- The document discusses Boolean algebra and its properties.
- Boolean algebra defines binary operators for logical operations like AND (.) and OR (+). It also defines identities and inverses.
- The algebra can be represented by truth tables to show it obeys properties like closure, identity, inverse, commutative, associative, and distributive laws.
- Boolean functions and expressions can be simplified, converted to canonical and standard forms using sums of minterms and products of maxterms.
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.
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.
1. Boolean algebra is a mathematical system used to specify and transform logic functions. It uses binary variables that take on values of 1 or 0 and logical operators like AND, OR, and NOT.
2. Logic gates implement logic functions physically using electronic components. Common gates are AND, OR, and NOT. Gates have a small but nonzero delay between input change and output change.
3. Boolean expressions can be represented using truth tables, logic diagrams, or algebraic expressions. Standard forms include sum-of-minterms and product-of-maxterms forms.
Theorems of Boolean algebra:
THEOREMS (a) (b)
1 Idempotent x + x = x x . x = x
2 Involution (x’)’ = x
3 Absorption x+ xy = x x (x+ y) = x
DeMorgan’s Theorems state that the complement of a product is equal to the sum of the complements and the complement of a sum is equal to the product of the complements. Boolean expressions can be simplified using properties, laws, and theorems of Boolean algebra such as consensus theorem.
This document discusses Boolean algebra and its relationship to digital logic gates. It begins with an introduction to Boolean algebra, defined by George Boole in the 19th century as a mathematical system to represent logical thought. Boolean algebra uses two values (true/false, on/off, 1/0) and binary operators like AND, OR, and NOT. The document then provides the axiomatic definition of Boolean algebra, describing its basic elements, operators, and properties. Finally, it discusses two-valued Boolean algebra specifically and how it represents binary logic used in digital circuits.
The document provides an overview of Boolean algebra, which is used to analyze and simplify digital circuits. It discusses Boolean algebra laws and operations, Boolean functions and their canonical forms, and methods for simplifying Boolean functions including algebraic simplification and Karnaugh maps. The key topics covered are Boolean algebra basics, laws and theorems, canonical forms such as SOP and POS, and simplification techniques including algebraic manipulation using laws and visualization using Karnaugh maps.
George Boole invented Boolean algebra in 1854 to represent logical operations using algebra. Boolean algebra uses two values, TRUE and FALSE, and defines operators like AND, OR, and NOT. It became useful for digital circuit design when Claude Shannon applied it to telephone switching circuits. Boolean algebra is defined by elements, operators, and axioms like closure, commutativity, identity, and inverse elements. The theorems of Boolean algebra can be proven using truth tables that list all possible combinations of input variables.
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.
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.
1. Boolean algebra is a mathematical system used to specify and transform logic functions. It uses binary variables that take on values of 1 or 0 and logical operators like AND, OR, and NOT.
2. Logic gates implement logic functions physically using electronic components. Common gates are AND, OR, and NOT. Gates have a small but nonzero delay between input change and output change.
3. Boolean expressions can be represented using truth tables, logic diagrams, or algebraic expressions. Standard forms include sum-of-minterms and product-of-maxterms forms.
Theorems of Boolean algebra:
THEOREMS (a) (b)
1 Idempotent x + x = x x . x = x
2 Involution (x’)’ = x
3 Absorption x+ xy = x x (x+ y) = x
DeMorgan’s Theorems state that the complement of a product is equal to the sum of the complements and the complement of a sum is equal to the product of the complements. Boolean expressions can be simplified using properties, laws, and theorems of Boolean algebra such as consensus theorem.
This document discusses Boolean algebra and its relationship to digital logic gates. It begins with an introduction to Boolean algebra, defined by George Boole in the 19th century as a mathematical system to represent logical thought. Boolean algebra uses two values (true/false, on/off, 1/0) and binary operators like AND, OR, and NOT. The document then provides the axiomatic definition of Boolean algebra, describing its basic elements, operators, and properties. Finally, it discusses two-valued Boolean algebra specifically and how it represents binary logic used in digital circuits.
The document provides an overview of Boolean algebra, which is used to analyze and simplify digital circuits. It discusses Boolean algebra laws and operations, Boolean functions and their canonical forms, and methods for simplifying Boolean functions including algebraic simplification and Karnaugh maps. The key topics covered are Boolean algebra basics, laws and theorems, canonical forms such as SOP and POS, and simplification techniques including algebraic manipulation using laws and visualization using Karnaugh maps.
George Boole invented Boolean algebra in 1854 to represent logical operations using algebra. Boolean algebra uses two values, TRUE and FALSE, and defines operators like AND, OR, and NOT. It became useful for digital circuit design when Claude Shannon applied it to telephone switching circuits. Boolean algebra is defined by elements, operators, and axioms like closure, commutativity, identity, and inverse elements. The theorems of Boolean algebra can be proven using truth tables that list all possible combinations of input variables.
Logic circuits are the basis of digital computer systems and operate using binary logic and Boolean algebra. Binary logic uses variables that can only have two values, 1 or 0, and logical operations on these variables. There are three basic logical operations: AND, OR, and NOT. Logic gates are electronic circuits that perform logical operations on inputs and produce an output. Boolean algebra uses rules and properties to describe logical relationships between binary variables. Logisim is a digital design tool that can be used to design and simulate logic circuits.
boolean_algebra.pdf for discrete mathematicssomnathmule3
This document discusses Boolean algebra and Boolean functions. It covers topics like Boolean operations, Boolean expressions, identities in Boolean algebra, and representing Boolean functions. Specifically, it shows that any Boolean function can be represented by a Boolean sum of Boolean products of the variables and their complements. It also introduces the concept of duality and defines a Boolean algebra in abstract terms.
George Boole developed Boolean algebra in 1854 to simplify and analyze complex logical expressions using binary logic. Boolean algebra uses logical symbols like 0 and 1 to represent digital circuit inputs and outputs, and laws and rules to reduce complex expressions to equivalent simpler forms using fewer logic gates. Some key laws and rules include the commutative, associative, distributive, absorption, and De Morgan's laws. Boolean functions describe logical relationships between variables and can be represented by algebraic expressions or truth tables. Methods like Karnaugh maps are used to minimize expressions.
George Boole developed Boolean algebra between 1815-1864 as an algebra of logic to represent logical statements as algebraic expressions. Boolean algebra uses two values, True and False (represented by 1 and 0 respectively) and logical operators like AND, OR, and NOT to represent logical statements and perform operations on them. Boolean algebra finds application in digital circuits where it is used to perform logical operations. Canonical forms and Karnaugh maps are techniques used to simplify Boolean expressions into their minimal forms.
The document discusses Boolean algebra and logic gates. It begins by defining an algebra as a mathematical system consisting of a set of elements, operators, and axioms. Boolean algebra was developed by George Boole in 1854 to model binary logic. It defines a set of two elements (0 and 1), and binary operators for AND, OR, and NOT. Huntington later formalized the postulates of Boolean algebra, including closure, identity elements, commutativity, distributivity, and complements. The document then proves several theorems of Boolean algebra, such as absorption and De Morgan's laws, using only the original postulates. It also discusses duality, consensus theorem, and operator precedence in Boolean expressions.
This document provides an introduction to Boolean algebra and its applications in digital logic. It discusses how Boolean algebra was developed by George Boole in the 1800s as an algebra of logic to represent logical statements as either true or false. The document then explains how Boolean algebra is used to perform logical operations in digital computers by representing true as 1 and false as 0. It introduces the basic logical operators of AND, OR, and NOT and provides their truth tables. The rest of the document discusses topics such as logic gates, truth tables, minterms, maxterms, and how to realize Boolean functions using sum of products and product of sums forms.
Boolean algebra is a system of logical operations developed by George Boole in the 19th century. It represents logical statements as expressions of binary variables, where true is represented by 1 and false by 0. The fundamental logical operations are AND, OR, and NOT. Boolean algebra finds application in digital circuits, where it is used to perform logical operations using electrical switches representing 1s and 0s. Boolean functions can be expressed in canonical forms such as Sum of Products (SOP) or Product of Sum (POS) and simplified using algebraic rules or Karnaugh maps to minimize the number of logic gates required.
Boolean algebra is an algebra of logic developed by George Boole between 1815-1864 to represent logical statements as an algebra of true and false. It is used to perform logical operations in digital computers by representing true as 1 and false as 0. The fundamental logical operators are AND, OR, and NOT. Boolean algebra expressions can be represented in sum of products (SOP) form or product of sums (POS) form and minimized using algebraic rules or Karnaugh maps. Minterms and maxterms are used to derive Boolean functions from truth tables in canonical SOP or POS form.
This document provides an overview of Boolean algebra and logic gates. It begins with an outline of topics covered, including Boolean algebra, logic operators, theorems, functions, and digital logic gates. It then introduces Boolean algebra as dealing with binary numbers and being important for designing computer logic circuits. Key concepts discussed include logical operators like AND, OR, and NOT; truth tables; Boolean functions; minterms and maxterms; canonical forms; and conversions between sum of products and product of sums forms. George Boole is identified as the founder of Boolean algebra. Truth tables and examples are provided to illustrate logical operators and evaluating Boolean expressions.
Lesson 2 : Logic Gates and Boolean Algebra
Part 1
Content:
1 .Boolean Theorem
2. Logic gates and Universal gates
Part 2
Content :
1. Standard SOP and POS
forms
2. Minterms and Maxterms
3. Karnaugh Map
P.S. Part 2 content will be uploaded later
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.
Digital logic circuits important question and answers for 5 unitsLekashri Subramanian
This document provides information about digital logic circuits and binary operations. It includes definitions of key terms like registers, register transfer, binary logic, logic gates, and parity bits. It also covers operations like subtraction using 2's and 1's complements, and reducing Boolean expressions using De Morgan's theorems, duality properties, and canonical 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.
The document discusses Boolean algebra and logic gates. It begins with an introduction to Boolean algebra and how it provides a mathematical framework for digital electronic systems. It then covers topics such as Boolean variables, expressions, logic gates like AND, OR, NOT, NAND, NOR, XOR and XNOR. It defines Boolean functions and discusses ways of representing functions using truth tables, logic diagrams and algebraic expressions. It also covers concepts like positive and negative logic, postulates and theorems of Boolean algebra, Venn diagrams, canonical and standard forms of logic functions including sum of products and product of sums forms.
The document provides information about digital logic circuits including definitions of binary logic, steps for binary to decimal and hexadecimal conversions, classification of binary codes, logic gates, combinational logic circuits like multiplexers, decoders, encoders, and comparators. It also includes properties of Boolean algebra and methods for minimizing Boolean functions using Karnaugh maps and Quine-McCluskey method. Various problems are given involving binary arithmetic, logic gate implementations, Boolean expressions and their simplification.
This document provides an overview of Boolean algebra and logic gates. It discusses basic logic gates like AND, OR, and NOT. It also covers other logic operations like NAND, NOR, EXOR and EXNOR. The document defines Boolean algebra and its postulates. It explains logic levels, positive and negative logic. It also discusses simplification of Boolean expressions, canonical and standard forms, and the use of Venn diagrams and minterms. The key topics covered are the basic concepts of Boolean algebra and digital logic that form the foundation for working with logic gates and circuits.
Boolean algebra is a mathematical system for specifying and transforming logic functions using binary operators. It defines an algebra structure on a set of elements with operators like AND and OR that satisfy certain axioms. Boolean algebra is used to study the design of digital systems by representing logic functions as sums of terms called minterms or products of terms called maxterms. Functions can be manipulated algebraically by techniques like complementing, using identities and theorems.
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.
This document provides an overview of digital circuits and logic concepts. It discusses number systems including binary, octal, hexadecimal and complements. It also covers logic gates, Boolean algebra, Karnaugh maps, logic families and integrated circuits. Specifically, it defines OR, AND, NOT and universal gates. It describes properties of Boolean algebra including commutative, associative and distributive properties. It also explains DeMorgan's theorems, sum of products and product of sums expressions. Finally, it discusses different logic families including transistor-transistor logic and metal-oxide semiconductor circuits.
E-sports refers to competitive video gaming. It began in 1972 and the first championship was held in 1981 with 10,000 participants. By 2013, an estimated 71.5 million people worldwide watched e-sports. The global e-sports market was worth $1.17 billion in 2021 and is expected to reach $5.74 billion by 2030. Popular e-sports games include Fortnite, BGMI, League of Legends, Dota 2, and Counter-Strike: Global Offensive. Careers in e-sports include content creation, event management, journalism, and more. However, excessive playing can lead to isolation, addiction, and health issues from lack of physical activity.
This document summarizes four main types of pollution: air, water, noise, and land pollution. It defines each type and discusses their causes and effects. For air pollution, it notes major sources are industries and automobiles, and effects include impacts on human health, animals, plants, and the atmosphere. For water pollution, it states 40% of deaths are caused by contaminated water from industrial and sewage waste. Noise pollution stems from traffic, construction, industries, and has health impacts like hearing loss and stress. Land pollution results from mining, household garbage, and industrial waste contaminating the earth. The document provides suggestions for preventing each type of pollution.
Logic circuits are the basis of digital computer systems and operate using binary logic and Boolean algebra. Binary logic uses variables that can only have two values, 1 or 0, and logical operations on these variables. There are three basic logical operations: AND, OR, and NOT. Logic gates are electronic circuits that perform logical operations on inputs and produce an output. Boolean algebra uses rules and properties to describe logical relationships between binary variables. Logisim is a digital design tool that can be used to design and simulate logic circuits.
boolean_algebra.pdf for discrete mathematicssomnathmule3
This document discusses Boolean algebra and Boolean functions. It covers topics like Boolean operations, Boolean expressions, identities in Boolean algebra, and representing Boolean functions. Specifically, it shows that any Boolean function can be represented by a Boolean sum of Boolean products of the variables and their complements. It also introduces the concept of duality and defines a Boolean algebra in abstract terms.
George Boole developed Boolean algebra in 1854 to simplify and analyze complex logical expressions using binary logic. Boolean algebra uses logical symbols like 0 and 1 to represent digital circuit inputs and outputs, and laws and rules to reduce complex expressions to equivalent simpler forms using fewer logic gates. Some key laws and rules include the commutative, associative, distributive, absorption, and De Morgan's laws. Boolean functions describe logical relationships between variables and can be represented by algebraic expressions or truth tables. Methods like Karnaugh maps are used to minimize expressions.
George Boole developed Boolean algebra between 1815-1864 as an algebra of logic to represent logical statements as algebraic expressions. Boolean algebra uses two values, True and False (represented by 1 and 0 respectively) and logical operators like AND, OR, and NOT to represent logical statements and perform operations on them. Boolean algebra finds application in digital circuits where it is used to perform logical operations. Canonical forms and Karnaugh maps are techniques used to simplify Boolean expressions into their minimal forms.
The document discusses Boolean algebra and logic gates. It begins by defining an algebra as a mathematical system consisting of a set of elements, operators, and axioms. Boolean algebra was developed by George Boole in 1854 to model binary logic. It defines a set of two elements (0 and 1), and binary operators for AND, OR, and NOT. Huntington later formalized the postulates of Boolean algebra, including closure, identity elements, commutativity, distributivity, and complements. The document then proves several theorems of Boolean algebra, such as absorption and De Morgan's laws, using only the original postulates. It also discusses duality, consensus theorem, and operator precedence in Boolean expressions.
This document provides an introduction to Boolean algebra and its applications in digital logic. It discusses how Boolean algebra was developed by George Boole in the 1800s as an algebra of logic to represent logical statements as either true or false. The document then explains how Boolean algebra is used to perform logical operations in digital computers by representing true as 1 and false as 0. It introduces the basic logical operators of AND, OR, and NOT and provides their truth tables. The rest of the document discusses topics such as logic gates, truth tables, minterms, maxterms, and how to realize Boolean functions using sum of products and product of sums forms.
Boolean algebra is a system of logical operations developed by George Boole in the 19th century. It represents logical statements as expressions of binary variables, where true is represented by 1 and false by 0. The fundamental logical operations are AND, OR, and NOT. Boolean algebra finds application in digital circuits, where it is used to perform logical operations using electrical switches representing 1s and 0s. Boolean functions can be expressed in canonical forms such as Sum of Products (SOP) or Product of Sum (POS) and simplified using algebraic rules or Karnaugh maps to minimize the number of logic gates required.
Boolean algebra is an algebra of logic developed by George Boole between 1815-1864 to represent logical statements as an algebra of true and false. It is used to perform logical operations in digital computers by representing true as 1 and false as 0. The fundamental logical operators are AND, OR, and NOT. Boolean algebra expressions can be represented in sum of products (SOP) form or product of sums (POS) form and minimized using algebraic rules or Karnaugh maps. Minterms and maxterms are used to derive Boolean functions from truth tables in canonical SOP or POS form.
This document provides an overview of Boolean algebra and logic gates. It begins with an outline of topics covered, including Boolean algebra, logic operators, theorems, functions, and digital logic gates. It then introduces Boolean algebra as dealing with binary numbers and being important for designing computer logic circuits. Key concepts discussed include logical operators like AND, OR, and NOT; truth tables; Boolean functions; minterms and maxterms; canonical forms; and conversions between sum of products and product of sums forms. George Boole is identified as the founder of Boolean algebra. Truth tables and examples are provided to illustrate logical operators and evaluating Boolean expressions.
Lesson 2 : Logic Gates and Boolean Algebra
Part 1
Content:
1 .Boolean Theorem
2. Logic gates and Universal gates
Part 2
Content :
1. Standard SOP and POS
forms
2. Minterms and Maxterms
3. Karnaugh Map
P.S. Part 2 content will be uploaded later
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.
Digital logic circuits important question and answers for 5 unitsLekashri Subramanian
This document provides information about digital logic circuits and binary operations. It includes definitions of key terms like registers, register transfer, binary logic, logic gates, and parity bits. It also covers operations like subtraction using 2's and 1's complements, and reducing Boolean expressions using De Morgan's theorems, duality properties, and canonical 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.
The document discusses Boolean algebra and logic gates. It begins with an introduction to Boolean algebra and how it provides a mathematical framework for digital electronic systems. It then covers topics such as Boolean variables, expressions, logic gates like AND, OR, NOT, NAND, NOR, XOR and XNOR. It defines Boolean functions and discusses ways of representing functions using truth tables, logic diagrams and algebraic expressions. It also covers concepts like positive and negative logic, postulates and theorems of Boolean algebra, Venn diagrams, canonical and standard forms of logic functions including sum of products and product of sums forms.
The document provides information about digital logic circuits including definitions of binary logic, steps for binary to decimal and hexadecimal conversions, classification of binary codes, logic gates, combinational logic circuits like multiplexers, decoders, encoders, and comparators. It also includes properties of Boolean algebra and methods for minimizing Boolean functions using Karnaugh maps and Quine-McCluskey method. Various problems are given involving binary arithmetic, logic gate implementations, Boolean expressions and their simplification.
This document provides an overview of Boolean algebra and logic gates. It discusses basic logic gates like AND, OR, and NOT. It also covers other logic operations like NAND, NOR, EXOR and EXNOR. The document defines Boolean algebra and its postulates. It explains logic levels, positive and negative logic. It also discusses simplification of Boolean expressions, canonical and standard forms, and the use of Venn diagrams and minterms. The key topics covered are the basic concepts of Boolean algebra and digital logic that form the foundation for working with logic gates and circuits.
Boolean algebra is a mathematical system for specifying and transforming logic functions using binary operators. It defines an algebra structure on a set of elements with operators like AND and OR that satisfy certain axioms. Boolean algebra is used to study the design of digital systems by representing logic functions as sums of terms called minterms or products of terms called maxterms. Functions can be manipulated algebraically by techniques like complementing, using identities and theorems.
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.
This document provides an overview of digital circuits and logic concepts. It discusses number systems including binary, octal, hexadecimal and complements. It also covers logic gates, Boolean algebra, Karnaugh maps, logic families and integrated circuits. Specifically, it defines OR, AND, NOT and universal gates. It describes properties of Boolean algebra including commutative, associative and distributive properties. It also explains DeMorgan's theorems, sum of products and product of sums expressions. Finally, it discusses different logic families including transistor-transistor logic and metal-oxide semiconductor circuits.
E-sports refers to competitive video gaming. It began in 1972 and the first championship was held in 1981 with 10,000 participants. By 2013, an estimated 71.5 million people worldwide watched e-sports. The global e-sports market was worth $1.17 billion in 2021 and is expected to reach $5.74 billion by 2030. Popular e-sports games include Fortnite, BGMI, League of Legends, Dota 2, and Counter-Strike: Global Offensive. Careers in e-sports include content creation, event management, journalism, and more. However, excessive playing can lead to isolation, addiction, and health issues from lack of physical activity.
This document summarizes four main types of pollution: air, water, noise, and land pollution. It defines each type and discusses their causes and effects. For air pollution, it notes major sources are industries and automobiles, and effects include impacts on human health, animals, plants, and the atmosphere. For water pollution, it states 40% of deaths are caused by contaminated water from industrial and sewage waste. Noise pollution stems from traffic, construction, industries, and has health impacts like hearing loss and stress. Land pollution results from mining, household garbage, and industrial waste contaminating the earth. The document provides suggestions for preventing each type of pollution.
The document describes a student feedback system that allows students to electronically rate and evaluate faculty performance. It reduces the manual effort of reviewing paper feedback forms. Students and faculty can access the system without being physically present. The system includes features for administrators to add/manage faculty and students, view feedback and analytics. It also includes features for faculty to view feedback and for students to register, provide feedback, and update profiles. The system is built using PHP, HTML, Bootstrap and stores data in a MySQL database.
This document introduces a fitness app called Blast N Burn Fitness created by a group of developers including Fenil Talaviya, Dhruv Desai, Naman Patel, and Darshak Ramani. It discusses the benefits of fitness apps for motivation and accessibility and highlights advantages like cost efficiency, personalized workouts and routines, safety, and connecting with online fitness communities. Contact information is provided for the website, YouTube channel, and Instagram account promoting the app which is still in development.
Networking and communication technologies have evolved greatly over time. The document discusses the history of the ARPANET and TCP/IP protocol leading to the modern Internet. It also defines important networking terms, describes the growth of broadband enabling new uses, and how organizations can implement internal and external networks. Cloud computing provides on-demand services over the Internet.
Point to Point Protocol (PPP) is a data link protocol commonly used to establish dial-up connections between a host and internet service provider. PPP encapsulates packets for transmission and provides error detection but not correction. It allows for negotiation of network layer addresses and establishment of a connection while keeping the protocol simple and streamlined. PPP frames contain flags for framing, an address field that is unused, a control field also currently unused, and a protocol field to indicate the upper layer protocol of the encapsulated data.
This is an overview of my current metallic design and engineering knowledge base built up over my professional career and two MSc degrees : - MSc in Advanced Manufacturing Technology University of Portsmouth graduated 1st May 1998, and MSc in Aircraft Engineering Cranfield University graduated 8th June 2007.
Data Communication and Computer Networks Management System Project Report.pdfKamal Acharya
Networking is a telecommunications network that allows computers to exchange data. In
computer networks, networked computing devices pass data to each other along data
connections. Data is transferred in the form of packets. The connections between nodes are
established using either cable media or wireless media.
An In-Depth Exploration of Natural Language Processing: Evolution, Applicatio...DharmaBanothu
Natural language processing (NLP) has
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as machine translation, email spam detection,
information extraction, summarization, healthcare,
and question answering. This paper first delineates
four phases by examining various levels of NLP and
components of Natural Language Generation,
followed by a review of the history and progression of
NLP. Subsequently, we delve into the current state of
the art by presenting diverse NLP applications,
contemporary trends, and challenges. Finally, we
discuss some available datasets, models, and
evaluation metrics in NLP.
We have designed & manufacture the Lubi Valves LBF series type of Butterfly Valves for General Utility Water applications as well as for HVAC applications.
1. Devang Patel Institute of Advance Technology and Research
CE252 : Digital Electronics
AY : 2022-23
Chapter 2
Boolean Algebra
Presented By :
Prof. Bhavika Patel
DEPSTAR, FTE
2.
3. Binary Operator: A Binary Operator defined on a set S of
elements is a rule that assigns to each pair of elements from S a
unique element from S.
i.e. Consider the relation, a * b = c
* is a binary operator, if it specifies a rule, for finding c from the
pair (a,b), and also a,b,c∈S.
However, * is not a binary operator if a, b ∈S , while the rule
finds c ∈ S
4. Postulates: Assumptions
1. Closure: A set is closed with respect to a binary operator if,
for every pair of elements of S, the binary operator specifies
a rule for obtaining a unique element of S.
for the set of Natural Numbers N = {1,2,3,4,5…….}
N is closed with binary operator (+) for a,b ∈ N we obtain
unique c ∈ N
But N is not closed with Binary operator (-)
because 2 – 3 = -1 when -1 ∈ N
5. 2. Associative Law : A binary operator (*) on a set S is said to be associative
whenever
(X * Y) * Z = X * (Y * Z) for all X,Y,Z ∈ S
3. Commutative Law: A binary operator (*) on a set S is said to be commutative
whenever
X * Y = Y * X for all X,Y ∈ S
4. Identity Element: A set S is to have an identity element with respect to a
binary operation (*) on S, if there exists an element E ∈ S with the property
E * X = X * E = X for every X ∈ S
i.e. 0 is identity element with respect to operation ( + )
0 + X = X + 0 = X for every X ∈ S
6. 5. Inverse: If a set S has the identity element E with respect to a binary
operator (*), there exists an element X ∈ S, which is called the inverse,
for every Y ∈ S, such that
X * Y = E
i.e. In the set of integers I with E = 0, the inverse of an element X is
(-X) since
X + (–X) = 0
6. Distributive Law: If (*) and (+) are two binary operators on a set S, (*)
is said to be distributive over (+), whenever
A * (B + C) = (A * B) + (A * C)
7. SUMMARY
The binary operator (+) defines addition.
The additive identity is 0.
The additive inverse defines subtraction.
The binary operator (.) defines multiplication.
The multiplication identity is 1.
The multiplication inverse of A is 1/A, defines division i.e., A . 1/A = 1
The only distribution law applicable is that of (.) over (+)
A . (B + C) = (A . B) + (A . C)
8. DEFINITION OF BOOLEAN ALGEBRA
In 1854 George Boole introduced a systematic approach of logic and developed an
algebraic system to treat the logic functions, which is now called Boolean
algebra.
In 1938 C.E. Shannon developed a two-valued Boolean algebra called Switching
algebra, and demonstrated that the properties of two-valued or bistable
electrical switching circuits can be represented by this algebra.
9. The following postulates are satisfied for the definition of Boolean algebra on a set
of elements S together with two binary operators (+) and (.)
1. (a) Closure with respect to the operator (+).
(b) Closure with respect to the operator (.).
2. (a) An identity element with respect to + is designated by 0 i.e.,
X + 0 = 0 + X = X
(b) An identity element with respect to . is designated by 1 i.e.,
X . 1 = 1 . X = X
3. (a) Commutative with respect to (+), i.e., X + Y = Y + X
(b) Commutative with respect to (.), i.e., X . Y = Y . X
10. 4. (a) (.) is distributive over (+), i.e., X . (Y + Z) = (X . Y) + (X . Z)
(b) (+) is distributive over (.), i.e., X + (Y .Z) = (X + Y) . (X + Z)
5. For every element X ∈ S, there exists an element X' ∈ S (called the
complement of X) such that X + X’ = 1 and X . X’ = 0
6. There exists at least two elements X,Y ∈ S, such that X is not equal to Y
i.e. In Boolean algebra 0 & 1
11. TWO-VALUED BOOLEAN ALGEBRA
Two-valued Boolean algebra is defined on a set of only two elements,
S={0,1}, with rules for two binary operators (+) and (.) and inversion or
complement as shown In the following operator tables at Figures 1, 2, and 3
respectively.
12. 1. Closure is obviously valid, as form the table it is observed that the result of each
operation is either 0 or 1 and 0,1 ∈ S.
2. From the tables, we can see that
(i) X + 0 = 0 + X = X 0 + 0 = 0 + 0 = 0 1 + 0 = 0 + 1 = 1
(ii) X . 1 = 1 . X = X 1 . 1 = 1 . 1 = 1 0 . 1 = 1 . 0 = 0
which verifies the two identity elements 0 for (+) and 1 for (.) as defined by
postulate 2.
3. The commutative laws are confirmed by the symmetry of binary operator tables.
X + Y = Y + X
1 + 0 = 0 + 1
13. 4. The distributive laws of (.) over (+) i.e., A . (B+C) = (A . B) + (A . C), and (+) over (.)
i.e., A + ( B . C) = (A+B) . (A+C) can be shown to be applicable with the help of the
truth tables considering all the possible values of A, B, and C as under. From the
complement table it can be observed that
14. (C) A + A’ = 1, since, 0 + 0’= 0 + 1 = 1 1 + 1’ = 1 + 0 = 1
(D) A . A’ = 0, since, 0 . 0’ = 0 . 1 = 0 1 . 1’ = 1 . 0 = 0
5. Postulate 6 also satisfies two-valued Boolean algebra that has two distinct
elements 0 and 1 where 0 is not equal to 1
15. BASIC PROPERTIES AND THEOREMS OF BOOLEAN ALGEBRA
Principle of Duality
From postulates, it is evident that they are grouped in pairs as part (a) and (b) and
every algebraic expression deductible from the postulates of Boolean algebra
remains valid if the operators and identity elements are interchanged.
This means one expression can be obtained from the other in each pair by
interchanging every element i.e., every 0 with 1, every 1 with 0, as well as
interchanging the operators i.e., every (+) with (.) and every (.) with (+). This
important property of Boolean algebra is called principle of duality.
i.e. 0 <->1 & + <-> .
(a) 0 . 1 = 0 <-> (b) 1 + 0 = 1
i.e. (b) is dual of (a)
16. The following is the complete list of postulates and theorems useful for two-valued
Boolean algebra
17. Theorem 1(a): A + A = A
A + A = (A + A).1 by postulate 2(b)
= (A + A) . ( A + A’) by postulate 5(a)
= A + (A.A’) by postulate 4(b)
= A + 0 by postulate 5(b)
= A by postulate 2(a)
Theorem 1(b): A . A = A
A . A = (A . A) + 0 by postulate 2(a)
= (A . A) + ( A . A’) by postulate 5(b)
= A (A + A’) by postulate 4(a)
= A . 1 by postulate 5(b)
= A by postulate 2(b)
18. Theorem 2(a): A + 1 = 1
Theorem 2(b): A . 0 = 0 by Duality
Theorem 6(a): A + A.B = A
A + A.B = A . 1 + A.B by postulate 2(b)
= A ( 1 + B) by postulate 4(a)
= A . 1 by theorem 2(a)
= A by postulate 2(b)
Theorem 6(b): A ( A + B ) = A by Duality
Venn Diagram
19. • Truth tables :
Both sides of the relation are checked to yield identical results for all possible
combinations of variables involved.
The following truth table verifies the 6(a) absorption theorem.
Following Truth Table verifies the De Morgan’s theorem (x+y)’=x’.y’
X Y XY X+XY
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
X Y X+Y (X+Y)’ X’ Y’ X’Y’
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0
20. • BOOLEAN FUNCTIONS
Binary variables have two values, either 0 or 1.
A Boolean function is an expression formed with binary variables, the two binary
operators AND and OR, one unary operator NOT, parentheses and equal sign.
The value of a function may be 0 or 1, depending on the values of variables present
in the Boolean function or expression. For example, if a Boolean function is
expressed algebraically as
F = AB’C
then the value of F will be 1, when A = 1, B = 0, and C = 1.
For other values of A, B, C the value of F is 0
21. Boolean functions can also be represented by Truth Tables. A truth table is the
tabular form of the values of a Boolean function according to the all possible
values of its variables. For an n variable, 2n combinations of 1s and 0s are listed
and one column represents function values according to the different
combinations of variables
For example, for three variables the Boolean function F = AB + C truth table can be
written as below in Figure
22. SIMPLIFICATION OF BOOLEAN EXPRESSIONS
• Simplify the Boolean function F = X + (X’ . Y)
= (X + X’) . (X + Y) (4(b))
= 1 . (X + Y) (5(b))
= X + Y (2(b))
• Simplify the Boolean function F = X . (X’+Y)
= (X . X’) + (X . Y)
= 0 + (X . Y)
= (X . Y)
• Simplify the Boolean function F = (X’ . Y’ . Z) + (X’ . Y . Z) + (X . Y’)
= (X’ . Z)(Y’ + Y) + (X . Y’)
= (X’ .Z) . 1 + (X . Y’)
Implementation of Boolean functions with gate (Circuit)
23. Simplify following Boolean Function into minimum
number of Literals and Draw Circuit Diagram using
AND, OR and NOT Gate
1. F=xy+x’z+yz
2. F=(x+y)(x’+z)(y+z)
3. F=xy+xy’
4. F=xyz+x’y+xyz’
5. F=y(wz’+wz)+xy
6. F=(a + b)' (a' + b')’
7. F = BC + AC' + AB + BCD
24. Canonical And Standard Forms
Boolean function expressed as a sum of minterms or product of
maxterms are said to be canonical form.
Standard form: Sum Of Product (SOP) : AB’ + AC’ + ABC
Product Of Sum (POS) : (A + B’) ▪ (A +C’) ▪ (A + B + C)
SUM -> OR gate
PRODUCT -> AND gate
25. Minterm
A product term containing all N variables of the function in either true or
complemented form is called the minterm. Each minterm is obtained by an AND
operation of the variables in their true form or complemented form.
For a two-variable function, four different combinations are possible, such as, A’B’,
A’B, AB’, and AB. These product terms are called the fundamental products or
standard products or minterms.
Note that, in the minterm, a variable will possess the value 1, if it is in true or
uncomplemented form, whereas, it contains the value 0, if it is in complemented
form .
Symbol for Minterm is mj, where j denotes the decimal equivalent of binary
number
26. • For three variables function, eight minterms are possible as listed in the following
table in Figure
27. Sum of Minterms
F = A + B’C
A = A (B + B’)
= AB + AB’
= AB(C +C’) + AB’(C + C’)
= ABC + ABC’ + AB’C + AB’C’
B’C = B’C (A +A’) = AB’C + A’B’C
F = ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C
= ABC + ABC’ + AB’C + AB’C’ + A’B’C
= A’B’C + AB’C’ + AB’C + ABC’ + ABC
= m1 + m4 + m5 + m6 + m7
F (A,B,C) = Σ (1, 4, 5, 6, 7)
28. Steps to obtain Sum of MINTERMS
• Find out how many variables is there in equation
• List out all minterms for given variables. i.e m0 to m7 for 3 variables
• Check the given equation is in SOP FORM or not? If not then convert it into SOP
form. i.e ab + bc + ac’ form
• Find out missing variable in each term & add that variable in that term. i.e for
A,B,C: B is missing in AC’. Add B in AC’ using AC’ = AC’(B+B’) = ABC’ + AB’C’
• ADD ALL MISSING VARIABLES IN EACH TERMS OF THE EQUATION
• ARRANGE ALL OBTAINED MINTERMS IN ASCENDING ORDERS. i.e from m0 to
m7 for 3 variables
29. MAXTERM
A sum term containing all n variables of the function in either true or
complemented form is called the maxterm. Each maxterm is obtained by an OR
operation of the variables in their true form or complemented form.
Four different combinations are possible for a two-variable function, such as, A’ +
B’, A’ + B, A + B', and A + B. These sum terms are called the standard sums or
maxterms.
Note that, in the maxterm, a variable will possess the value 0, if it is in true or
uncomplemented form, whereas, it contains the value 1, if it is in complemented
form.
Symbol for Maxterm is Mj, where j denotes the decimal equivalent of binary
number
30. • Like minterms, for a three-variable function, eight maxterms are also possible as
listed in the following table
31. Product of Maxterms
F = xy + x’z
= (xy + x’)(xy + z)
= (x + x’) (y + x’) (x + z) (y + z)
= (x’ + y) (x + z) (y + z)
x’ + y = x’ + y + zz’ = (x’ + y + z) (x’ + y + z’)
x + z = x + z + yy’ = (x + z + y) (x+ z + y’)
y + z = y + z + xx’ = (y + z + x) (y + z + x’)
F = (x + y + z) (x + y’ + z) (x’ + y + z) (x’ + y + z’)
= M0M2M4M5
F (x, y, z) = П (0, 2, 4, 5)
32. Steps to obtain PRODUCT OF MAXTERMS
• Find out how many variables is there in given equation
• List out all maxterms for given variables. i.e M0 to M7 for 3 variables
• Check the given equation is in POS FORM or not? If not then convert it into POS
form. i.e (a + b)(b + c)(a +c’) form
• Find out missing variable in each term & add that variable in that term. i.e for
A,B,C: B is missing in (A + C’). Add B in (A + C’) using (A + C’) = (A + C’)+BB’
= (A + B + C)(A + B’ + C)
• ADD ALL MISSING VARIABLES IN EACH TERMS OF THE EQUATION
• ARRANGE ALL OBTAINED MAXTERMS IN ASCENDING ORDERS. i.e from M0 to
M7 for 3 variables
34. • Express the following functions in a sum of minterms and product of
maxterms.
1. F(A,B,C,D) = D(A' + B) + B’D
2. F(W,X,Y,Z) = Y'Z + WXY' + WXZ' + W'X’Z
3. F(A,B,C) = (A' + B)(B' + C)
4. F(W,X,Y,Z) = (W + Y + Z) (X + Y') (W' + Z)
35. Conversion Between Canonical Forms
F(A, B, C) = Σ (1, 4, 5, 6, 7) (Sum Of Product) (Sum Of Minterms)
F’(A, B, C) = Σ (0, 2, 3) = m0 + m2 + m3
complement of F’ = F = (m0 + m2 + m3)’
= m0’ . m2’ . m3’
= M0.M2.M3
= П (0, 2, 3) (Product Of Sum) (Product Of Maxterms)
mj’ = Mj
F (A, B, C) = Σ (1, 4, 5, 6, 7) = П (0, 2, 3)
37. IC DIGITAL LOGIC FAMILIES
TTL Transistor – Transistor Logic
ECL Emitter – Coupled Logic
MOS Metal – Oxide Semiconductor
CMOS Complementary Metal – Oxide Semiconductor
IIL Integrated – Injection Logic
TTL has an extensive list of digital functions and is currently the most popular logic
family.
ECL is used in system requiring high speed operations.
MOS & IIL are used in circuits requiring high component density (Large Number of
Components like Transistors)
CMOS is used in system requiring low power consumption.
38. TTL - 5400 & 7400 SERIES i.e. 7400 7402 7432
ECL – 10000 SERIES i.e. 10102(Quad 2-Input NOR Gate) & 10107(triple–2 input
exclusive OR/NOR gate)
CMOS – 4000 SERIES i.e. 4002(dual 4-input NOR gate)
39. Positive and Negative Logic
Logic Value Signal Value Logic Value Signal Value
1 H 0 H
0 L 1 L
POSITIVE LOGIC NEGATIVE LOGIC
40. Special Characteristics:
Fan out: specifies the number of standard loads(means amount of current needed
by an input of another gate in same IC) that the output of a gate can drive
without impairing its normal operation.
Power dissipation: is the supplied power required to operate the gate. This
parameter is expressed in milliwatts (mW). It represents the power delivered to
the gate from power supply.
Propagation delay: is the average transition delay time for a signal to propagate
from input to output when the binary signals change in value. This parameter is
expressed in nanoseconds (ns)
Noise Margin: is the maximum noise voltage added to the input signal of a digital
circuit that does not cause an undesirable change in the circuit output.
41. SUMMARY OF CHAPTER 2
Basic Definitions
Axiomatic definitions of Boolean Algebra
Basic Theorems & Properties of Boolean Algebra
Boolean Functions
Canonical & Standard Forms
Minterms & Maxterms
Digital Logic Gates
IC Digital Logic Families