Minterm, Maxterm and Karnaugh Maps:
Introduction, minterms and sum of minterm form, maxterm and Product
of maxterm form, Reduction technique using Karnaugh maps –
2/3/4/5/6 variable K-maps, Grouping of variables in K-maps, K-maps
for product of sum form, minimize Boolean expression using K-map
and obtain K-map from Boolean expression, Quine Mc Cluskey
Method.
FYBSC IT Digital Electronics Unit I Chapter I Number System and Binary Arithm...Arti Parab Academics
Number System:
Analog System, digital system, numbering system, binary number
system, octal number system, hexadecimal number system, conversion
from one number system to another, floating point numbers, weighted
codes binary coded decimal, non-weighted codes Excess – 3 code, Gray
code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code,
Hollerith Code, Morse Code, Teletypewriter (TTY), Error detection
and correction, Universal Product Code, Code conversion.
FYBSC IT Digital Electronics Unit II Chapter I Boolean Algebra and Logic GatesArti Parab Academics
Boolean Algebra and Logic Gates:
Introduction, Logic (AND OR NOT), Boolean theorems, Boolean
Laws, De Morgan’s Theorem, Perfect Induction, Reduction of Logic
expression using Boolean Algebra, Deriving Boolean expression from
given circuit, exclusive OR and Exclusive NOR gates, Universal Logic
gates, Implementation of other gates using universal gates, Input
bubbled logic, Assertion level.
FYBSC IT Digital Electronics Unit IV Chapter II Sequential Circuits- Flip-FlopsArti Parab Academics
Sequential Circuits: Flip-Flop:
Introduction, Terminologies used, S-R flip-flop, D flip-fop, JK flipflop, Race-around condition, Master – slave JK flip-flop, T flip-flop, conversion from one type of flip-flop to another, Application of flipflops.
This document discusses digital registers and counters. It defines latches and flip flops, which are basic memory elements that can store single bits of data. Registers are groups of flip flops that can store multiple bits and are used to hold information in digital systems. Shift registers can shift data in one or both directions, while cyclic registers can shift in both directions. Parallel-in serial-out registers load data in parallel and output it serially. Counters are registers that sequence through states upon each input pulse and are used to count events. Asynchronous counters have external clocks connected to each flip flop, while synchronous counters receive a common external clock.
This document discusses binary coded decimal (BCD). It defines BCD as a numerical code that assigns a 4-bit binary code to each decimal digit from 0 to 9. Numbers larger than 9 are expressed digit by digit in BCD. BCD is used because it is easy to encode/decode decimals and useful for digital systems that display decimal outputs. The document also describes how addition and subtraction are performed in BCD through binary addition rules and handling carries.
1) Binary codes represent numbers, letters, and other data using groups of bits or symbols. Weighted binary codes follow a positional weighting principle where each bit position represents a specific weight.
2) Non-weighted codes like excess-3 code and Gray code do not assign positional weights. Gray code is used in shaft position encoders to prevent multiple bit changes that can cause problems.
3) BCD (binary coded decimal) represents each decimal digit with a 4-bit binary number, allowing representation of numbers from 0-9. BCD addition can result in numbers outside the valid 0-9 range, requiring carries between digits.
FYBSC IT Digital Electronics Unit I Chapter I Number System and Binary Arithm...Arti Parab Academics
Number System:
Analog System, digital system, numbering system, binary number
system, octal number system, hexadecimal number system, conversion
from one number system to another, floating point numbers, weighted
codes binary coded decimal, non-weighted codes Excess – 3 code, Gray
code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code,
Hollerith Code, Morse Code, Teletypewriter (TTY), Error detection
and correction, Universal Product Code, Code conversion.
FYBSC IT Digital Electronics Unit II Chapter I Boolean Algebra and Logic GatesArti Parab Academics
Boolean Algebra and Logic Gates:
Introduction, Logic (AND OR NOT), Boolean theorems, Boolean
Laws, De Morgan’s Theorem, Perfect Induction, Reduction of Logic
expression using Boolean Algebra, Deriving Boolean expression from
given circuit, exclusive OR and Exclusive NOR gates, Universal Logic
gates, Implementation of other gates using universal gates, Input
bubbled logic, Assertion level.
FYBSC IT Digital Electronics Unit IV Chapter II Sequential Circuits- Flip-FlopsArti Parab Academics
Sequential Circuits: Flip-Flop:
Introduction, Terminologies used, S-R flip-flop, D flip-fop, JK flipflop, Race-around condition, Master – slave JK flip-flop, T flip-flop, conversion from one type of flip-flop to another, Application of flipflops.
This document discusses digital registers and counters. It defines latches and flip flops, which are basic memory elements that can store single bits of data. Registers are groups of flip flops that can store multiple bits and are used to hold information in digital systems. Shift registers can shift data in one or both directions, while cyclic registers can shift in both directions. Parallel-in serial-out registers load data in parallel and output it serially. Counters are registers that sequence through states upon each input pulse and are used to count events. Asynchronous counters have external clocks connected to each flip flop, while synchronous counters receive a common external clock.
This document discusses binary coded decimal (BCD). It defines BCD as a numerical code that assigns a 4-bit binary code to each decimal digit from 0 to 9. Numbers larger than 9 are expressed digit by digit in BCD. BCD is used because it is easy to encode/decode decimals and useful for digital systems that display decimal outputs. The document also describes how addition and subtraction are performed in BCD through binary addition rules and handling carries.
1) Binary codes represent numbers, letters, and other data using groups of bits or symbols. Weighted binary codes follow a positional weighting principle where each bit position represents a specific weight.
2) Non-weighted codes like excess-3 code and Gray code do not assign positional weights. Gray code is used in shaft position encoders to prevent multiple bit changes that can cause problems.
3) BCD (binary coded decimal) represents each decimal digit with a 4-bit binary number, allowing representation of numbers from 0-9. BCD addition can result in numbers outside the valid 0-9 range, requiring carries between digits.
The document discusses minimizing Boolean expressions using Karnaugh maps. It explains that Karnaugh maps provide a graphical way to simplify logic circuits by grouping adjacent 1s in the map. The steps for minimization using Karnaugh maps are outlined, including drawing the map, entering values, forming the largest possible groups of 1s, and selecting the fewest groups needed to cover all 1s. Rules for grouping such as group size and overlap are also covered.
The document discusses Boolean expressions and their use in computer programming. It defines Boolean expressions as expressions that evaluate to true or false. Boolean expressions are composed of logical operators like AND, OR, and NOT. The document then discusses different logical operators and their truth tables. It also covers Boolean algebra identities and theorems. Finally, it introduces concepts like minterms, maxterms, sum of products, and product of sums and how Karnaugh maps can be used to simplify Boolean expressions.
This document discusses multiplexers and demultiplexers. It defines them as devices that allow digital information from several sources to be routed onto a single line (multiplexers) or distributed to multiple output lines (demultiplexers). The key properties of multiplexers and demultiplexers are described, including the relationship between the number of inputs, outputs, and selection lines. Examples of implementing multiplexers and demultiplexers using logic gates are provided.
This document provides an introduction to digital electronics and digital signals. It discusses the basics of analog and digital signals, with digital signals taking on discrete voltage levels compared to the continuous variation of analog signals. The advantages of digital techniques are explained, such as increased noise immunity and reliability. Common number systems are introduced, including binary, octal, hexadecimal and decimal, along with methods for converting between them. The key concepts of bytes, coding and voltage assignments in digital circuits are also covered at a high level.
Shift registers are constructed using flip-flops connected in a way to store and transfer digital data. Data is stored at the Q output of D flip-flops during a clock pulse. Shift registers allow data to be transferred between flip-flops upon a clock edge. There are four types of data movement: serial in serial out, serial in parallel out, parallel in serial out, and parallel in parallel out. Shift registers can be loaded serially or in parallel and are used in applications like pseudo random pattern generators, ring counters, and Johnson counters.
This document discusses counters, which are digital circuits used for counting pulses. It describes asynchronous and synchronous counters, and different types including up/down counters, decade counters, ring counters, and Johnson counters. Examples of counter applications are given such as in kitchen appliances, washing machines, microwaves, and programmable logic controllers. Counters are used for tasks like time measurement, frequency division, and digital signal generation.
Fixed-point and floating-point numbers can be represented in computers using binary numbers. Floating-point numbers represent numbers in scientific notation with a sign, mantissa, and exponent. In 8-bit floating point, numbers use 1 bit for sign, 3 bits for exponent, and 4 bits for mantissa, such as 0.001 x 21 = 2.25. Larger precision formats such as 32-bit and 64-bit floating point according to the IEEE standard use more bits for exponent and mantissa.
The document discusses string operations in 8086 assembly language. String operations allow copying, searching, storing, and comparing strings of characters in memory. The direction flag (DF) determines whether string operations proceed from left to right or right to left. Instructions like MOVSB, STOSB, LODSB, CMPSB operate on bytes or words in strings and update index registers accordingly based on the DF value. Examples demonstrate using these instructions to copy, store, load, compare, and search strings.
The document discusses various number systems including decimal, binary, octal, and hexadecimal. It provides examples of converting between these different bases using techniques like dividing by the base, tracking remainders, and grouping bits. Common powers are also defined for bases 10 and 2. The key concepts covered are representation of quantities in different number systems, conversion between number systems, addition and multiplication in binary, and representing fractions in binary.
The Reason Why we use master slave JK flip flop instead of simple level triggered flip flop is Racing condition which can be successfully avoided using two SR latches fed with inverted clocks.
This document discusses different types of codes used to encode information for transmission and storage. It begins by explaining that encoding is required to send information unambiguously over long distances and that decoding is needed to retrieve the original information. It then provides reasons for using coding, such as increasing transmission efficiency and enabling error correction. The document proceeds to describe binary coding and how increasing the number of bits allows more items to be uniquely represented. It also discusses properties of good codes like ease of use and error detection. Specific code types are then outlined, including binary coded decimal codes, unit distance codes, error detection codes, and alphanumeric codes. Gray code and excess-3 code are explained as examples.
FYBSC IT Digital Electronics Unit I Chapter II Number System and Binary Arith...Arti Parab Academics
Binary Arithmetic:
Binary addition, Binary subtraction, Negative number representation,
Subtraction using 1’s complement and 2’s complement, Binary
multiplication and division, Arithmetic in octal number system,
Arithmetic in hexadecimal number system, BCD and Excess – 3
arithmetic.
The document discusses a 4-bit binary coded decimal (BCD) adder. A BCD adder can add two 4-bit BCD numbers and produce a 4-bit BCD sum. If the sum exceeds 9 in decimal, a carry is generated. The document also mentions full adders, 7-segment displays, and 74LS47 BCD to 7-segment decoder/driver ICs. It provides an explanation of why 6 is added to BCD sums that exceed 9.
The document discusses various number systems including binary, decimal, octal and hexadecimal. It covers how to convert between these number systems using techniques like dividing by the base, tracking remainders, and grouping bits. Examples are provided for converting between the different systems. Common number prefixes like kilo, mega and giga are also explained in the context of computing.
The document provides an introduction to Karnaugh maps (K-maps), which are a graphical method for simplifying Boolean logic expressions. It outlines the basics of K-maps, including their advantages, different types (2, 3, 4 and 5 variable maps), and how to group cells and derive simplified logic expressions. The document also discusses sum-of-products (SOP) form, product-of-sums (POS) form, don't care conditions, prime implicants, and includes examples of mapping truth tables to K-maps and simplifying expressions.
Booth's multiplication algorithm was invented by Andrew D. Booth in 1951 while studying crystallography at Birkbeck College in London. It improves the speed of computer multiplication by reducing the number of additions or subtractions needed. The algorithm uses a grid with the multiplicand in the top row, the negative multiplicand in the middle row, and the multiplier in the bottom row. It then iteratively shifts and adds or subtracts based on the last two bits of the product to build up the final result in fewer steps than standard addition methods. Several examples are provided to demonstrate how the algorithm works.
This presentation summarizes Karnaugh maps, which are a graphical technique for simplifying Boolean expressions. Karnaugh maps arrange the terms of a truth table in a two-dimensional grid, making common factors between terms visible. They can be used for functions with up to five variables. Examples show how to identify groupings of terms and simplify expressions using Karnaugh maps for two, three, and four variables. The presentation concludes with an example of a five variable Karnaugh map.
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 information on combinational logic circuits and summarizes steps for analyzing combinational logic problems using truth tables and Karnaugh maps. It begins by defining combinational circuits as those whose outputs solely depend on current inputs, as opposed to sequential circuits which use memory elements. It then provides examples of writing truth tables and deriving Boolean expressions from problem statements. The document also covers standard forms of sum of products and product of sums, and methods for simplifying expressions using Karnaugh maps including grouping cells and rules for grouping.
The document discusses minimizing Boolean expressions using Karnaugh maps. It explains that Karnaugh maps provide a graphical way to simplify logic circuits by grouping adjacent 1s in the map. The steps for minimization using Karnaugh maps are outlined, including drawing the map, entering values, forming the largest possible groups of 1s, and selecting the fewest groups needed to cover all 1s. Rules for grouping such as group size and overlap are also covered.
The document discusses Boolean expressions and their use in computer programming. It defines Boolean expressions as expressions that evaluate to true or false. Boolean expressions are composed of logical operators like AND, OR, and NOT. The document then discusses different logical operators and their truth tables. It also covers Boolean algebra identities and theorems. Finally, it introduces concepts like minterms, maxterms, sum of products, and product of sums and how Karnaugh maps can be used to simplify Boolean expressions.
This document discusses multiplexers and demultiplexers. It defines them as devices that allow digital information from several sources to be routed onto a single line (multiplexers) or distributed to multiple output lines (demultiplexers). The key properties of multiplexers and demultiplexers are described, including the relationship between the number of inputs, outputs, and selection lines. Examples of implementing multiplexers and demultiplexers using logic gates are provided.
This document provides an introduction to digital electronics and digital signals. It discusses the basics of analog and digital signals, with digital signals taking on discrete voltage levels compared to the continuous variation of analog signals. The advantages of digital techniques are explained, such as increased noise immunity and reliability. Common number systems are introduced, including binary, octal, hexadecimal and decimal, along with methods for converting between them. The key concepts of bytes, coding and voltage assignments in digital circuits are also covered at a high level.
Shift registers are constructed using flip-flops connected in a way to store and transfer digital data. Data is stored at the Q output of D flip-flops during a clock pulse. Shift registers allow data to be transferred between flip-flops upon a clock edge. There are four types of data movement: serial in serial out, serial in parallel out, parallel in serial out, and parallel in parallel out. Shift registers can be loaded serially or in parallel and are used in applications like pseudo random pattern generators, ring counters, and Johnson counters.
This document discusses counters, which are digital circuits used for counting pulses. It describes asynchronous and synchronous counters, and different types including up/down counters, decade counters, ring counters, and Johnson counters. Examples of counter applications are given such as in kitchen appliances, washing machines, microwaves, and programmable logic controllers. Counters are used for tasks like time measurement, frequency division, and digital signal generation.
Fixed-point and floating-point numbers can be represented in computers using binary numbers. Floating-point numbers represent numbers in scientific notation with a sign, mantissa, and exponent. In 8-bit floating point, numbers use 1 bit for sign, 3 bits for exponent, and 4 bits for mantissa, such as 0.001 x 21 = 2.25. Larger precision formats such as 32-bit and 64-bit floating point according to the IEEE standard use more bits for exponent and mantissa.
The document discusses string operations in 8086 assembly language. String operations allow copying, searching, storing, and comparing strings of characters in memory. The direction flag (DF) determines whether string operations proceed from left to right or right to left. Instructions like MOVSB, STOSB, LODSB, CMPSB operate on bytes or words in strings and update index registers accordingly based on the DF value. Examples demonstrate using these instructions to copy, store, load, compare, and search strings.
The document discusses various number systems including decimal, binary, octal, and hexadecimal. It provides examples of converting between these different bases using techniques like dividing by the base, tracking remainders, and grouping bits. Common powers are also defined for bases 10 and 2. The key concepts covered are representation of quantities in different number systems, conversion between number systems, addition and multiplication in binary, and representing fractions in binary.
The Reason Why we use master slave JK flip flop instead of simple level triggered flip flop is Racing condition which can be successfully avoided using two SR latches fed with inverted clocks.
This document discusses different types of codes used to encode information for transmission and storage. It begins by explaining that encoding is required to send information unambiguously over long distances and that decoding is needed to retrieve the original information. It then provides reasons for using coding, such as increasing transmission efficiency and enabling error correction. The document proceeds to describe binary coding and how increasing the number of bits allows more items to be uniquely represented. It also discusses properties of good codes like ease of use and error detection. Specific code types are then outlined, including binary coded decimal codes, unit distance codes, error detection codes, and alphanumeric codes. Gray code and excess-3 code are explained as examples.
FYBSC IT Digital Electronics Unit I Chapter II Number System and Binary Arith...Arti Parab Academics
Binary Arithmetic:
Binary addition, Binary subtraction, Negative number representation,
Subtraction using 1’s complement and 2’s complement, Binary
multiplication and division, Arithmetic in octal number system,
Arithmetic in hexadecimal number system, BCD and Excess – 3
arithmetic.
The document discusses a 4-bit binary coded decimal (BCD) adder. A BCD adder can add two 4-bit BCD numbers and produce a 4-bit BCD sum. If the sum exceeds 9 in decimal, a carry is generated. The document also mentions full adders, 7-segment displays, and 74LS47 BCD to 7-segment decoder/driver ICs. It provides an explanation of why 6 is added to BCD sums that exceed 9.
The document discusses various number systems including binary, decimal, octal and hexadecimal. It covers how to convert between these number systems using techniques like dividing by the base, tracking remainders, and grouping bits. Examples are provided for converting between the different systems. Common number prefixes like kilo, mega and giga are also explained in the context of computing.
The document provides an introduction to Karnaugh maps (K-maps), which are a graphical method for simplifying Boolean logic expressions. It outlines the basics of K-maps, including their advantages, different types (2, 3, 4 and 5 variable maps), and how to group cells and derive simplified logic expressions. The document also discusses sum-of-products (SOP) form, product-of-sums (POS) form, don't care conditions, prime implicants, and includes examples of mapping truth tables to K-maps and simplifying expressions.
Booth's multiplication algorithm was invented by Andrew D. Booth in 1951 while studying crystallography at Birkbeck College in London. It improves the speed of computer multiplication by reducing the number of additions or subtractions needed. The algorithm uses a grid with the multiplicand in the top row, the negative multiplicand in the middle row, and the multiplier in the bottom row. It then iteratively shifts and adds or subtracts based on the last two bits of the product to build up the final result in fewer steps than standard addition methods. Several examples are provided to demonstrate how the algorithm works.
This presentation summarizes Karnaugh maps, which are a graphical technique for simplifying Boolean expressions. Karnaugh maps arrange the terms of a truth table in a two-dimensional grid, making common factors between terms visible. They can be used for functions with up to five variables. Examples show how to identify groupings of terms and simplify expressions using Karnaugh maps for two, three, and four variables. The presentation concludes with an example of a five variable Karnaugh map.
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 information on combinational logic circuits and summarizes steps for analyzing combinational logic problems using truth tables and Karnaugh maps. It begins by defining combinational circuits as those whose outputs solely depend on current inputs, as opposed to sequential circuits which use memory elements. It then provides examples of writing truth tables and deriving Boolean expressions from problem statements. The document also covers standard forms of sum of products and product of sums, and methods for simplifying expressions using Karnaugh maps including grouping cells and rules for grouping.
This document provides information on combinational logic circuits and techniques for analyzing them, including:
1. Combinational circuits have outputs that solely depend on current inputs, unlike sequential circuits which use memory elements.
2. Truth tables are used to represent the relationships between inputs and outputs, and techniques like Karnaugh maps can simplify Boolean expressions.
3. Karnaugh maps arrange minterms or maxterms in a grid, allowing groups of redundant variables to be identified and simplified. Standard forms like sum of products can be plotted and minimized on the map.
This document provides an overview of digital electronics and Boolean algebra topics, including:
- Boolean algebra deals with binary variables and logical operations. It originated from George Boole's 1854 book.
- Logic gates are basic building blocks of digital systems. Common logic gates include AND, OR, NOT, NAND, NOR gates.
- Boolean laws like commutative, associative, distributive, De Morgan's theorems are used to simplify logic expressions.
- Karnaugh maps are used to minimize logic expressions into sum of products or product of sums form. Don't care conditions allow for further simplification.
- Universal gates like NAND and NOR can be used to construct all other logic gates
1. The document discusses digital logic circuits and their use of binary logic with 0 representing false and 1 representing true.
2. It explains that complex digital logic circuits like computers can be built using basic logic gates that perform operations like NOT, AND, OR, etc.
3. Boolean algebra is used as a formal tool to describe and design complex binary logic circuits using gates.
The document discusses Boolean function minimization using Karnaugh maps. It begins by introducing Karnaugh maps and how they are used to simplify Boolean functions into logic circuits with the fewest gates and inputs. Different sized Karnaugh maps are demonstrated, including two-variable, three-variable, and four-variable maps. Techniques for simplifying functions based on the number of adjacent squares in the map are described. Several examples of using Karnaugh maps to minimize Boolean functions are provided.
K map in digital electronics for engineering studentsbanjareankita1
1. The document discusses K-maps, which are a method for minimizing Boolean expressions with 3-4 variables.
2. A K-map is a table that represents variable combinations and can be used to find Sum of Products (SOP) or Product of Sums (POS) expressions.
3. The process of using a K-map involves placing 1s and 0s in the cells based on maxterms or minterms, then grouping cells to find product terms that are summed or multiplied to find the minimal SOP or POS expression.
This document provides information about minimizing Boolean functions using Karnaugh maps. It discusses how Karnaugh maps can be used to simplify Boolean expressions into sums of products. Different examples are provided to demonstrate how to minimize functions with 2, 3, 4, and 5 variables using Karnaugh maps. Additional topics covered include don't care conditions, implementing logic with NAND and NOR gates, and exclusive OR functions.
The document discusses minterms, maxterms, and their representation using shorthand notation in digital logic. It also covers the steps to obtain the shorthand notation for minterms and maxterms. Standard forms such as SOP and POS are introduced along with methods to simplify boolean functions into canonical forms using Karnaugh maps. The implementation of boolean functions using NAND and NOR gates is also described through examples.
This document discusses Karnaugh maps (K-maps), which are a tool for simplifying Boolean expressions into their minimum sum-of-products (SOP) or product-of-sums (POS) forms. It explains how to map standard and non-standard SOP expressions onto K-maps by placing 1s in the cells corresponding to the terms. Groups of adjacent 1s can then be identified to find the minimum SOP expression. The same process applies to POS expressions, except 0s are placed on the map instead of 1s. Examples are provided to demonstrate how to group cells and determine the minimum expressions from K-maps.
Digital electronics k map comparators and their functionkumarankit06875
This document provides an overview of a digital electronics presentation covering K-maps, comparators, and their applications. The agenda includes an introduction to K-maps and how they are used to simplify Boolean expressions. It also covers comparators, their operation and function. Examples are given of using K-maps to minimize logic expressions and identify prime implicants. The applications of K-maps in digital circuit design optimization are discussed. Comparators and their use in examples is briefly outlined.
The document discusses Karnaugh map methods for minimizing Boolean functions. It introduces Karnaugh maps as a tool for representing Boolean functions with up to six variables. The key points covered are:
1. Karnaugh maps arrange the variables in a grid so that logically adjacent cells correspond to inputs that differ in only one variable.
2. Cells are marked with 1s or 0s based on the function's truth table. Adjacent 1s can be combined to eliminate variables and simplify the function.
3. Examples show how to use Karnaugh maps to minimize Boolean functions expressed as sums of products (SOP) or products of sums (POS).
4. "Don't care"
Canonical forms express Boolean functions as a sum of minterms or product of maxterms. Standard forms, like canonical forms, can have variables that do not appear in each term. Canonical forms are not usually minimal, while standard forms are simplified versions of canonical forms. Boolean functions can be expressed in Sum of Products or Product of Sum forms, and minimized using algebraic methods or Karnaugh maps.
The document discusses Karnaugh maps (K-maps), which are a tool for representing and simplifying Boolean functions with up to six variables. K-maps arrange the variables in a grid according to their binary values. Adjacent cells that differ in only one variable can be combined to simplify the function by eliminating that variable. The document provides examples of using K-maps to minimize Boolean functions in sum of products and product of sums form. It also discusses techniques like combining cells into the largest groups possible and handling don't-care conditions to further simplify expressions.
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.
This document discusses simplification of Boolean functions using Karnaugh maps. It describes two methods for simplification - algebraic and graphical. The graphical method uses Karnaugh maps, which arrange variables in a two-dimensional grid with 2n cells. Each cell represents a minterm. Adjacent minterms that are identical except for one variable can be combined. Several examples demonstrate constructing K-maps and simplifying functions down to prime implicants by grouping adjacent 1s. Don't care conditions are also introduced, where certain input combinations do not affect the output. The document concludes by showing a two-stage logic network example where K-maps are used to design the logic for each stage.
This document outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
The Karnaugh map method provides a graphical way to simplify logic equations or convert truth tables into logic circuits. It arranges variables in a grid so that adjacent squares differ in only one variable. Loops of adjacent 1s can then be identified to eliminate variables from the logic expression. Larger loops eliminate more variables - pairs eliminate one variable, quads eliminate two variables, and octets eliminate three variables. The method is demonstrated through examples of constructing Karnaugh maps from truth tables and simplifying the resulting logic expressions through looping.
Similar to FYBSC IT Digital Electronics Unit II Chapter II Minterm, Maxterm and Karnaugh Maps (20)
Data Security and Privacy:
Introduction to Data Security: Importance, common security threats.
Data Privacy: Privacy concerns in the digital age, protecting personal information online.
Introduction to Computer Fundamentals:
Overview of Computer Fundamentals: Definition, importance, and evolution of computers.
Computer Hardware: Central Processing Unit (CPU), memory (RAM and ROM), input and output devices, storage devices.
Computer Software: Operating systems, application software, programming languages. Computer Applications in psychology
Computer Ethics and Emerging Technologies:
Computer Ethics: Ethical considerations in computer usage, intellectual property rights, and plagiarism.
Emerging Technologies: Artificial Intelligence (AI), Internet of Things (IoT), Blockchain Technology.
Introduction to Computer Fundamentals:
Overview of Computer Fundamentals: Definition, importance, and evolution of computers.
Computer Hardware: Central Processing Unit (CPU), memory (RAM and ROM), input and output devices, storage devices.
Computer Software: Operating systems, application software, programming languages. Computer Applications in Healthcare
Computer Networks and Internet Basics:
Computer Networks: Introduction to networks, types of networks (LAN, WAN, WLAN), network topologies.
Networking Basics: Network components (routers, switches, hubs), IP addressing (IPv4, IPv6), TCP/IP Protocol.
Internet and World Wide Web: Understanding the Internet, web browsers, search engines, online research techniques.
The document discusses operating systems and software applications. It provides an overview of operating systems, including their functions, types, popular systems like Windows, macOS and Linux, and user interfaces. It describes file management with file systems, directory structures and common file operations. Key software applications are also mentioned like word processing, spreadsheets, presentations and databases.
Bioinformatics: Bioinformatics, Healthcare Informatics and Analytics for Improved Healthcare System, Intelligent Monitoring and Control for Improved Healthcare System.
Protocols and Evidence based Healthcare: information technology tools to support best practices in health care, information technology tools that inform and empower patients.
Clinical Decision Support Systems: Making Decisions, the impact health information technology on the delivery of care in a rapidly changing healthcare marketplace.
Design and Evaluation of Information Systems and Services: principles of designing information systems, strategies for Information system evaluation, Information Systems Effectiveness Measures.
This document provides information on quality improvement strategies, protocols, and evidence-based healthcare. It discusses principles of designing information systems and strategies for evaluating them. It also covers quality improvement tools like the PDCA cycle and factors that help create and sustain healthcare informatics as a new field. The learning objectives are outlined on quality improvement tools, factors to create healthcare informatics, and understanding the PDCA cycle. The introduction defines quality and different approaches to defining it. Six criteria for right healthcare are also mentioned.
Information Privacy and Security: The Value and Importance of Health Information Privacy, security of health data, potential technical approaches to health data privacy and security.
Electronic Health Records: purpose of electronic health records, popular electronic health record system, advantages of electronic records, challenges of electronic health records, the key players involved.
Overview of Health Informatics: survey of fundamentals of health information technology, Identify the forces behind health informatics, educational and career opportunities in health informatics.
Information System Acquisition & Lifecycle: system acquisition process, phases: Initiation, Planning, Procurement, System Development, System Implementation, Maintenance & Operations, and Closeout. development models.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
FYBSC IT Digital Electronics Unit II Chapter II Minterm, Maxterm and Karnaugh Maps
1. BOOLEAN ALGEBRA
AND LOGIC GATES
&
MINTERM, MAXTERM
AND KARNAUGH
MAPS
UNIT II
DIGITAL
ELECTRONICS
PROF.ARTI GAVAS-PARAB
ANNA LEELA COLLEGE OF COMMERCE AND ECONOMICS, SHOBHA
JAYARAM SHETTY COLLEGE FOR BMS
CHAPTER II
2. UNIT II: CONTENTS
Boolean Algebra and Logic Gates:
Introduction, Logic (AND OR NOT),
Boolean theorems, Boolean Laws, De Morgan’s Theorem,
Perfect Induction, Reduction of Logic expression using Boolean Algebra,
Deriving Boolean expression from given circuit, exclusive OR and Exclusive NOR gates,
Universal Logic gates, Implementation of other gates using universal gates, Input bubbled logic,Assertion level.
Minterm, Maxterm and Karnaugh Maps:
Introduction, minterms and sum of minterm form, maxterm and Product of maxterm form,
Reduction technique using Karnaugh maps – 2/3/4/5/6 variable K-maps, Grouping of variables in K-maps, K-maps for product
of sum form,
Mminimize Boolean expression using K-map and obtain K-map from Boolean expression,
Quine Mc Cluskey Method.
C
H
A
P
T
E
R
I
C
H
A
P
T
E
R
II
3. MINTERM, MAXTERM AND KARNAUGH MAPS: INTRODUCTION
MinTerm:-Product term
which contains each of the n
variables in either
complemented or
uncomplimented form.
MaxTerm:-Sum term which
contains each of the n
variables in either
complemented or
uncomplimented form.
4. BOOLEAN FUNCTION
A Boolean function is an algebraic form of Boolean expression.A Boolean expression is an expression which
consists of variables, constants (0-false and 1-true) and logical operators which results in true or false.
A Boolean function of n-variables is represented by f(x1, x2, x3….xn).
By using Boolean laws and theorems, we can simplify the Boolean functions of digital circuits.
A brief note of different ways of representing a Boolean function is shown below.
Sum-of-Products (SOP) Form
Product-of-sums (POS) form
Canonical forms
There are two types of canonical forms:
Sum-of-min terms or Canonical SOP
Product-of- max terms or Canonical POS
5. SUM OF MINTERM (SUM OF PRODUCTS (SOP))
The sum-of-products (SOP) form is a method (or form) of simplifying the Boolean expressions of logic gates.
In this SOP form of Boolean function representation, the variables are operated by AND (product) to form a
product term and all these product terms are ORed (summed or added) together to get the final function.
Examples
AB + ABC + CDE
(AB) ̅ + ABC + CD E ̅
SOP form can be obtained by
Writing an AND term for each input combination, which produces HIGH output.
Writing the input variables if the value is 1, and write the complement of the variable if its value is 0.
OR the AND terms to obtain the output function.
6. SUM OF PRODUCTS (SOP): EXAMPLE
Boolean expression for majority function F = A’BC + AB’C + ABC ‘ + ABC
Truth table:
For a 3-variable (x, y and z) Boolean function, the possible
minterms are:
x’y’z’, x’y’z, x’yz’, x’yz, xy’z’, xy’z, xyz’ and xyz.
•1 – Minterms = minterms for which the function F = 1.
•0 – Minterms = minterms for which the function F = 0.
7. PRODUCT OF MAXTERM (PRODUCT OF SUM (POS))
In this POS form, all the variables are ORed, i.e. written as sums to form sum terms.
All these sum terms are ANDed (multiplied) together to get the product-of-sum form.This form is exactly
opposite to the SOP form. So this can also be said as “Dual of SOP form”.
Examples
(A+B) * (A + B + C) * (C +D)
(A+B) ̅ * (C + D + E ̅)
POS form can be obtained by
Writing an OR term for each input combination, which produces LOW output.
Writing the input variables if the value is 0, and write the complement of the variable if its value is 1.
AND the OR terms to obtain the output function.
8. PRODUCT OF SUM (POS): EXAMPLE
Boolean expression for majority function F = (A + B + C) (A + B + C ‘) (A + B’ + C) (A’ + B + C)
Truth table:
For a 3-variable (x, y and z) Boolean function, the possible
maxterms are:
x + y + z, x + y + z’, x + y’ + z, x + y’ + z’, x’ + y + z, x’ + y + z’,
x’ + y’ + z and x’ + y’ + z’.
•1 – Max terms = max terms for which the function F = 1.
•0 – max terms = max terms for which the function F = 0.
9. CANONICAL FORM (STANDARD SOP FORM)
Any Boolean function that is expressed as a sum of minterms or as a product of max terms is said to
be in its “canonical form”.
Any Boolean function can be expressed as the sum (OR) of its 1- min terms.The representation of the
equation will be
F(list of variables) = Σ(list of 1-min term indices)
Ex: F (x, y, z) = Σ (3, 5, 6, 7)
The inverse of the function can be expressed as a sum (OR) of its 0- min terms.The representation of
the equation will be
F(list of variables) = Σ(list of 0-min term indices)
Ex: F’ (x, y, z) = Σ (0,1, 2, 4)
10. CANONICAL FORM (STANDARD POS FORM)
In max term, each variable is complimented, if its value is assigned to 1, and each variable is un-
complimented if its value is assigned to 0.
Any Boolean function can be expressed the product (AND) of its 0 – max terms.The representation
of the equation will be
F(list of variables) = Π (list of 0-max term indices)
Ex: F (x, y, z) = Π (0, 1, 2, 4)
The inverse of the function can be expressed as a product (AND) of its 1 – max terms.The
representation of the equation will be
F(list of variables) = Π (list of 1-max term indices)
Ex: F’ (x, y, z) = Π (3, 5, 6, 7)
11. REDUCTION TECHNIQUE USING KARNAUGH MAPS (K-MAPS) RULES
Karnaugh mapping is a graphic technique for reducing a Sum-of-Products (SOP) and Product-of-Sums (POS)
expression to its minimum form.
Two, three and four variable k-maps will have 4, 8 and 16 cells respectively.
Each cell of the k-map corresponds to a particular combination of the input variable and between adjacent cells
only one variable is allowed to change (As in Gray Code Sequence).
Use the following steps to reduce an expression using a k-map.
Mark each term of the SOP/POS expression in the correct cell of the k-map. (kind of like the game Battleship)
Circle adjacent cells in groups of 2, 4 or 8 making the circles as large as possible. (NO DIAGONALS!)
Write a term for each circle in a final SOP/POS expression.The variables in a term are the ones that remain constant
across a circle.
The cells of a k-map are continuous left-to-right and top-to-bottom.The wraparound feature can be used to draw
the circles as large as possible.
When a variable does not appear in the original equation, the equation must be plotted so that all combinations
of the missing variable(s) are covered.
12. K-MAPS FOR 2 TO 5VARIABLES
Karnaugh introduced a method for simplification of Boolean functions in an easy way.
This method is known as Karnaugh map method or K-map method. It is a graphical method, which consists of 2n
cells for ‘n’ variables.The adjacent cells are differed only in single bit position.
K-Map method is most suitable for minimizing Boolean functions of 2 variables to 5 variables. Now, let us discuss
about the K-Maps for 2 to 5 variables one by one.
would be 00, 01, 10, 11. It is 00, 01, 11 10, which is Gray code sequence. Gray code sequence only changes one
binary bit as we go from one number to the next in the sequence, unlike binary.
13. 2VARIABLE K-MAP
The number of cells in 2 variable K-map is four, since the number of variables is two.The following figure shows 2
variable K-Map.
There is only one possibility of grouping 4 adjacent min terms.
The possible combinations of grouping 2 adjacent min terms are {(m0, m1), (m2, m3), (m0, m2) and (m1, m3)}.
14. 3VARIABLE K-MAP
The number of cells in 3 variable K-map is eight, since the number of variables is three.The following figure shows
3 variable K-Map.
There is only one possibility of grouping 8 adjacent min terms.
The possible combinations of grouping 4 adjacent min terms are {(m0, m1, m3, m2), (m4, m5, m7, m6), (m0, m1, m4,
m5), (m1, m3, m5, m7), (m3, m2, m7, m6) and (m2, m0, m6, m4)}.
The possible combinations of grouping 2 adjacent min terms are {(m0, m1), (m1, m3), (m3, m2), (m2, m0), (m4, m5),
(m5, m7), (m7, m6), (m6, m4), (m0, m4), (m1, m5), (m3, m7) and (m2, m6)}.
If x=0, then 3 variable K-map becomes 2 variable K-map.
15. 4VARIABLE K-MAP
The number of cells in 4 variable K-map is sixteen, since the number of variables is four.The following figure
shows 4 variable K-Map.
There is only one possibility of grouping 16 adjacent min terms.
Let R1, R2, R3 and R4 represents the min terms of first row, second row, third row and fourth row respectively.
Similarly, C1, C2, C3 and C4 represents the min terms of first column, second column, third column and fourth
column respectively.The possible combinations of grouping 8 adjacent min terms are {(R1, R2), (R2, R3), (R3, R4),
(R4, R1), (C1, C2), (C2, C3), (C3, C4), (C4, C1)}.
If w=0, then 4 variable K-map becomes 3 variable K-map.
16. 5VARIABLE K-MAP
The number of cells in 5 variable K-map is thirty-two, since the number of variables is 5.The following figure
shows 5 variable K-Map.
There is only one possibility of grouping 32 adjacent min terms.
There are two possibilities of grouping 16 adjacent min terms.
i.e., grouping of min terms from m0 to m15 and m16 to m31.
If v=0, then 5 variable K-map becomes 4 variable K-map.
17. MINIMIZATION OF BOOLEAN FUNCTIONS USING K-MAPS
Follow these rules for simplifying K-maps in order to get standard sum of products form.
Select the respective K-map based on the number of variables present in the Boolean function.
If the Boolean function is given as sum of min terms form, then place the ones at respective min term cells in the
K-map. If the Boolean function is given as sum of products form, then place the ones in all possible cells of K-map
for which the given product terms are valid.
Check for the possibilities of grouping maximum number of adjacent ones. It should be powers of two. Start from
highest power of two and upto least power of two. Highest power is equal to the number of variables considered
in K-map and least power is zero.
Each grouping will give either a literal or one product term. It is known as prime implicant.The prime implicant
is said to be essential prime implicant, if atleast single ‘1’ is not covered with any other groupings but only that
grouping covers.
Note down all the prime implicants and essential prime implicants.The simplified Boolean function contains all
essential prime implicants and only the required prime implicants.
18. MINIMIZATION OF BOOLEAN FUNCTIONS USING K-MAPS
Note 1 − If outputs are not defined for some combination of inputs, then those
output values will be represented with don’t care symbol ‘x’.That means, we can
consider them as either ‘0’ or ‘1’.
Note 2 − If don’t care terms also present, then place don’t cares ‘x’ in the respective
cells of K-map. Consider only the don’t cares ‘x’ that are helpful for grouping
maximum number of adjacent ones. In those cases, treat the don’t care value as ‘1’.
19. GROUPING OF K-MAPVARIABLES
The square that contains ‘1’ should
be taken in simplifying, at least once.
The square that contains ‘1’ can be
considered as many times as the
grouping is possible with it.
Group shouldn’t include any zeros
(0).
A group should be the as large as
possible.
Groups can be horizontal or vertical.
Grouping of variables in diagonal
manner is not allowed.
20. 2VARIABLE K-MAP: EXAMPLE
A general representation
of a 2 variable K-map
plot is shown below.
Example
Simplify the given 2-variable
Boolean equation by using
K-map.
F = XY’ + X’Y + X’Y’
Here the lower right cell is used in both
groups.
3) After grouping the variables, the next
step is determining the minimized
expression.
By reducing each group, we obtain a
conjunction of the minimized expression
such as by taking out the common terms
from two groups, i.e. X’ andY’.
4) So the reduced equation will be X’ +Y’.
1) We put 1 at the output terms given
in equation.
In this K-map, we can create 2 groups
by following the rules for grouping,
one is by combining (X’,Y) and (X’,Y’)
terms and the other is by combining
(X,Y’) and (X’,Y’) terms.
21. 2VARIABLE K-MAP: EXAMPLE
Example 1:
Consider the following map.The
function plotted is:
Z = f(A,B) = A + AB
Z=A
22. 3VARIABLE K-MAP: EXAMPLE
A typical plot of a 3-variable K-map is
shown below. It can be observed that the
positions of columns 10 and 11 are
interchanged so that there is only change
in one variable across adjacent cells.This
modification will allow in minimizing the
logic.
We put 1 at the output terms
given in equation.
Example
Simplify the given 3-variable Boolean equation by
using k-map.
F = X’Y Z + X’Y’ Z + XY Z’ + X’Y’ Z’ + XY Z + XY’ Z’
And in both the terms, we have ‘Y’ in common. So the group of size
4 is reduced as the conjunctionY.To consume every cell which has
1 in it, we group the rest of cells to form size 2 group, as shown
below.
The 2 size group has no common variables, so they are written
with their variables and its conjugates.
So the reduced equation will be X Z’ +Y’ + X’ Z. In this equation,
no further minimization is possible.
23. 3VARIABLE K-MAP: EXAMPLE
By using the rules of simplification and ringing
of adjacent cells in order to make as many
variables redundant, the minimised result
obtained is
By using the rules of simplification and ringing
of adjacent cells in order to make as many
variables redundant, the minimised result
obtained is
24. K-MAP OF 3VARIABLES (SOP)
Z= ∑A,B,C(1,3,6,7)
From red group we get product term—
A’C
From green group we get product term—
AB
Summing these product terms we get-
Final expression (A’C+AB)
25. K-MAP OF 3VARIABLES (POS)
From red group
we find terms
A B C’
Taking complement
of these two
A’ B’ C
Now sum up them
(A’ + B’ + C)
From brown group
we find terms
A’ B’ C’
Taking complement of
these two
A B C
Now sum up them
(A + B + C)
We will take product of these three terms
:Final expression (A’ + B’ + C) (B’ + C’)
(A + B + C)
From green group
we find terms
B C
Taking complement
of these two terms
B’ C’
Now sum up them
(B’+C’)
Example:
F(A,B,C)=π(0,3,6,7)
26. This Boolean expression has seven product terms.They
are mapped top to bottom and left to right on the K-
map above.
For example, the first P-term A’B’CD is the first
row, 3rd cell, corresponding to map location A=0, B=0,
C=1, D=1.
The other product terms are placed in a similar manner.
Encircling the largest groups possible, two groups of
four are shown above.
The dashed horizontal group corresponds to the
simplified product term AB.The vertical group
corresponds to Boolean CD. Since there are two
groups, there will be two product terms in the Sum-Of-
Products result of Out=AB+CD.
Example:
f(A,B,C,D)=
4VARIABLE K-MAP: EXAMPLE
27. 4VARIABLE K-MAP: EXAMPLE
Fold up the corners of the map here like it is a napkin to make the
four cells physically adjacent.
The four cells here are a group of four because they all have the
Boolean variables B’ and D’ in common. In other words, B=0 for
the four cells, and D=0 for the four cells.
The other variables (A, C) are 0 in some cases, 1 in other cases
with respect to the four corner cells.
Thus, these variables (A, C) are not involved with this group of
four.This single group comes out of the map as one product term
for the simplified result: Out=B’D’
Example:
f(A,B,C,D)=
28. 4VARIABLE K-MAP: EXAMPLE
For the K-map here, roll the top and bottom edges
into a cylinder forming eight adjacent cells.
This group of eight has one Boolean variable in
common: B=0.
Therefore, the one group of eight is covered by one
p-term: B’.
The original eight-term Boolean expression
simplifies to Out=B’
Example:
f(A,B,C,D)=
29. MISSING-TERMS IN 4VARIABLE K MAPS
The Boolean expression here has nine p-terms, three
of which have three Booleans instead of four.The
difference is that while four Boolean variable product
terms cover one cell, the three Boolean p-terms
cover a pair of cells each.
The six product terms of four Boolean variables map
in the usual manner above as single cells.The three
Boolean variable terms (three each) map as cell pairs,
which is shown above.
Note that we are mapping p-terms into the K-map,
not pulling them out at this point.
For the simplification, we form two groups of eight.
Cells in the corners are shared with both groups.This
is fine. In fact, this leads to a better solution than
forming a group of eight and a group of four without
sharing any cells. Final Solution is Out=B’+D’
Example:
f(A,B,C,D)=
30. 4VARIABLE K-MAP: EXAMPLE
Here we map the un-simplified Boolean
expression to the Karnaugh map.
Three of the cells form into groups of two
cells.
A fourth cell cannot be combined with
anything, which often happens in “real world”
problems. In this case, the Boolean p-
term ABCD is unchanged in the simplification
process.
Result:
Out= B’C’D’+A’B’D’+ABCD
Example:
f(A,B,C,D)=
31. 4VARIABLE K-MAP: EXAMPLE
Often times there is more than one
minimum cost solution to a simplification
problem. Such is the case illustrated below.
Both results above have four product terms
of three Boolean variable each. Both are
equally valid minimal cost solutions.
The difference in the final solution is due to
how the cells are grouped as shown here.
A minimal cost solution is a valid logic
design with the minimum number of gates
with the minimum number of inputs.
Example:
f(A,B,C,D)=
32. 4VARIABLE K-MAP: EXAMPLE
Below we map the un-simplified
Boolean equation as usual and form a
group of four as a first simplification
step. It may not be obvious how to
pick up the remaining cells.
Pick up three more cells in a group of
four, center above.There are still two
cells remaining. the minimal cost
method to pick up those is to group
them with neighboring cells as groups
of four as at above right.
On a cautionary note, do not attempt
to form groups of three.
Groupings must be powers of 2, that
is, 1, 2, 4, 8 ...
Example:
f(A,B,C,D)=
33. K-MAP FOR 4VARIABLES (SOP)
F(P,Q,R,S)=∑(0,2,5,7,8,10,13,15)
From red group we get product term—
QS
From green group we get product
term—
Q’S’
Summing these product terms we get-
Final expression (QS+Q’S’)
34. K-MAP FOR 4VARIABLES (POS)
F(A,B,C,D)=π(3,5,7,8,10,11,12,13)
From green group we find terms C’ D B
Taking their complement and summing them (C+D’+B’)
From red group we find terms C D A’
Taking their complement and summing them (C’+D’+A)
From blue group we find terms A C’ D’
Taking their complement and summing them (A’+C+D)
From brown group we find terms A B’ C
Taking their complement and summing them (A’+B+C’)
Finally we express these as product –
(C+D’+B’).(C’+D’+A).(A’+C+D).(A’+B+C’)
35. K-MAP FOR 4VARIABLES (DON’T CARES)
Don’t cares in a Karnaugh map, or truth table, may be either 1s or 0s, as long as we don’t care what the output is
for an input condition we never expect to see.We plot these cells with an asterisk, *, among the normal 1s
and 0s.
When forming groups of cells, treat the don’t care cell as either a 1 or a 0, or ignore the don’t cares.
This is helpful if it allows us to form a larger group than would otherwise be possible without the don’t cares.
There is no requirement to group all or any of the don’t cares.
Only use them in a group if it simplifies the logic.
36. QUINE-MCCLUSKEY TABULAR METHOD
Quine-McClukey tabular method is a tabular method based on the concept of prime implicants.
The prime implicant is a product or sum term, which can’t be further reduced by combining with any other
product or sum terms of the given Boolean function.
Quine-McClukey tabular method is a tabular method based on the concept of prime implicants.We know that
prime implicant is a product or sum term, which can’t be further reduced by combining with any other product
or sum terms of the given Boolean function.
37. QUINE-MCCLUSKEY TABULAR METHOD: STEPS FOR SIMPLIFYING BOOLEAN FUNCTIONS
Step 1 − Arrange the given min terms in an ascending order and make the groups based on the number of ones
present in their binary representations. So, there will be at most ‘n+1’ groups if there are ‘n’ Boolean variables in a
Boolean function or ‘n’ bits in the binary equivalent of min terms.
Step 2 − Compare the min terms present in successive groups. If there is a change in only one-bit position, then take
the pair of those two min terms. Place this symbol ‘_’ in the differed bit position and keep the remaining bits as it is.
Step 3 − Repeat step2 with newly formed terms till we get all prime implicants.
Step 4 − Formulate the prime implicant table. It consists of set of rows and columns. Prime implicants can be placed
in row wise and min terms can be placed in column wise. Place ‘1’ in the cells corresponding to the min terms that are
covered in each prime implicant.
Step 5 − Find the essential prime implicants by observing each column. If the min term is covered only by one prime
implicant, then it is essential prime implicant.Those essential prime implicants will be part of the simplified Boolean
function.
Step 6 − Reduce the prime implicant table by removing the row of each essential prime implicant and the columns
corresponding to the min terms that are covered in that essential prime implicant. Repeat step 5 for Reduced prime
implicant table. Stop this process when all min terms of given Boolean function are over.
38. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 1
simplify the following Boolean function, f(W,X,Y,Z)=∑m(2,6,8,9,10,11,14,15) using Quine-McClukey tabular method.
Group Name Min terms W X Y Z
GA1
2 0 0 1 0
8 1 0 0 0
GA2
6 0 1 1 0
9 1 0 0 1
10 1 0 1 0
GA3
11 1 0 1 1
14 1 1 1 0
GA4 15 1 1 1 1
The given Boolean function is
in sum of min terms form.
It is having 4 variables W, X,Y & Z.
The given min terms are 2, 6, 8,
9, 10, 11, 14 and 15.
The ascending order of these
min terms based on the
number of ones present in
their binary equivalent is 2, 8, 6,
9, 10, 11, 14 and 15.
The following table shows
these min terms and their
equivalent
binary representations.
Min
Term
Binary
2 0010
6 0110
8 1000
9 1001
10 1010
11 1011
14 1110
15 1111
39. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 2
The given min terms are arranged into 4 groups based on the number of ones present in their binary equivalents.
The following table shows the possible merging of min terms from adjacent groups.
Group Name Min terms W X Y Z
GB1
2,6 0 - 1 0
2,10 - 0 1 0
8,9 1 0 0 -
8,10 1 0 - 0
GB2
6,14 - 1 1 0
9,11 1 0 - 1
10,11 1 0 1 -
10,14 1 - 1 0
GB3
11,15 1 - 1 1
14,15 1 1 1 -
40. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 3
The min terms, which are differed in only one-bit position from adjacent groups are merged.That differed bit is
represented with this symbol,‘-‘. In this case, there are three groups and each group contains combinations of two
min terms.The following table shows the possible merging of min term pairs from adjacent groups.
Group Name Min terms W X Y Z
GB1
2,6,10,14 - - 1 0
2,10,6,14 - - 1 0
8,9,10,11 1 0 - -
8,10,9,11 1 0 - -
GB2
10,11,14,15 1 - 1 -
10,14,11,15 1 - 1 -
41. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 4
The successive groups of min term pairs, which are differed in only one-bit position are merged.That differed bit
is represented with this symbol,‘-‘. In this case, there are two groups and each group contains combinations of
four min terms. Here, these combinations of 4 min terms are available in two rows. So, we can remove the
repeated rows.The reduced table after removing the redundant rows is shown below.
Group Name Min terms W X Y Z
GC1 2,6,10,14 - - 1 0
8,9,10,11 1 0 - -
GC2 10,11,14,15 1 - 1 -
42. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 5
Further merging of the combinations of min terms from adjacent groups is not possible, since they are differed in
more than one-bit position.There are three rows in the above table. So, each row will give one prime implicant.
Therefore, the prime implicants areYZ’,WX’ & WY.
The prime implicant table is shown below.
Min terms / Prime
Implicants
2 6 8 9 10 11 14 15
YZ’ 1 1 1 1
WX’ 1 1 1 1
WY 1 1 1 1
43. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 6
The prime implicants are placed in row wise and min terms are placed in column wise. 1s are placed in the
common cells of prime implicant rows and the corresponding min term columns.
The min terms 2 and 6 are covered only by one prime implicant YZ’. So, it is an essential prime implicant.
This will be part of simplified Boolean function. Now, remove this prime implicant row and the corresponding min
term columns.The reduced prime implicant table is shown below.
Min terms / Prime
Implicants
8 9 11 15
WX’ 1 1 1
WY 1 1
44. QUINE-MCCLUSKEY TABULAR METHOD: EXAMPLE: STEP 7
The min terms 8 and 9 are covered only by one prime implicant WX’. So, it is an essential prime implicant.
This will be part of simplified Boolean function. Now, remove this prime implicant row and the corresponding min
term columns.The reduced prime implicant table is shown below.
Min terms / Prime Implicants 15
WY 1
The min term 15 is covered only by one prime implicant WY. So, it is an essential prime implicant.This will be
part of simplified Boolean function.
In this example problem, we got three prime implicants and all the three are essential.
Therefore, the simplified Boolean function is
F(W,X,Y,Z) =YZ’ + WX’ + WY.