This document discusses Boolean algebra and logic simplifications. It covers binary logic and gates, Boolean algebra properties and manipulation techniques, standard and canonical forms such as sum of products (SOP) and product of sums (POS), and Karnaugh maps. Minterms and maxterms represent the canonical forms, denoting each combination in the truth table. Boolean algebra allows simplifying logic functions through algebraic manipulation and putting them in standard forms.
- Boolean algebra uses binary values (1/0) to represent true/false in digital circuits.
- The basic Boolean operations are AND, OR, and NOT. Truth tables and Boolean expressions can both be used to represent the functions of circuits.
- Boolean expressions can be simplified using algebraic rules like commutative, distributive, DeMorgan's, and absorption laws. This allows simpler circuit implementations.
Boolean algebra can be used to simplify digital circuit expressions. A Boolean function can be represented as either a Boolean expression or a truth table. There are two main methods to convert between these representations: (1) using a sum of products to get a Boolean expression from a truth table by including all variable combinations that evaluate to 1, and (2) using a product of sums to get a Boolean expression by including all variable combinations that evaluate to 0.
Boolean algebra can be used to simplify digital circuit expressions. A Boolean function can be represented as either a logical expression using AND, OR, and NOT operators or as a truth table. There are standard methods to convert between these representations, such as using sums of minterms or products of maxterms. Boolean algebra postulates and theorems allow logical expressions to be simplified in ways that result in equivalent but simpler circuits.
This document provides an overview of digital logic circuits and data representation. It defines basic Boolean algebra concepts like binary operators, truth tables, and theorems. It also covers standard forms for representing Boolean functions like sum of products and product of sums. Finally, it discusses logic gates, multiple input gates, and De Morgan's theorem as it applies to gate transformations.
B.sc cs-ii-u-1.5 digital logic circuits, digital componentRai University
This document provides an overview of digital logic circuits and Boolean algebra. It defines basic logic operators like AND, OR, and NOT. It introduces Boolean algebra concepts such as Boolean variables, algebraic manipulation, and theorems. It also discusses Boolean functions using truth tables and logic circuits. Finally, it covers standard forms like Sum of Products and Product of Sums, logic gates, and De Morgan's theorem.
Bca 2nd sem-u-1.5 digital logic circuits, digital componentRai University
This document provides an overview of digital logic circuits and Boolean algebra. It defines basic logic operators like AND, OR, and NOT. It introduces Boolean algebra concepts such as Boolean variables, algebraic manipulation, and theorems. It also discusses Boolean functions using truth tables and logic circuits. Finally, it covers standard forms like Sum of Products and Product of Sums, and logic gates including AND, OR, XOR, and their equivalents.
This document discusses Boolean algebra and logic simplifications. It covers binary logic and gates, Boolean algebra properties and manipulation techniques, standard and canonical forms such as sum of products (SOP) and product of sums (POS), and Karnaugh maps. Minterms and maxterms represent the canonical forms, denoting each combination in the truth table. Boolean algebra allows simplifying logic functions through algebraic manipulation and putting them in standard forms.
- Boolean algebra uses binary values (1/0) to represent true/false in digital circuits.
- The basic Boolean operations are AND, OR, and NOT. Truth tables and Boolean expressions can both be used to represent the functions of circuits.
- Boolean expressions can be simplified using algebraic rules like commutative, distributive, DeMorgan's, and absorption laws. This allows simpler circuit implementations.
Boolean algebra can be used to simplify digital circuit expressions. A Boolean function can be represented as either a Boolean expression or a truth table. There are two main methods to convert between these representations: (1) using a sum of products to get a Boolean expression from a truth table by including all variable combinations that evaluate to 1, and (2) using a product of sums to get a Boolean expression by including all variable combinations that evaluate to 0.
Boolean algebra can be used to simplify digital circuit expressions. A Boolean function can be represented as either a logical expression using AND, OR, and NOT operators or as a truth table. There are standard methods to convert between these representations, such as using sums of minterms or products of maxterms. Boolean algebra postulates and theorems allow logical expressions to be simplified in ways that result in equivalent but simpler circuits.
This document provides an overview of digital logic circuits and data representation. It defines basic Boolean algebra concepts like binary operators, truth tables, and theorems. It also covers standard forms for representing Boolean functions like sum of products and product of sums. Finally, it discusses logic gates, multiple input gates, and De Morgan's theorem as it applies to gate transformations.
B.sc cs-ii-u-1.5 digital logic circuits, digital componentRai University
This document provides an overview of digital logic circuits and Boolean algebra. It defines basic logic operators like AND, OR, and NOT. It introduces Boolean algebra concepts such as Boolean variables, algebraic manipulation, and theorems. It also discusses Boolean functions using truth tables and logic circuits. Finally, it covers standard forms like Sum of Products and Product of Sums, logic gates, and De Morgan's theorem.
Bca 2nd sem-u-1.5 digital logic circuits, digital componentRai University
This document provides an overview of digital logic circuits and Boolean algebra. It defines basic logic operators like AND, OR, and NOT. It introduces Boolean algebra concepts such as Boolean variables, algebraic manipulation, and theorems. It also discusses Boolean functions using truth tables and logic circuits. Finally, it covers standard forms like Sum of Products and Product of Sums, and logic gates including AND, OR, XOR, and their equivalents.
This document discusses Boolean algebra and logic gates. It begins with an introduction to binary logic and Boolean variables that can take on values of 0 or 1. It describes logical operators like AND, OR, and NOT. Boolean algebra provides a mathematical system for specifying and transforming logic functions. The document provides examples of Boolean functions and logic gates. It discusses topics like Boolean variables and values, Boolean functions, logical operators, Boolean arithmetic, theorems, and algebraic proofs. George Boole is credited with developing Boolean algebra. Truth tables and Karnaugh maps are shown as ways to analyze Boolean functions.
This document discusses Boolean algebra properties and concepts. It defines Boolean variables and constants, and lists common Boolean algebra properties like idempotence, complement, duality, commutativity, associativity, distributivity, De Morgan's laws, and absorption. It also discusses Boolean expressions, truth tables, canonical forms, minterms, maxterms, and how to derive the complement and canonical forms of a Boolean function.
This document provides an overview of logic gates, Boolean algebra, and digital circuits. It defines basic logic gates like AND, OR, and NOT. It introduces Boolean algebra concepts such as binary variables, algebraic manipulation using laws and theorems, and canonical forms. Standard logic implementations including sum of products, product of sums, and universal gates using NAND and NOR are discussed. De Morgan's theorems and their application to logic gate equivalents are also covered.
This document discusses gate level minimization and optimization of Boolean functions. It covers cost criteria for logic circuits including literal cost, gate input cost, and gate input cost with NOTs. Common techniques for Boolean function optimization are also described, including algebraic manipulation and Karnaugh maps. Karnaugh maps allow visual grouping of minterms to minimize gate inputs and simplify Boolean expressions. The concepts of implicants, prime implicants, and essential prime implicants are introduced in the context of Karnaugh map analysis. Don't care conditions are also discussed.
Unit 2 Boolean Algebra and Logic Gates.pdfShirazHusain4
This document discusses Boolean algebra and logic gates. It introduces Boolean postulates and laws such as identity, complement, commutative, associative, distributive, and absorption. It also covers simplification of Boolean expressions using these laws and theorems. Karnaugh maps are presented as a visual method to simplify logic expressions into their minimal forms. Implementation of Boolean functions using logic gates like AND, OR, NAND, and NOR is demonstrated through examples.
This document discusses canonical and standard forms in digital logic circuits. It defines minterms and maxterms, and describes how to convert between sum of products and product of sums forms. Procedures for converting Boolean functions to sum of minterms and product of maxterms are provided with examples. Other logic operations such as XOR and XNOR are also defined.
digital logic design Chapter 2 boolean_algebra_&_logic_gatesImran Waris
The document discusses Boolean algebra and logic gates. It defines binary operators like AND, OR, and NOT. It covers Boolean algebra postulates and theorems including duality, DeMorgan's theorem, and absorption. Standard forms like sum of products and product of sums are presented. Common logic gates such as AND, OR, NAND, NOR, XOR, and XNOR are defined. Homework problems from a textbook are listed involving simplifying Boolean expressions, drawing logic diagrams, and converting expressions between canonical forms.
This document provides an overview of Boolean algebra and logic gates. It introduces Boolean logic operations like AND, OR, and NOT. It covers Boolean algebra laws and De Morgan's theorems. It also discusses logic gate types like AND, OR, NOT, NAND, NOR, XOR and XNOR. Karnaugh maps are introduced as a method to simplify Boolean expressions.
18 pc09 1.2_ digital logic gates _ boolean algebra_basic theoremsarunachalamr16
Digital logic gates are basic building blocks of digital circuits that make logical decisions based on input combinations. There are three basic logic gates: OR, AND, and NOT. Other common gates such as NAND, NOR, XOR, and XNOR are derived from these. Boolean algebra uses variables that can be 1 or 0, and logical operators like AND, OR, and NOT to represent logic functions. Logic functions can be expressed in canonical forms such as sum of minterms or product of maxterms. Standard forms like SOP and POS are also used. Conversions between these forms allow simplifying logic functions.
boolean algebra and logic simplificationUnsa Shakir
The document provides an overview of Boolean algebra and logic simplification. It covers topics such as Boolean variables that can take true/false or 1/0 values, basic logic gates like AND, OR, NOT, NAND and NOR gates, canonical forms including sum-of-products and product-of-sums, De Morgan's laws, and examples of simplifying Boolean expressions and implementing logic circuits.
The document discusses Boolean algebra and its application to digital logic design. It defines Boolean algebra as a mathematical system used to represent binary variables and logical relationships. The key aspects covered include:
- The axiomatic definition of Boolean algebra using Huntington's postulates.
- The representation of Boolean functions using truth tables and logic gate diagrams. Boolean functions express logical relationships between binary variables.
- Techniques for manipulating and minimizing Boolean expressions through algebraic rules to simplify logic circuits.
The document discusses properties and theorems of Boolean algebra. It defines key concepts like Boolean variables, constants, minterms, maxterms, and duality. Properties covered include idempotence, complement, absorption, consensus, commutative, associative, distributive, De Morgan's laws. Truth tables and canonical forms are also discussed. Duality and complementation are shown to derive equivalent expressions.
This document defines functions and related terminology. It discusses:
- The definition of a function, domain, codomain, range, and related terms
- Properties of functions including one-to-one, onto, and bijective functions
- Inverse functions and how they relate to bijective functions
- Examples are provided to illustrate injective, surjective and bijective functions as well as calculating inverse functions.
George Boole first introduced Boolean algebra in 1854 as a way to systematically analyze logic circuits. Boolean algebra uses variables and operations like AND, OR and NOT to represent the behavior of digital logic gates. A key insight was Claude Shannon's 1938 application of Boolean algebra to the analysis and design of logic circuits. Boolean algebra provides a concise way to represent the operation of any logic circuit and determine its output for all combinations of inputs.
Boolean algebra deals with logical operations on binary variables that have two possible values, typically represented as 1 and 0. George Boole first introduced Boolean algebra in 1854. Boolean algebra uses logic gates like AND, OR, and NOT as basic building blocks. Positive logic represents 1 as high and 0 as low, while negative logic uses the opposite. Boolean algebra laws and Karnaugh maps are used to simplify logical expressions. Don't care conditions allow for groupings in K-maps that further reduce expressions.
This document provides an overview of digital logic circuits and digital systems. It discusses binary logic, logic gates like NAND and NOR, Boolean algebra, decoders, adders, and the differences between analog and digital signals. It also covers representations of digital designs using truth tables, Boolean algebra, logic gate schematics, and logic simulations. Common logic gates, functions, identities, simplification techniques, and the duality principle of Boolean algebra are described.
This document provides an overview of combinational logic circuits and Boolean algebra. It discusses binary logic, logic gates, Boolean expressions, canonical forms such as sum of minterms and product of maxterms, and other concepts relevant to combinational logic design. Key topics covered include binary variables, logical operations, truth tables, logic gate symbols and behavior, Boolean algebra identities, simplifying Boolean expressions, and representing functions in canonical forms.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
This document discusses Boolean algebra and logic gates. It begins with an introduction to binary logic and Boolean variables that can take on values of 0 or 1. It describes logical operators like AND, OR, and NOT. Boolean algebra provides a mathematical system for specifying and transforming logic functions. The document provides examples of Boolean functions and logic gates. It discusses topics like Boolean variables and values, Boolean functions, logical operators, Boolean arithmetic, theorems, and algebraic proofs. George Boole is credited with developing Boolean algebra. Truth tables and Karnaugh maps are shown as ways to analyze Boolean functions.
This document discusses Boolean algebra properties and concepts. It defines Boolean variables and constants, and lists common Boolean algebra properties like idempotence, complement, duality, commutativity, associativity, distributivity, De Morgan's laws, and absorption. It also discusses Boolean expressions, truth tables, canonical forms, minterms, maxterms, and how to derive the complement and canonical forms of a Boolean function.
This document provides an overview of logic gates, Boolean algebra, and digital circuits. It defines basic logic gates like AND, OR, and NOT. It introduces Boolean algebra concepts such as binary variables, algebraic manipulation using laws and theorems, and canonical forms. Standard logic implementations including sum of products, product of sums, and universal gates using NAND and NOR are discussed. De Morgan's theorems and their application to logic gate equivalents are also covered.
This document discusses gate level minimization and optimization of Boolean functions. It covers cost criteria for logic circuits including literal cost, gate input cost, and gate input cost with NOTs. Common techniques for Boolean function optimization are also described, including algebraic manipulation and Karnaugh maps. Karnaugh maps allow visual grouping of minterms to minimize gate inputs and simplify Boolean expressions. The concepts of implicants, prime implicants, and essential prime implicants are introduced in the context of Karnaugh map analysis. Don't care conditions are also discussed.
Unit 2 Boolean Algebra and Logic Gates.pdfShirazHusain4
This document discusses Boolean algebra and logic gates. It introduces Boolean postulates and laws such as identity, complement, commutative, associative, distributive, and absorption. It also covers simplification of Boolean expressions using these laws and theorems. Karnaugh maps are presented as a visual method to simplify logic expressions into their minimal forms. Implementation of Boolean functions using logic gates like AND, OR, NAND, and NOR is demonstrated through examples.
This document discusses canonical and standard forms in digital logic circuits. It defines minterms and maxterms, and describes how to convert between sum of products and product of sums forms. Procedures for converting Boolean functions to sum of minterms and product of maxterms are provided with examples. Other logic operations such as XOR and XNOR are also defined.
digital logic design Chapter 2 boolean_algebra_&_logic_gatesImran Waris
The document discusses Boolean algebra and logic gates. It defines binary operators like AND, OR, and NOT. It covers Boolean algebra postulates and theorems including duality, DeMorgan's theorem, and absorption. Standard forms like sum of products and product of sums are presented. Common logic gates such as AND, OR, NAND, NOR, XOR, and XNOR are defined. Homework problems from a textbook are listed involving simplifying Boolean expressions, drawing logic diagrams, and converting expressions between canonical forms.
This document provides an overview of Boolean algebra and logic gates. It introduces Boolean logic operations like AND, OR, and NOT. It covers Boolean algebra laws and De Morgan's theorems. It also discusses logic gate types like AND, OR, NOT, NAND, NOR, XOR and XNOR. Karnaugh maps are introduced as a method to simplify Boolean expressions.
18 pc09 1.2_ digital logic gates _ boolean algebra_basic theoremsarunachalamr16
Digital logic gates are basic building blocks of digital circuits that make logical decisions based on input combinations. There are three basic logic gates: OR, AND, and NOT. Other common gates such as NAND, NOR, XOR, and XNOR are derived from these. Boolean algebra uses variables that can be 1 or 0, and logical operators like AND, OR, and NOT to represent logic functions. Logic functions can be expressed in canonical forms such as sum of minterms or product of maxterms. Standard forms like SOP and POS are also used. Conversions between these forms allow simplifying logic functions.
boolean algebra and logic simplificationUnsa Shakir
The document provides an overview of Boolean algebra and logic simplification. It covers topics such as Boolean variables that can take true/false or 1/0 values, basic logic gates like AND, OR, NOT, NAND and NOR gates, canonical forms including sum-of-products and product-of-sums, De Morgan's laws, and examples of simplifying Boolean expressions and implementing logic circuits.
The document discusses Boolean algebra and its application to digital logic design. It defines Boolean algebra as a mathematical system used to represent binary variables and logical relationships. The key aspects covered include:
- The axiomatic definition of Boolean algebra using Huntington's postulates.
- The representation of Boolean functions using truth tables and logic gate diagrams. Boolean functions express logical relationships between binary variables.
- Techniques for manipulating and minimizing Boolean expressions through algebraic rules to simplify logic circuits.
The document discusses properties and theorems of Boolean algebra. It defines key concepts like Boolean variables, constants, minterms, maxterms, and duality. Properties covered include idempotence, complement, absorption, consensus, commutative, associative, distributive, De Morgan's laws. Truth tables and canonical forms are also discussed. Duality and complementation are shown to derive equivalent expressions.
This document defines functions and related terminology. It discusses:
- The definition of a function, domain, codomain, range, and related terms
- Properties of functions including one-to-one, onto, and bijective functions
- Inverse functions and how they relate to bijective functions
- Examples are provided to illustrate injective, surjective and bijective functions as well as calculating inverse functions.
George Boole first introduced Boolean algebra in 1854 as a way to systematically analyze logic circuits. Boolean algebra uses variables and operations like AND, OR and NOT to represent the behavior of digital logic gates. A key insight was Claude Shannon's 1938 application of Boolean algebra to the analysis and design of logic circuits. Boolean algebra provides a concise way to represent the operation of any logic circuit and determine its output for all combinations of inputs.
Boolean algebra deals with logical operations on binary variables that have two possible values, typically represented as 1 and 0. George Boole first introduced Boolean algebra in 1854. Boolean algebra uses logic gates like AND, OR, and NOT as basic building blocks. Positive logic represents 1 as high and 0 as low, while negative logic uses the opposite. Boolean algebra laws and Karnaugh maps are used to simplify logical expressions. Don't care conditions allow for groupings in K-maps that further reduce expressions.
This document provides an overview of digital logic circuits and digital systems. It discusses binary logic, logic gates like NAND and NOR, Boolean algebra, decoders, adders, and the differences between analog and digital signals. It also covers representations of digital designs using truth tables, Boolean algebra, logic gate schematics, and logic simulations. Common logic gates, functions, identities, simplification techniques, and the duality principle of Boolean algebra are described.
This document provides an overview of combinational logic circuits and Boolean algebra. It discusses binary logic, logic gates, Boolean expressions, canonical forms such as sum of minterms and product of maxterms, and other concepts relevant to combinational logic design. Key topics covered include binary variables, logical operations, truth tables, logic gate symbols and behavior, Boolean algebra identities, simplifying Boolean expressions, and representing functions in canonical forms.
Similar to Basic Digital Logic(Notes-DE) for engineering .ppt (20)
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
2. Overview
• Binary logic and Gates
• Boolean Algebra
– Basic Properties
– Algebraic Manipulation
• Standard and Canonical Forms
– Minterms and Maxterms (Canonical forms)
– SOP and POS (Standard forms)
• Karnaugh Maps (K-Maps)
– 2, 3, 4, and 5 variable maps
– Simplification using K-Maps
• K-Map Manipulation
– Implicants: Prime, Essential
– Don’t Cares
• More Logic Gates
2024/5/13 Boolean Algebra PJF - 2
3. Binary Logic
• Deals with binary variables that take 2 discrete
values (0 and 1), and with logic operations
• Three basic logic operations:
– AND, OR, NOT
• Binary/logic variables are typically
represented as letters: A,B,C,…,X,Y,Z
2024/5/13 Boolean Algebra PJF - 3
5. Basic Logic Operators
• AND
• OR
• NOT
• F(a,b) = a•b, F is 1 if and only if a=b=1
• G(a,b) = a+b, G is 1 if either a=1 or b=1
• H(a) = a’, H is 1 if a=0
2024/5/13 Boolean Algebra PJF - 5
Binary
Unary
7. Truth Tables for logic operators
Truth table: tabular form that uniguely represents the relationship
between the input variables of a function and its output
A B F=A•B
0 0 0
0 1 0
1 0 0
1 1 1
2024/5/13 PJF - 7
Boolean Algebra
2-Input AND
A B F=A+B
0 0 0
0 1 1
1 0 1
1 1 1
2-Input OR
A F=A’
0 1
1 0
NOT
8. Truth Tables (cont.)
• Q: Let a function F() depend on n variables.
How many rows are there in the truth table of
F() ?
2024/5/13 Boolean Algebra PJF - 8
A: 2n
rows, since there are 2n
possible
binary patterns/combinations for the n
variables
9. Logic Gates
• Logic gates are abstractions of electronic circuit
components that operate on one or more input
signals to produce an output signal.
2024/5/13 Boolean Algebra PJF - 9
2-Input AND 2-Input OR NOT (Inverter)
A A
A
B B
F G H
F = A•B G = A+B H = A’
10. Timing Diagram
2024/5/13 Boolean Algebra PJF - 10
A
B
F=A•B
G=A+B
H=A’
1
1
1
1
1
0
0
0
0
0
t0 t1 t2 t3 t4 t5 t6
Input
signals
Gate
Output
Signals
Basic
Assumption:
Zero time for
signals to
propagate
Through gates
Transitions
11. Combinational Logic Circuit
from Logic Function
• Consider function F = A’ + B•C’ + A’•B’
• A combinational logic circuit can be constructed to implement F, by
appropriately connecting input signals and logic gates:
– Circuit input signals from function variables (A, B, C)
– Circuit output signal function output (F)
– Logic gates from logic operations
2024/5/13 PJF - 11
Boolean Algebra
A
B
C
F
12. Combinational Logic Circuit
from Logic Function (cont.)
• In order to design a cost-effective
and efficient circuit, we must
minimize the circuit’s size (area) and
propagation delay (time required for
an input signal change to be
observed at the output line)
• Observe the truth table of F=A’ + B•C’
+ A’•B’ and G=A’ + B•C’
• Truth tables for F and G are identical
same function
• Use G to implement the logic circuit
(less components)
A B C F G
0 0 0 1 1
0 0 1 1 1
0 1 0 1 1
0 1 1 1 1
1 0 0 0 0
1 0 1 0 0
1 1 0 1 1
1 1 1 0 0
2024/5/13 PJF - 12
Boolean Algebra
14. Boolean Algebra
• VERY nice machinery used to manipulate
(simplify) Boolean functions
• George Boole (1815-1864): “An investigation
of the laws of thought”
• Terminology:
– Literal: A variable or its complement
– Product term: literals connected by •
– Sum term: literals connected by +
2024/5/13 Boolean Algebra PJF - 14
15. Boolean Algebra Properties
Let X: boolean variable, 0,1: constants
1. X + 0 = X -- Zero Axiom
2. X • 1 = X -- Unit Axiom
3. X + 1 = 1 -- Unit Property
4. X • 0 = 0 -- Zero Property
2024/5/13 Boolean Algebra PJF - 15
16. Boolean Algebra Properties (cont.)
Let X: boolean variable, 0,1: constants
5. X + X = X -- Idepotence
6. X • X = X -- Idepotence
7. X + X’ = 1 -- Complement
8. X • X’ = 0 -- Complement
9. (X’)’ = X -- Involution
2024/5/13 Boolean Algebra PJF - 16
17. Duality
• The dual of an expression is obtained by exchanging
(• and +), and (1 and 0) in it, provided that the
precedence of operations is not changed.
• Cannot exchange x with x’
• Example:
– Find H(x,y,z), the dual of F(x,y,z) = x’yz’ + x’y’z
– H = (x’+y+z’) (x’+y’+ z)
2024/5/13 Boolean Algebra PJF - 17
18. Duality (cont’d)
2024/5/13 Boolean Algebra PJF - 18
With respect to duality, Identities 1 – 8
have the following relationship:
1. X + 0 = X 2. X • 1 = X (dual of 1)
3. X + 1 = 1 4. X • 0 = 0 (dual of 3)
5. X + X = X 6. X • X = X (dual of 5)
7. X + X’ = 1 8. X • X’ = 0 (dual of 8)
19. More Boolean Algebra Properties
Let X,Y, and Z: boolean variables
10. X + Y = Y + X 11. X • Y = Y • X -- Commutative
12. X + (Y+Z) = (X+Y) + Z 13. X•(Y•Z) = (X•Y)•Z -- Associative
14. X•(Y+Z) = X•Y + X•Z 15. X+(Y•Z) = (X+Y) • (X+Z)
-- Distributive
16. (X + Y)’ = X’ • Y’ 17. (X • Y)’ = X’ + Y’ -- DeMorgan’s
In general,
( X1 + X2 + … + Xn )’ = X1’•X2’ • … •Xn’, and
( X1•X2•… •Xn )’ = X1’ + X2’ + … + Xn’
2024/5/13 Boolean Algebra PJF - 19
20. Absorption Property
1. x + x•y = x
2. x•(x+y) = x (dual)
• Proof:
x + x•y = x•1 + x•y
= x•(1+y)
= x•1
= x
QED (2 true by duality, why?)
2024/5/13 Boolean Algebra PJF - 20
21. Power of Duality
1. x + x•y = x is true, so (x + x•y)’=x’
2. (x + x•y)’=x’•(x’+y’)
3. x’•(x’+y’) =x’
4. Let X=x’, Y=y’
5. X•(X+Y) =X, which is the dual of x + x•y = x.
6. The above process can be applied to any formula. So
if a formula is valid, then its dual must also be valid.
7. Proving one formula also proves its dual.
2024/5/13 Boolean Algebra PJF - 21
23. Truth Tables (revisited)
• Enumerates all possible
combinations of variable values
and the corresponding function
value
• Truth tables for some arbitrary
functions
F1(x,y,z), F2(x,y,z), and F3(x,y,z) are
shown to the right.
2024/5/13 Boolean Algebra PJF - 23
x y z F1 F2 F3
0 0 0 0 1 1
0 0 1 0 0 1
0 1 0 0 0 1
0 1 1 0 1 1
1 0 0 0 1 0
1 0 1 0 1 0
1 1 0 0 0 0
1 1 1 1 0 1
24. Truth Tables (cont.)
• Truth table: a unique representation of a Boolean
function
• If two functions have identical truth tables, the
functions are equivalent (and vice-versa).
• Truth tables can be used to prove equality theorems.
• However, the size of a truth table grows
exponentially with the number of variables involved,
hence unwieldy. This motivates the use of Boolean
Algebra.
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25. Boolean expressions-NOT unique
• Unlike truth tables, expressions
representing a Boolean function are NOT
unique.
• Example:
– F(x,y,z) = x’•y’•z’ + x’•y•z’ + x•y•z’
– G(x,y,z) = x’•y’•z’ + y•z’
• The corresponding truth tables for F() and
G() are to the right. They are identical.
• Thus, F() = G()
2024/5/13 Boolean Algebra PJF - 25
x y z F G
0 0 0 1 1
0 0 1 0 0
0 1 0 1 1
0 1 1 0 0
1 0 0 0 0
1 0 1 0 0
1 1 0 1 1
1 1 1 0 0
26. Algebraic Manipulation
• Boolean algebra is a useful tool for simplifying
digital circuits.
• Why do it? Simpler can mean cheaper, smaller,
faster.
• Example: Simplify F = x’yz + x’yz’ + xz.
F = x’yz + x’yz’ + xz
= x’y(z+z’) + xz
= x’y•1 + xz
= x’y + xz
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28. Complement of a Function
• The complement of a function is derived by
interchanging (• and +), and (1 and 0), and
complementing each variable.
• Otherwise, interchange 1s to 0s in the truth
table column showing F.
• The complement of a function IS NOT THE
SAME as the dual of a function.
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29. Complementation: Example
• Find G(x,y,z), the complement of
F(x,y,z) = xy’z’ + x’yz
• G = F’ = (xy’z’ + x’yz)’
= (xy’z’)’ • (x’yz)’ DeMorgan
= (x’+y+z) • (x+y’+z’) DeMorgan again
• Note: The complement of a function can also be
derived by finding the function’s dual, and then
complementing all of the literals
2024/5/13 Boolean Algebra PJF - 29
30. Canonical and Standard Forms
• We need to consider formal techniques for the
simplification of Boolean functions.
– Identical functions will have exactly the same
canonical form.
– Minterms and Maxterms
– Sum-of-Minterms and Product-of- Maxterms
– Product and Sum terms
– Sum-of-Products (SOP) and Product-of-Sums (POS)
2024/5/13 Boolean Algebra PJF - 30
31. Definitions
• Literal: A variable or its complement
• Product term: literals connected by •
• Sum term: literals connected by +
• Minterm: a product term in which all the variables
appear exactly once, either complemented or
uncomplemented
• Maxterm: a sum term in which all the variables
appear exactly once, either complemented or
uncomplemented
2024/5/13 Boolean Algebra PJF - 31
32. Minterm
• Represents exactly one combination in the truth table.
• Denoted by mj, where j is the decimal equivalent of
the minterm’s corresponding binary combination (bj).
• A variable in mj is complemented if its value in bj is 0,
otherwise is uncomplemented.
• Example: Assume 3 variables (A,B,C), and j=3. Then, bj
= 011 and its corresponding minterm is denoted by mj
= A’BC
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33. Maxterm
• Represents exactly one combination in the truth table.
• Denoted by Mj, where j is the decimal equivalent of
the maxterm’s corresponding binary combination (bj).
• A variable in Mj is complemented if its value in bj is 1,
otherwise is uncomplemented.
• Example: Assume 3 variables (A,B,C), and j=3. Then, bj
= 011 and its corresponding maxterm is denoted by
Mj = A+B’+C’
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34. Truth Table notation for Minterms and
Maxterms
• Minterms and
Maxterms are easy
to denote using a
truth table.
• Example:
Assume 3 variables
x,y,z
(order is fixed)
2024/5/13 Boolean Algebra PJF - 34
x y z Minterm Maxterm
0 0 0 x’y’z’ = m0 x+y+z = M0
0 0 1 x’y’z = m1 x+y+z’ = M1
0 1 0 x’yz’ = m2 x+y’+z = M2
0 1 1 x’yz = m3 x+y’+z’= M3
1 0 0 xy’z’ = m4 x’+y+z = M4
1 0 1 xy’z = m5 x’+y+z’ = M5
1 1 0 xyz’ = m6 x’+y’+z = M6
1 1 1 xyz = m7 x’+y’+z’ = M7
35. Canonical Forms (Unique)
• Any Boolean function F( ) can be expressed as a
unique sum of minterms and a unique product
of maxterms (under a fixed variable ordering).
• In other words, every function F() has two
canonical forms:
– Canonical Sum-Of-Products (sum of minterms)
– Canonical Product-Of-Sums (product of
maxterms)
2024/5/13 Boolean Algebra PJF - 35
36. Canonical Forms (cont.)
• Canonical Sum-Of-Products:
The minterms included are those mj such that
F( ) = 1 in row j of the truth table for F( ).
• Canonical Product-Of-Sums:
The maxterms included are those Mj such that
F( ) = 0 in row j of the truth table for F( ).
2024/5/13 Boolean Algebra PJF - 36
37. Example
• Truth table for f1(a,b,c) at right
• The canonical sum-of-products form for f1
is
f1(a,b,c) = m1 + m2 + m4 + m6
= a’b’c + a’bc’ + ab’c’ + abc’
• The canonical product-of-sums form for f1 is
f1(a,b,c) = M0 • M3 • M5 • M7
= (a+b+c)•(a+b’+c’)•
(a’+b+c’)•(a’+b’+c’).
• Observe that: mj = Mj’
2024/5/13 Boolean Algebra PJF - 37
a b c f1
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 0
38. Shorthand: ∑ and ∏
• f1(a,b,c) = ∑ m(1,2,4,6), where ∑ indicates that this is
a sum-of-products form, and m(1,2,4,6) indicates
that the minterms to be included are m1, m2, m4, and
m6.
• f1(a,b,c) = ∏ M(0,3,5,7), where ∏ indicates that this
is a product-of-sums form, and M(0,3,5,7) indicates
that the maxterms to be included are M0, M3, M5,
and M7.
• Since mj = Mj’ for any j,
∑ m(1,2,4,6) = ∏ M(0,3,5,7) = f1(a,b,c)
2024/5/13 Boolean Algebra PJF - 38
39. Conversion Between Canonical Forms
• Replace ∑ with ∏ (or vice versa) and replace those j’s that
appeared in the original form with those that do not.
• Example:
f1(a,b,c) = a’b’c + a’bc’ + ab’c’ + abc’
= m1 + m2 + m4 + m6
= ∑(1,2,4,6)
= ∏(0,3,5,7)
= (a+b+c)•(a+b’+c’)•(a’+b+c’)•(a’+b’+c’)
2024/5/13 Boolean Algebra PJF - 39
40. Standard Forms (NOT Unique)
• Standard forms are “like” canonical forms,
except that not all variables need appear in
the individual product (SOP) or sum (POS)
terms.
• Example:
f1(a,b,c) = a’b’c + bc’ + ac’
is a standard sum-of-products form
• f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’)
is a standard product-of-sums form.
2024/5/13 Boolean Algebra PJF - 40
41. Conversion of SOP from standard to
canonical form
• Expand non-canonical terms by inserting
equivalent of 1 in each missing variable x:
(x + x’) = 1
• Remove duplicate minterms
• f1(a,b,c) = a’b’c + bc’ + ac’
= a’b’c + (a+a’)bc’ + a(b+b’)c’
= a’b’c + abc’ + a’bc’ + abc’ + ab’c’
= a’b’c + abc’ + a’bc + ab’c’
2024/5/13 Boolean Algebra PJF - 41
42. Conversion of POS from standard to
canonical form
• Expand noncanonical terms by adding 0 in terms of
missing variables (e.g., xx’ = 0) and using the
distributive law
• Remove duplicate maxterms
• f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’)
= (a+b+c)•(aa’+b’+c’)•(a’+bb’+c’)
= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•
(a’+b+c’)•(a’+b’+c’)
= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•(a’+b+c’)
2024/5/13 Boolean Algebra PJF - 42
43. Karnaugh Maps
• Karnaugh maps (K-maps) are graphical
representations of boolean functions.
• One map cell corresponds to a row in the
truth table.
• Also, one map cell corresponds to a minterm
or a maxterm in the boolean expression
• Multiple-cell areas of the map correspond to
standard terms.
2024/5/13 Boolean Algebra PJF - 43
44. Two-Variable Map
2024/5/13 Boolean Algebra PJF - 44
m3
m2
1
m1
m0
0
1
0
x1
x2
0 1
2 3
NOTE: ordering of variables is IMPORTANT
for f(x1,x2), x1 is the row, x2 is the column.
Cell 0 represents x1’x2’; Cell 1 represents
x1’x2; etc. If a minterm is present in the
function, then a 1 is placed in the
corresponding cell.
m3
m1
1
m2
m0
0
1
0
x2
x1
0 2
1 3
OR
45. Two-Variable Map (cont.)
• Any two adjacent cells in the map differ by
ONLY one variable, which appears
complemented in one cell and
uncomplemented in the other.
• Example:
m0 (=x1’x2’) is adjacent to m1 (=x1’x2) and m2
(=x1x2’) but NOT m3 (=x1x2)
2024/5/13 Boolean Algebra PJF - 45
46. 2-Variable Map -- Example
• f(x1,x2) = x1’x2’+ x1’x2 + x1x2’
= m0 + m1 + m2
= x1’ + x2’
• 1s placed in K-map for specified
minterms m0, m1, m2
• Grouping (ORing) of 1s allows
simplification
• What (simpler) function is
represented by each dashed
rectangle?
– x1’ = m0 + m1
– x2’ = m0 + m2
• Note m0 covered twice
2024/5/13 Boolean Algebra PJF - 46
x1 0 1
0 1 1
1 1 0
x2
0 1
2 3
47. Minimization as SOP using K-map
• Enter 1s in the K-map for each product term in
the function
• Group adjacent K-map cells containing 1s to
obtain a product with fewer variables. Group
size must be in power of 2 (2, 4, 8, …)
• Handle “boundary wrap” for K-maps of 3 or
more variables.
• Realize that answer may not be unique
2024/5/13 Boolean Algebra PJF - 47
48. Three-Variable Map
2024/5/13 Boolean Algebra PJF - 48
m6
m7
m5
m4
1
m2
m3
m1
m0
0
10
11
01
00
yz
x
0 1 3 2
4 5 7 6
-Note: variable ordering is (x,y,z); yz specifies
column, x specifies row.
-Each cell is adjacent to three other cells (left or
right or top or bottom or edge wrap)
49. Three-Variable Map (cont.)
2024/5/13 Boolean Algebra PJF - 49
The types of structures
that are either minterms or
are generated by repeated
application of the
minimization theorem on a
three variable map are
shown at right.
Groups of 1, 2, 4, 8 are
possible.
minterm
group of 2 terms
group of 4 terms
50. Simplification
• Enter minterms of the Boolean function into
the map, then group terms
• Example: f(a,b,c) = a’c + abc + bc’
• Result: f(a,b,c) = a’c+ b
2024/5/13 Boolean Algebra PJF - 50
1 1 1
1 1
a
bc
1 1 1
1 1
51. More Examples
• f1(x, y, z) = ∑ m(2,3,5,7)
• f2(x, y, z) = ∑ m (0,1,2,3,6)
2024/5/13 Boolean Algebra PJF - 51
f1(x, y, z) = x’y + xz
f2(x, y, z) = x’+yz’
yz
X 00 01 11 10
0 1 1
1 1 1
1 1 1 1
1
52. Four-Variable Maps
• Top cells are adjacent to bottom cells. Left-edge cells
are adjacent to right-edge cells.
• Note variable ordering (WXYZ).
2024/5/13 PJF - 52
Boolean Algebra
m10
m11
m9
m8
10
m14
m15
m13
m12
11
m6
m7
m5
m4
01
m2
m3
m1
m0
00
10
11
01
00
WX
YZ
53. Four-variable Map Simplification
• One square represents a minterm of 4 literals.
• A rectangle of 2 adjacent squares represents a
product term of 3 literals.
• A rectangle of 4 squares represents a product term
of 2 literals.
• A rectangle of 8 squares represents a product term
of 1 literal.
• A rectangle of 16 squares produces a function that is
equal to logic 1.
2024/5/13 Boolean Algebra PJF - 53
54. Example
• Simplify the following Boolean function (A,B,C,D) =
∑m(0,1,2,4,5,7,8,9,10,12,13).
• First put the function g( ) into the map, and then
group as many 1s as possible.
2024/5/13 Boolean Algebra PJF - 54
cd
ab
1
1
1
1
1
1
1
1
1
1
1
g(A,B,C,D) = c’+b’d’+a’bd
1
1
1
1
1
1
1
1
1
1
1
55. Don't Care Conditions
• There may be a combination of input values which
– will never occur
– if they do occur, the output is of no concern.
• The function value for such combinations is called a don't
care.
• They are denoted with x or –. Each x may be arbitrarily
assigned the value 0 or 1 in an implementation.
• Don’t cares can be used to further simplify a function
2024/5/13 Boolean Algebra PJF - 55
56. Minimization using Don’t Cares
• Treat don't cares as if they are 1s to generate
PIs.
• Delete PI's that cover only don't care
minterms.
• Treat the covering of remaining don't care
minterms as optional in the selection process
(i.e. they may be, but need not be, covered).
2024/5/13 Boolean Algebra PJF - 56
57. Example
• Simplify the function f(a,b,c,d)
whose K-map is shown at the
right.
• f = a’c’d+ab’+cd’+a’bc’
or
• f = a’c’d+ab’+cd’+a’bd’
2024/5/13 Boolean Algebra PJF - 57
x
x
1
1
x
x
0
0
1
0
1
1
1
0
1
0
x
x
1
1
x
x
0
0
1
0
1
1
1
0
1
0
0 1 0 1
1 1 0 1
0 0 x x
1 1 x x
ab
cd
00
01
11
10
00 01 11 10
58. Another Example
• Simplify the function
g(a,b,c,d) whose K-map is
shown at right.
• g = a’c’+ ab
or
• g = a’c’+b’d
2024/5/13 Boolean Algebra PJF - 58
x 1 0 0
1 x 0 x
1 x x 1
0 x x 0
x 1 0 0
1 x 0 x
1 x x 1
0 x x 0
x 1 0 0
1 x 0 x
1 x x 1
0 x x 0
ab
cd
59. Algorithmic minimization
• What do we do for functions with more
variables?
• You can “code up” a minimizer (Computer-
Aided Design, CAD)
– Quine-McCluskey algorithm
– Iterated consensus
• We won’t discuss these techniques here
2024/5/13 Boolean Algebra PJF - 59
60. More Logic Gates
• NAND and NOR Gates
– NAND and NOR circuits
– Two-level Implementations
– Multilevel Implementations
• Exclusive-OR (XOR) Gates
– Odd Function
– Parity Generation and Checking
2024/5/13 Boolean Algebra PJF - 60
61. More Logic Gates
• We can construct any combinational circuit with
AND, OR, and NOT gates
• Additional logic gates are used for practical reasons
2024/5/13 PJF - 61
Boolean Algebra
63. NAND Gate
• Known as a “universal” gate because ANY
digital circuit can be implemented with NAND
gates alone.
• To prove the above, it suffices to show that
AND, OR, and NOT can be implemented using
NAND gates only.
2024/5/13 Boolean Algebra PJF - 63
64. NAND Gate Emulation
2024/5/13 Boolean Algebra PJF - 64
X
X
F = (X•X)’
= X’+X’
= X’
X
Y
Y
F = ((X•Y)’)’
= (X’+Y’)’
= X’’•Y’’
= X•Y
F = (X’•Y’)’
= X’’+Y’’
= X+Y
X
X
F = X’
X
Y
Y
F X•Y
F = X+Y
65. NAND Circuits
• To easily derive a NAND implementation of a
boolean function:
– Find a simplified SOP
– SOP is an AND-OR circuit
– Change AND-OR circuit to a NAND circuit
– Use the alternative symbols below
2024/5/13 PJF - 65
Boolean Algebra
66. AND-OR (SOP) Emulation
Using NANDs
2024/5/13 Boolean Algebra PJF - 66
a) Original SOP
b) Implementation with NANDs
Two-level implementations
67. AND-OR (SOP) Emulation
Using NANDs (cont.)
2024/5/13 Boolean Algebra PJF - 67
Verify:
(a) G = WXY + YZ
(b) G = ( (WXY)’ • (YZ)’ )’
= (WXY)’’ + (YZ)’’ = WXY + YZ
68. SOP with NAND
(a) Original SOP
(b) Double inversion and grouping
(c) Replacement with NANDs
2024/5/13 Boolean Algebra PJF - 68
AND-NOT
NOT-OR
69. Two-Level NAND Gate
Implementation - Example
F (X,Y,Z) = m(0,6)
1. Express F in SOP form:
F = X’Y’Z’ + XYZ’
2. Obtain the AND-OR implementation for F.
3. Add bubbles and inverters to transform AND-
OR to NAND-NAND gates.
2024/5/13 Boolean Algebra PJF - 69
71. Multilevel NAND Circuits
Starting from a multilevel circuit:
1. Convert all AND gates to NAND gates with AND-NOT
graphic symbols.
2. Convert all OR gates to NAND gates with NOT-OR
graphic symbols.
3. Check all the bubbles in the diagram. For every
bubble that is not counteracted by another bubble
along the same line, insert a NOT gate or
complement the input literal from its original
appearance.
2024/5/13 Boolean Algebra PJF - 71
72. Example
Use NAND gates
and NOT gates to
implement
Z=E’F(AB+C’+D’)+GH
AB
AB+C’+D’
E’F(AB+C’+D’)
E’F(AB+C’+D’)+GH
2024/5/13 Boolean Algebra PJF - 72
74. NOR Gate
• Also a “universal” gate because ANY digital
circuit can be implemented with NOR gates
alone.
• This can be similarly proven as with the NAND
gate.
2024/5/13 PJF - 74
Boolean Algebra
75. NOR Circuits
• To easily derive a NOR implementation of a boolean
function:
– Find a simplified POS
– POS is an OR-AND circuit
– Change OR-AND circuit to a NOR circuit
– Use the alternative symbols below
2024/5/13 PJF - 75
Boolean Algebra
76. Two-Level NOR Gate
Implementation - Example
F(X,Y,Z) = m(0,6)
1. Express F’ in SOP form:
1. F’ = m(1,2,3,4,5,7)
= X’Y’Z + X’YZ’ + X’YZ + XY’Z’ + XY’Z + XYZ
2. F’ = XY’ + X’Y + Z
2. Take the complement of F’ to get F in the POS form:
F = (F’)' = (X'+Y)(X+Y')Z'
3. Obtain the OR-AND implementation for F.
4. Add bubbles and inverters to transform OR-AND
implementation to NOR-NOR implementation.
2024/5/13 Boolean Algebra PJF - 76
78. XOR and XNOR
X Y F = XY
0 0 0
0 1 1
1 0 1
1 1 0
2024/5/13 PJF - 78
Boolean Algebra
Y
F
XOR: “not-equal” gate
X Y F = XY
0 0 1
0 1 0
1 0 0
1 1 1
X
Y
F
XNOR: “equal” gate
X
79. Exclusive-OR (XOR) Function
• XOR (also ) : the “not-equal” function
• XOR(X,Y) = X Y = X’Y + XY’
• Identities:
– X 0 = X
– X 1 = X’
– X X = 0
– X X’ = 1
• Properties:
– X Y = Y X
– (X Y) W = X ( Y W)
2024/5/13 Boolean Algebra PJF - 79