean 
Algebra
Introduction
Boolean algebra forms the mathematical framework for analyzing and designing digital 
circuits using binary variables (0, 1). It serves as the foundation for all digital systems 
including computers and microprocessors.
The key concepts include variables that represent binary values, operations such as AND 
(·), OR (+), and NOT (¬), and truth values of True (1) and False (0). These elements work 
together to enable the analysis and design of digital logic systems.
Boolean algebra finds applications in simplifying logic circuits, designing combinational 
and sequential circuits, and minimizing hardware resources - all essential aspects of 
modern digital electronics.
Logic Gates & Truth Tables
Basic Gates
AND: Output = 1 if all inputs are 1
OR: Output = 1 if any input is 1
NOT: Inverts input (0³1, 1³0)
Derived Gates
NAND: NOT of AND operation
NOR: NOT of OR operation
XOR: Output = 1 if odd number of 
inputs are 1
XNOR: Output = 1 if even number 
of inputs are 1
Truth Tables
Lists all possible input combinations and corresponding outputs
Example (AND): A=0, B=0 ³ 0; A=1, B=1 ³ 1
Logic gates are the basic building blocks of digital circuits that perform 
Boolean operations. They implement Boolean functions in hardware, allowing 
for the creation of complex digital systems from simple components.
Laws of Boolean Algebra
Basic Laws
Identity Laws: A + 0 = A, A · 1 = A
Null Laws: A + 1 = 1, A · 0 = 0
Idempotent Laws: A + A = A, A · A = A
Complement Laws: A + ¬A = 1, A · ¬A = 0
Boolean algebra follows specific laws that allow for the manipulation and simplification of logical expressions. 
These fundamental laws provide the mathematical foundation for working with binary variables and operations.
Understanding these laws is crucial for simplifying complex Boolean expressions and designing efficient digital 
circuits. They enable engineers to reduce the number of gates required and optimize circuit performance.
Page 5


Boolean 
Algebra
Introduction
Boolean algebra forms the mathematical framework for analyzing and designing digital 
circuits using binary variables (0, 1). It serves as the foundation for all digital systems 
including computers and microprocessors.
The key concepts include variables that represent binary values, operations such as AND 
(·), OR (+), and NOT (¬), and truth values of True (1) and False (0). These elements work 
together to enable the analysis and design of digital logic systems.
Boolean algebra finds applications in simplifying logic circuits, designing combinational 
and sequential circuits, and minimizing hardware resources - all essential aspects of 
modern digital electronics.
Logic Gates & Truth Tables
Basic Gates
AND: Output = 1 if all inputs are 1
OR: Output = 1 if any input is 1
NOT: Inverts input (0³1, 1³0)
Derived Gates
NAND: NOT of AND operation
NOR: NOT of OR operation
XOR: Output = 1 if odd number of 
inputs are 1
XNOR: Output = 1 if even number 
of inputs are 1
Truth Tables
Lists all possible input combinations and corresponding outputs
Example (AND): A=0, B=0 ³ 0; A=1, B=1 ³ 1
Logic gates are the basic building blocks of digital circuits that perform 
Boolean operations. They implement Boolean functions in hardware, allowing 
for the creation of complex digital systems from simple components.
Laws of Boolean Algebra
Basic Laws
Identity Laws: A + 0 = A, A · 1 = A
Null Laws: A + 1 = 1, A · 0 = 0
Idempotent Laws: A + A = A, A · A = A
Complement Laws: A + ¬A = 1, A · ¬A = 0
Boolean algebra follows specific laws that allow for the manipulation and simplification of logical expressions. 
These fundamental laws provide the mathematical foundation for working with binary variables and operations.
Understanding these laws is crucial for simplifying complex Boolean expressions and designing efficient digital 
circuits. They enable engineers to reduce the number of gates required and optimize circuit performance.
More Boolean Algebra Laws
Commutative Laws
A + B = B + A, A · B = B · A
Associative Laws
(A + B) + C = A + (B + C), (A · B) · C = A · (B · C)
Distributive Laws
A · (B + C) = A · B + A · C
A + (B · C) = (A + B) · (A + C)
De Morgan's Laws
¬(A + B) = ¬A · ¬B, ¬(A · B) = ¬A + ¬B
These advanced Boolean laws are essential tools for simplifying complex logical expressions. They provide the 
mathematical framework needed to transform and optimize Boolean functions, enabling more efficient circuit 
designs with fewer components.