(1) The document discusses applying Quine-McCluskey method with don't cares to minimize Boolean functions.
(2) Key steps include finding all prime implicants including don't cares, creating a prime implicant chart to identify essential prime implicants, and choosing prime implicants to cover all minterms.
(3) An example applies the method to F(a,b,c) = Σm(2,4) + Σd(1,5,6), identifying (2,6) and (4,5) as essential prime implicants to minimize to F = bc' + ab'.
This chapter defines and investigates exponential and logarithmic functions. Exponential functions have a variable exponent and constant base, and are important due to their wide variety of applications including compound interest and radioactive decay. Logarithmic functions are defined as the inverse functions of exponential functions. The chapter explores properties of these functions such as their graphs and how to solve exponential and logarithmic equations. Objectives include defining the functions, investigating their properties, introducing applications, and solving related equations.
Computer Aided Assessment (CAA) for mathematicstelss09
Computer aided assessment (CAA) uses computer algebra systems to automatically mark mathematical work, allowing for immediate feedback. It can check student answers algebraically for equivalence rather than just matching answers. This addresses issues with multiple choice questions. Well-designed CAA questions can test for conceptual understanding and properties of functions. The system provides data on student misconceptions to inform feedback. Authoring questions requires balancing expressive power and ease of creation.
This document contains a mathematics exam for a 5th year primary student in Sri Aman, Sarawak, Malaysia. The exam consists of 20 multiple choice questions testing various math skills like fractions, percentages, time, money, word problems, and geometry. It provides the student with instructions, notifies them that the exam is out of 40 total marks, and includes an answer sheet for them to write their responses.
1. This document contains examples of finding antiderivatives (indefinite integrals) of various functions.
2. The examples demonstrate using basic integration rules like power rule, reverse derivative rule, trigonometric integral formulas to find antiderivatives.
3. Additional examples show using initial conditions to determine an unknown constant term when an antiderivative is given in terms of an arbitrary constant.
Quiz 2 will be held on January 27 covering sections 1.4, 1.5, 1.7, and 1.8. Test 1 is scheduled for February 1. The document then provides steps to find the inverse of a 2x2 matrix, discusses invertibility if the determinant is 0, and gives an example of finding the inverse of a 3x3 matrix using row reduction of the augmented matrix.
1. The document provides examples and explanations of key concepts in geometry including Cartesian coordinates, distance between points, types of triangles, area of triangles and polygons, division of line segments, slope and inclination of lines, and angle between two lines.
2. One example shows that the points (-2, 0), (2, 3) and (5, -1) are the vertices of a right triangle by applying the Pythagorean theorem.
3. Another example finds the area of the triangle with vertices (5, 4), (-2, 1) and (2, -3) to be 20 square units using the area formula.
The document discusses various topics in theoretical computer science including:
1) Fibonacci numbers and their recurrence relation. Fibonacci numbers appear in patterns in nature like pinecones and flower petals.
2) Polynomial division and how it relates to geometric series and Fibonacci numbers. The power series expansion of x/(1-x-x^2) equals the Fibonacci sequence.
3) Other mathematical concepts like continued fractions, the golden ratio, and their connections to algorithms and computer science problems.
The abc's of cbm for maths, spelling and writingi4ppis
Here are 3 sample math IEP goals and objectives:
1. In one year, Jose will solve single-digit addition problems within 5 seconds with 90% accuracy as measured weekly using grade 1 math CBM material.
2. In 6 months, Maria will correctly write the digits for multi-digit subtraction problems with regrouping from grade 3 math CBM material, earning a score of 24 correct digits in 2 minutes on bi-weekly probes.
3. By the end of the I-EP year, David will accurately solve multiplication facts for numbers 1-5 within 3 seconds as measured monthly using grade 2 math mastery measurement material.
This chapter defines and investigates exponential and logarithmic functions. Exponential functions have a variable exponent and constant base, and are important due to their wide variety of applications including compound interest and radioactive decay. Logarithmic functions are defined as the inverse functions of exponential functions. The chapter explores properties of these functions such as their graphs and how to solve exponential and logarithmic equations. Objectives include defining the functions, investigating their properties, introducing applications, and solving related equations.
Computer Aided Assessment (CAA) for mathematicstelss09
Computer aided assessment (CAA) uses computer algebra systems to automatically mark mathematical work, allowing for immediate feedback. It can check student answers algebraically for equivalence rather than just matching answers. This addresses issues with multiple choice questions. Well-designed CAA questions can test for conceptual understanding and properties of functions. The system provides data on student misconceptions to inform feedback. Authoring questions requires balancing expressive power and ease of creation.
This document contains a mathematics exam for a 5th year primary student in Sri Aman, Sarawak, Malaysia. The exam consists of 20 multiple choice questions testing various math skills like fractions, percentages, time, money, word problems, and geometry. It provides the student with instructions, notifies them that the exam is out of 40 total marks, and includes an answer sheet for them to write their responses.
1. This document contains examples of finding antiderivatives (indefinite integrals) of various functions.
2. The examples demonstrate using basic integration rules like power rule, reverse derivative rule, trigonometric integral formulas to find antiderivatives.
3. Additional examples show using initial conditions to determine an unknown constant term when an antiderivative is given in terms of an arbitrary constant.
Quiz 2 will be held on January 27 covering sections 1.4, 1.5, 1.7, and 1.8. Test 1 is scheduled for February 1. The document then provides steps to find the inverse of a 2x2 matrix, discusses invertibility if the determinant is 0, and gives an example of finding the inverse of a 3x3 matrix using row reduction of the augmented matrix.
1. The document provides examples and explanations of key concepts in geometry including Cartesian coordinates, distance between points, types of triangles, area of triangles and polygons, division of line segments, slope and inclination of lines, and angle between two lines.
2. One example shows that the points (-2, 0), (2, 3) and (5, -1) are the vertices of a right triangle by applying the Pythagorean theorem.
3. Another example finds the area of the triangle with vertices (5, 4), (-2, 1) and (2, -3) to be 20 square units using the area formula.
The document discusses various topics in theoretical computer science including:
1) Fibonacci numbers and their recurrence relation. Fibonacci numbers appear in patterns in nature like pinecones and flower petals.
2) Polynomial division and how it relates to geometric series and Fibonacci numbers. The power series expansion of x/(1-x-x^2) equals the Fibonacci sequence.
3) Other mathematical concepts like continued fractions, the golden ratio, and their connections to algorithms and computer science problems.
The abc's of cbm for maths, spelling and writingi4ppis
Here are 3 sample math IEP goals and objectives:
1. In one year, Jose will solve single-digit addition problems within 5 seconds with 90% accuracy as measured weekly using grade 1 math CBM material.
2. In 6 months, Maria will correctly write the digits for multi-digit subtraction problems with regrouping from grade 3 math CBM material, earning a score of 24 correct digits in 2 minutes on bi-weekly probes.
3. By the end of the I-EP year, David will accurately solve multiplication facts for numbers 1-5 within 3 seconds as measured monthly using grade 2 math mastery measurement material.
1) The document discusses multiplication and division facts and operations. It provides a multiplication/division facts table and examples of multiplication diagrams and shortcuts to help children learn these concepts.
2) Vocabulary terms are defined, including product, factor, and quotient. Do-anytime activities and games are suggested to practice the concepts, such as using pennies to demonstrate the commutative property of multiplication.
3) Answers to sample homework problems are provided as a guide for parents to check their child's work.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
The document is a mathematics exam paper for Form Four students in Negeri Sembilan, Malaysia. It contains 14 questions testing concepts in algebra, geometry, trigonometry and calculus. Formulas that may be useful for answering the questions are provided. The first section, Section A, requires students to answer all 12 multiple choice and short answer questions. Section B requires answers for 4 out of 6 longer form questions.
The document discusses binary number systems and binary arithmetic. It begins by explaining that binary uses only two digits, 0 and 1, with a base or radix of 2. Each digit has a value depending on its position in the number. The document then provides reasons why binary is used in computers, such as being digital and using simple on/off circuits. It proceeds to explain the rules and processes for binary addition, subtraction, and provides examples of carrying and borrowing in binary arithmetic.
Memorial High School is equipping its math department with new graphing calculators to help students learn mathematics concepts and be successful on standardized tests. The new TI-NSpire calculators will provide students with tools to model mathematical situations, solve problems using multiple representations and approaches, and reduce test anxiety. Teachers will be trained on using the technology to reinforce operations and help students who are deficient in particular areas.
This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.
This document discusses search problems and exploring state spaces. It provides examples of search problems like the 8-puzzle and 15-puzzle. It explains that search involves exploring a state space to find a path from an initial state to a goal state. The state space grows enormously large for puzzles like the 15-puzzle. Effective search requires constructing solutions by exploring only a small portion of the total state space, which is typically represented as a search tree. The document outlines the key components of formulating a problem as a search problem and searching the state space to find a solution.
1. This document summarizes key concepts in numerical analysis related to floating point arithmetic, including: the floating point number system; relative error; rounding; IEEE standard; precision versus accuracy; exceptional results; cancellation; and fused multiply-add instructions.
2. It discusses common misconceptions around floating point arithmetic, including the ideas that innocuous calculations are always accurate; increasing precision improves accuracy; and cancellation eliminates rounding errors.
3. The document uses examples to illustrate concepts like loss of accuracy from repeated operations, sensitivity to precision levels, and how cancellation can expose existing errors.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
This document summarizes selection algorithms and the 1-center problem.
The selection algorithm uses a prune and search approach. It recursively partitions the dataset into subsets based on the median, pruning away elements that are guaranteed to be outside the desired rank. This results in a linear time complexity of O(n).
The 1-center problem finds the smallest circle enclosing a set of points. A constrained version restricts the center to a given line. The algorithm works by forming point pairs, computing bisectors, and recursively pruning points outside the optimal region.
By tracking the sign of distances to farthest points, the full 2D solution can also be obtained in linear time by recursively considering constrained subproblems on the x
This document provides an overview of adding and subtracting rational expressions. It begins with instructions on finding the least common denominator (LCD) and provides examples of adding fractions with unlike denominators. It then demonstrates subtracting fractions with unlike denominators by first making the denominators the same. Several examples of adding and subtracting rational expressions are worked through step-by-step. Special cases involving factoring denominators are also discussed.
1. The student success sheet provides guidance and practice for concepts related to graphing and writing equations of lines in Algebra 1.
2. It includes examples and practice problems for graphing lines using an x-y table, verifying if a point lies on a line, identifying the slope and y-intercept of a line, and writing equations of lines given a point and the slope.
3. Support resources are provided for students who need additional help, including sign up for extra assistance and tutorial videos.
This document discusses various methods for minimizing switching functions, including:
1. The Karnaugh map method, which represents truth tables graphically to find logically adjacent terms that can be combined.
2. Prime implicants and essential prime implicants, which are product terms that cover minterms. The essential ones must be included in the minimal expression.
3. Don't care conditions, which allow further simplification by treating unspecified minterms as don't cares.
4. The Quine-McCluskey tabulation method, which systematically generates prime implicants and finds the essential ones and minimal cover.
The document contains a mathematics exam paper with 14 pages. It consists of two sections - Section A with 52 marks and Section B with 14 marks. Section A contains 12 multiple choice questions. Various formulas are provided that may help in answering the questions, including formulas for algebra, geometry, trigonometry and calculus.
[END SUMMARY]
1. The document provides instructions and examples for various math topics ranging from ordering fractions and decimals to solving proportions and working with percentages.
2. Examples are given for multiplying and dividing by 10, 100, 1000 as well as rounding decimals. Properties of 2D and 3D shapes, types of angles, and solving proportions using direct variation are also covered.
3. Different methods are demonstrated for operations like long division, multiplying two-digit numbers, working with negative numbers, simplifying ratios, and finding fractions of quantities using a calculator.
1) Edexcel is an examining and awarding body that provides qualifications worldwide. It supports centers that offer education programs to learners through a network of UK and international offices.
2) Candidates' work will be marked according to principles such as marking positively and awarding all marks deserved according to the mark scheme. Subject specialists are available to help with specific content questions.
3) The document provides notes on marking principles for a GCSE mathematics exam, including how to apply the mark scheme and address various student responses.
The document provides examples and explanations of various math concepts including place value, multiples, fractions, decimals, addition, subtraction, multiplication, division, measuring units, classifying shapes, and representing data visually using charts, diagrams and tables. It covers essential math topics for levels 3-4 such as number sequences, operations, geometry, measurement, and data handling.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
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 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.
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.
1) The document discusses multiplication and division facts and operations. It provides a multiplication/division facts table and examples of multiplication diagrams and shortcuts to help children learn these concepts.
2) Vocabulary terms are defined, including product, factor, and quotient. Do-anytime activities and games are suggested to practice the concepts, such as using pennies to demonstrate the commutative property of multiplication.
3) Answers to sample homework problems are provided as a guide for parents to check their child's work.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
The document is a mathematics exam paper for Form Four students in Negeri Sembilan, Malaysia. It contains 14 questions testing concepts in algebra, geometry, trigonometry and calculus. Formulas that may be useful for answering the questions are provided. The first section, Section A, requires students to answer all 12 multiple choice and short answer questions. Section B requires answers for 4 out of 6 longer form questions.
The document discusses binary number systems and binary arithmetic. It begins by explaining that binary uses only two digits, 0 and 1, with a base or radix of 2. Each digit has a value depending on its position in the number. The document then provides reasons why binary is used in computers, such as being digital and using simple on/off circuits. It proceeds to explain the rules and processes for binary addition, subtraction, and provides examples of carrying and borrowing in binary arithmetic.
Memorial High School is equipping its math department with new graphing calculators to help students learn mathematics concepts and be successful on standardized tests. The new TI-NSpire calculators will provide students with tools to model mathematical situations, solve problems using multiple representations and approaches, and reduce test anxiety. Teachers will be trained on using the technology to reinforce operations and help students who are deficient in particular areas.
This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.
This document discusses search problems and exploring state spaces. It provides examples of search problems like the 8-puzzle and 15-puzzle. It explains that search involves exploring a state space to find a path from an initial state to a goal state. The state space grows enormously large for puzzles like the 15-puzzle. Effective search requires constructing solutions by exploring only a small portion of the total state space, which is typically represented as a search tree. The document outlines the key components of formulating a problem as a search problem and searching the state space to find a solution.
1. This document summarizes key concepts in numerical analysis related to floating point arithmetic, including: the floating point number system; relative error; rounding; IEEE standard; precision versus accuracy; exceptional results; cancellation; and fused multiply-add instructions.
2. It discusses common misconceptions around floating point arithmetic, including the ideas that innocuous calculations are always accurate; increasing precision improves accuracy; and cancellation eliminates rounding errors.
3. The document uses examples to illustrate concepts like loss of accuracy from repeated operations, sensitivity to precision levels, and how cancellation can expose existing errors.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
This document summarizes selection algorithms and the 1-center problem.
The selection algorithm uses a prune and search approach. It recursively partitions the dataset into subsets based on the median, pruning away elements that are guaranteed to be outside the desired rank. This results in a linear time complexity of O(n).
The 1-center problem finds the smallest circle enclosing a set of points. A constrained version restricts the center to a given line. The algorithm works by forming point pairs, computing bisectors, and recursively pruning points outside the optimal region.
By tracking the sign of distances to farthest points, the full 2D solution can also be obtained in linear time by recursively considering constrained subproblems on the x
This document provides an overview of adding and subtracting rational expressions. It begins with instructions on finding the least common denominator (LCD) and provides examples of adding fractions with unlike denominators. It then demonstrates subtracting fractions with unlike denominators by first making the denominators the same. Several examples of adding and subtracting rational expressions are worked through step-by-step. Special cases involving factoring denominators are also discussed.
1. The student success sheet provides guidance and practice for concepts related to graphing and writing equations of lines in Algebra 1.
2. It includes examples and practice problems for graphing lines using an x-y table, verifying if a point lies on a line, identifying the slope and y-intercept of a line, and writing equations of lines given a point and the slope.
3. Support resources are provided for students who need additional help, including sign up for extra assistance and tutorial videos.
This document discusses various methods for minimizing switching functions, including:
1. The Karnaugh map method, which represents truth tables graphically to find logically adjacent terms that can be combined.
2. Prime implicants and essential prime implicants, which are product terms that cover minterms. The essential ones must be included in the minimal expression.
3. Don't care conditions, which allow further simplification by treating unspecified minterms as don't cares.
4. The Quine-McCluskey tabulation method, which systematically generates prime implicants and finds the essential ones and minimal cover.
The document contains a mathematics exam paper with 14 pages. It consists of two sections - Section A with 52 marks and Section B with 14 marks. Section A contains 12 multiple choice questions. Various formulas are provided that may help in answering the questions, including formulas for algebra, geometry, trigonometry and calculus.
[END SUMMARY]
1. The document provides instructions and examples for various math topics ranging from ordering fractions and decimals to solving proportions and working with percentages.
2. Examples are given for multiplying and dividing by 10, 100, 1000 as well as rounding decimals. Properties of 2D and 3D shapes, types of angles, and solving proportions using direct variation are also covered.
3. Different methods are demonstrated for operations like long division, multiplying two-digit numbers, working with negative numbers, simplifying ratios, and finding fractions of quantities using a calculator.
1) Edexcel is an examining and awarding body that provides qualifications worldwide. It supports centers that offer education programs to learners through a network of UK and international offices.
2) Candidates' work will be marked according to principles such as marking positively and awarding all marks deserved according to the mark scheme. Subject specialists are available to help with specific content questions.
3) The document provides notes on marking principles for a GCSE mathematics exam, including how to apply the mark scheme and address various student responses.
The document provides examples and explanations of various math concepts including place value, multiples, fractions, decimals, addition, subtraction, multiplication, division, measuring units, classifying shapes, and representing data visually using charts, diagrams and tables. It covers essential math topics for levels 3-4 such as number sequences, operations, geometry, measurement, and data handling.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
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 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.
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.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
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.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
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.
Film vocab for eal 3 students: Australia the movie
Quine Mc-Cluskey
1. ECE 474A/57A
Computer-Aided Logic Design
Lecture 8
Qunie-McCluskey with Don’t Cares, Iterated
Consensus, Row/Column Dominance
ECE 474a/575a 1 of 39
Susan Lysecky
K-map with Don’t Cares
Consider F(a, b, c) = Σm(2, 4) + Σd(1, 5, 6)
What should we do with the don’t cares?
Include d.c. if it helps to further minimize the cover (m5, m6)
Don’t need to include d.c. if it doesn’t help (better to exclude m1)
bc bc bc
a 00 01 11 10 a 00 01 11 10 a 00 01 11 10
0 1 3 2 0 1 3 2 0 1 3 2
0 0 X 0 1 0 0 X 0 1 0 0 X 0 1
4 5 7 6 4 5 7 6 4 5 7 6
1 1 X 0 X 1 1 X 0 X 1 1 X 0 X
F = ab’c’ + a’bc’ F = ab’ + b’c + bc’ F = ab’ + bc’
How do we apply these ideas to Quine-McCluskey?
ECE 474a/575a 2 of 39
Susan Lysecky
Quine-McCluskey with Don’t Cares
Example 1
F(a, b, c) = Σm(2, 4) + Σd(1, 5, 6)
Step 1: Find all the prime implicants
List all elements of on-set and don’t care set, represented as a binary number
Mark don’t cares with “D”
G1 (1) 001 D
(2) 010
(4) 100
G2 (5) 101 D
(6) 110 D
ECE 474a/575a 3 of 39
Susan Lysecky
1
2. Quine-McCluskey with Don’t Cares
Example 1
Step 1: Find all the prime implicants (cont’)
Compare each entry in Gi to each entry in Gi+1
If they differ by 1 bit, we can apply the uniting theorem and eliminate a literal
If both values are don’t cares, retain “D”, otherwise no need to mark
Add check to implicant to remind us that it is not a prime implicant
G1 (1) 001 D G1 (1,5) -01 D no new implicants are generated – end
of step 1
(2) 010 (2,6) -10
(4) 100 (4,5) 10-
we have found all prime implicants
G2 (5) 101 D (4,6) 1-0 (ones without check marks)
(6) 110 D
ECE 474a/575a 4 of 39
Susan Lysecky
Quine-McCluskey with Don’t Cares
Example 1
Step 2: Create Prime Implicant Chart to find all essential prime implicants
Minterms are added as columns in the table
Prime implicants not marked as “D” are added as rows
Original Equation
F = Σm(2, 4) + Σd(1, 5, 6)
Derived in Step1
(1,5) -01 D
2 4
(2,6) -10
(4,5) 10- (2,6) P1
(4,6) 1-0
(4,5) P2
(4,6) P3
ECE 474a/575a 5 of 39
Susan Lysecky
Quine-McCluskey with Don’t Cares
Example 1
Step 2: Create Prime Implicant Chart to find all essential prime implicants
Place “X” in a row if the prime implicant covers the minterm
Essential prime implicants are found by looking for rows with a single “X”
If minterm is covered by one and only one prime implicant – it’s an essential prime implicant
Add essential prime implicants to the cover
P1 is essential, need to
include
Choose between P2 and P3 to
cover remaining minterm
2 4
Option 1
(2,6) P1 F = P1 + P2
F = bc’ + ab’
(4,5) P2
(4,6) P3 Option 2
F = P1 + P3
F = bc’ + ac’
ECE 474a/575a 6 of 39
Susan Lysecky
2
3. Quine-McCluskey with Don’t Cares
Example 2
F = Σm(0, 3, 10, 15) + Σd(1, 2, 7, 8, 11, 14)
F cd
ab 00 01 11 10
Using a K-map we get
0 1 3 2
00 1 X 1 X
4 5 7 6
01 X
12 13 15 14 F = a’b’ + ac
11 1 X
8 9 11 10
2 product terms, 2 variables each
10 X X 1
Can we do just was well with Q.M.?
ECE 474a/575a 7 of 39
Susan Lysecky
Quine-McCluskey with Don’t Cares
Example 2
F = Σm(0, 3, 10, 15) + Σd(1, 2, 7, 8, 11, 14)
G0 (0) 0000 G1 (0,1) 000- G0 (0,1,2,3) 00--
G1 (1) 0001 D (0,2) 00-0 (0,2,8,10) -0-0
(2) 0010 D (0,8) -000 G1 (2,3,10,11) -01-
(8) 1000 D (1,3) 00-1 G2 (3,7,11,15) --11
G2 (3) 0011 (10,11,14,15) 1-1-
combined on-set
(10) 1010 minterm and don’t
care, drop the “D”
G3 (7) 0111 D
(11) 1011 D combined two don’t
cares, keep the “D”
(14) 1110 D These are your prime implicants
G4 (15) 1111 (all smaller product terms have
been combine into larger terms)
ECE 474a/575a 8 of 39
Susan Lysecky
Quine-McCluskey with Don’t Cares
Example 2
F = Σm(0, 3, 10, 15) + Σd(1, 2, 7, 8, 11, 14)
0 3 10 15
(0,1,2,3) P1
(0,2,8,10) P2
(2,3,10,11) P3
No essentials, how do we choose?
(3,7,11,15) P4
(10,11,14,15) P5 TRY PETRICKS!
F = (m0)(m3)(m10)(m15)
F = (P1+P2)(P1+P3+P4)(P2+P3+P5)(P4+P5)
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4. Quine-McCluskey with Don’t Cares
Example 2
F = (P1+P2)(P1+P3+P4)(P2+P3+P5)(P4+P5)
P1P1 + P1P3 + P1P4 + P1P2 + P2P3 + P2P4
P1
F = (P1+P2P3+P2P4)(P2+P3+P5)(P4+P5)
P2P4 + P2P5 + P3P4 + P3P5 + P4P5 + P5P5
P5
F = (P1+P2P3+P2P4)(P2P4+P3P4+P5)
F = P1P2P4+P1P3P4+P1P5+P2P2P3P4+P2P3P3P4+P2P3P5+P2P2P4P4+P2P3P4P4+P2P4P5
P2P4
F = P1P3P4+P1P5+P2P3P5+P2P4
Best Options
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Quine-McCluskey with Don’t Cares
Example 2
F = Σm(0, 3, 10, 15) + Σd(1, 2, 7, 8, 11, 14)
cd cd cd
ab 00 01 11 10 ab 00 01 11 10 ab 00 01 11 10
0 1 3 2 0 1 3 2 0 1 3 2
00 1 X 1 X 00 1 X 1 X 00 1 X 1 X
4 5 7 6 4 5 7 6 4 5 7 6
01 X 01 X 01 X
12 13 15 14 12 13 15 14 12 13 15 14
11 1 X 11 1 X 11 1 X
8 9 11 10 8 9 11 10 8 9 11 10
10 X X 1 10 X X 1 10 X X 1
Original K-map Q.M. Solution 1 Q.M. Solution2
F = P1P5 F=P2P4
F = a’b’ + ac F = a’b’ + ac F= b’d’ + cd
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Quine-McCluskey Overview
How is each step Are there alternatives?
Quine-McCluskey Algorithm currently done?
(1) Find all prime implicants Tabular Iterated Consensus to find
Minimization complete sum
(2) Find all essential prime Prime Implicant Chart Constraint Matrix
implicants (row with single “X”) (basically same thing
except axis switched)
(3) Select a minimal set of remaining Petrick’s Method Row/Column
prime implicants that covers the Dominance
on-set of the function
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5. Iterated Consensus/Complete Sum
Consider F(x, y, z) = yz + x’y + y’z’ + xyz + x’z’
According to tabular minimization
First expanded product term into minterms
yz
Then start comparing pairs to determine prime implicants
xyz x’yz
Some of the work already done!
Instead we can take existing expression and determine the complete sum
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Iterated Consensus/Complete Sum
Def: A complete sum is a SOP formula composed of all prime implicants of
the function
Thm: A SOP formula is a complete sum if and only if
(1) No term includes any other term
(2) The consensus of any two terms of the formula either does
not exist or is contained in some other term of the formula
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Iterated Consensus/Complete Sum
What is consensus?
In Boolean Algebra, consensus is defined as
(a) xy + x’z + yz = xy + x’z
x’z yz
(b) (x+y)(x’+z)(y+z) = (x+y)(x’+z) bc
a 00 01 11 10
0 1 3 2
0 1 1
4 5 7 6
1 1 1 xy
Proof: xy + x’z + yz = xy + x’z
K-map shows yz already
= xy + x’z + (x + x’)yz covered by other two primes
= xy + x’z + xyz + x’yz
= (xy + xyz) + (x’z + x’yz)
= xy(1 + z) + x’z(1 + y)
= xy(1) + x’z(1)
= xy + x’z
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6. Iterated Consensus/Complete Sum
Typically consensus theorem used to simplify
xy + x’z + yz = xy + x’z
Boolean equations
Removed redundant terms (x+y)(x’+z)(y+z) = (x+y)(x’+z)
Ex abc + a’bd + bcd = abc + a’bd
x y x’ z yz x y x’ z
Ex abc’d + c’d’e + abc’e = c’(abd + d’e + abe) = c’(abd + de)
y x x’ z yz y x x’ z
Ex (a+b)(a’+c)(b+c) = (a+b)(a’+c)
x y x’ z yz x y x’ z
Ex (a+b)(c’+d)(a+c’) = cannot simplify
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Iterated Consensus/Complete Sum
We’ll use consensus backwards
Add redundant terms
xy + x’z + yz = xy + x’z
Generate complete sum
Is SOP not a complete sum, it’s missing prime implicants
Missing prime must be covered by two or more implicants x’z yz
Find the term spanning these implicants (consensus term), bc
find the complete sum a 00 01 11 10
0 1 3 2
0 1 1
4 5 7 6
F = xy + x’z // not a complete sum 1 1 1 xy
F = xy + x’z + yz // a complete sum
K-map shows yz already
covered by other two primes
Why is it important to start with complete sum?
Better opportunity to apply absorption [x (x + y) = x]
with one of the original terms to obtain simpler expression
Step 1 of Quine McCluskey
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Quine-McCluskey Overview
How is each step Are there alternatives?
Quine-McCluskey Algorithm currently done?
(1) Find all prime implicants Tabular Iterated Consensus to find
Minimization complete sum
(2) Find all essential prime Prime Implicant Chart Constraint Matrix
implicants (row with single “X”) (basically same thing
except axis switched)
(3) Select a minimal set of remaining Petrick’s Method Row/Column
prime implicants that covers the Dominance
on-set of the function
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7. Iterated Consensus to Find Complete Sum
Methodology to convert SOP function to complete sum
1. Start with arbitrary SOP form
2. Add consensus pair of all terms not contained in any other term
3. Compare new terms with existing and among other new terms to see if any new
consensus terms can be generated
4. Remove all terms contained in some other term
Repeat 2 – 4 until no change occurs
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Iterated Consensus to Find Complete Sum
Example 3
F = yz + x’y + y’z’ + xyz + x’z’ 1. Start with arbitrary SOP form
2. Add consensus pair of all terms not contained in any other term
yz + x’y = NO x’y + y’z’ = x’z’ (INCL) y’z’ + xyz = xzz’ 0, NO
yz + y’z’ = yy’ 0, NO x’y + xyz = yz (INCL) y’z’ + x’z’ = NO
yz + xyz = NO x’y + x’z’ = NO
yz + x’z’ = x’y (INCL) xyz + x’z’ = xx’y 0, NO
3. Compare new terms with existing and among other new terms to see if any new consensus terms
can be generated
No new terms generated
4. Remove all terms contained in some other term
yz + x’y + y’z’ + x’z’
yz + x’y + y’z’ + xyz + x’z’
since there is a change you will need to start
again – you will find in the next iteration no
change occurs
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Iterative vs. Recursive
Iterative approach
Repetitive procedure used to add new Iterated Consensus Methodology
consensus terms 1. Start with arbitrary SOP form
2. Add consensus pair of all terms
Recursive approach not contained in any other term
Also repetitive, but we are trying to 3. Compare new terms with existing
keep simplifying problem until solution and among other new terms to
is easy see if any new consensus terms
can be generated
4. Remove all terms contained in
some other term
Repeat 2 – 4 until no change occurs
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8. Recursive Consensus Methodology
Break down equation until it is trivial to find complete sum
Boole’s expansion Theorem (a.k.a. Shannon Expansion)
Get down to 1 term, the complete sum of this term is itself
f(x1, x2, …, xn) = [x1’ f(0, x2, …, xn)] + [x1 f(1, x2, …, xn)]
= [x1’ + f(1, x2, …, xn)] [x1 + f(0, x2, …, xn)]
Reconstruct equation, or equation’s complete sum, using Thm 4.6.1 (Hatchel pg.138)
The SOP obtained from the two complete sums F1 and F2 by the
following is a complete sum for F1 F2
1. Multiply out F1 and F2 using the idempotent property (a+a=a,
a a=a), distributive properties, and x x’ = 0
2. Eliminate all terms contained in some other terms
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Recursive Consensus Methodology
Example 4
F = a’b’ + a’bc’ + ac
a=0 a=1
b’ + bc’ c Done! We are down to 1 term
b=0 b=1 CS(c) = c
1 c’
CS(1) = 1 CS(c’) = c’
Now how do we put it all back
together again?
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Recursive Consensus Methodology
Example 4
F = a’b’ + a’bc’ + ac
= ABS[(a + b’ + c’) (a’ + c)]
= ABS[aa’ + ac + a’b’ + b’c + a’c’ + cc’]
= ABS[(b + 1) (b’ + c’)] a=0 a=1
= ABS[ac + a’b’ + b’c + a’c’]
= ABS[(bb’ + bc’ + b’ + c’)]
= ac + a’b’ + b’c + a’c’
= ABS[(0 + bc’ + b’ + c’)] b’ + bc’ c
= b’ + c’ CS(c) = c
b=0 b=1
1 c’
CS(1) = 1 CS(c’) = c’
CS(F1 F2) = ABS [CS(F1) CS(F2)]
f(x1, x2, …, xn) = [x1’ + f(1, x2, …, xn)] [x1 + f(0, x2, …, xn)]
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9. Recursive Consensus Methodology
Example 4
Started with F = a’b’ + a’bc’ + ac
Ended with CS(F) = ac + a’b’ + b’c + a’c’
Did it work?
a’b’ a’bc’ a’b’ a’c’
bc bc
a 00 01 11 10 a 00 01 11 10
0 1 3 2 0 1 3 2
0 1 1 1 0 1 1 1
4 5 7 6 4 5 7 6
1 1 1 ac 1 1 1 ac
b’c
Not a complete sum – missing Complete sum achieved
some prime implicants
Recursive method beneficial when dealing with larger equations
Book example F = v’xyz +v’w’x + v’x’z’ + v’wxz + w’yz’ + vw’z + vwx’z
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Quine-McCluskey Overview
How is each step Are there alternatives?
Quine-McCluskey Algorithm currently done?
(1) Find all prime implicants Tabular Iterated Consensus to find
Minimization complete sum
(2) Find all essential prime Prime Implicant Chart Constraint Matrix
implicants (row with single “X”) (basically same thing
except axis switched)
(3) Select a minimal set of remaining Petrick’s Method Row/Column
prime implicants that covers the Dominance
on-set of the function
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Constraint Matrix
Describes conditions or constraints a cover must satisfy
Each column corresponds to a prime implicant
Each row correspond to a minterm
P1 P2 Pn
M1 List of prime implicants Note:
(complete sum) Similar to prime implicants
M2 chart. However, this textbook
List of minterms swaps the rows/cols
Mm
GOAL – choose minimal subset of primes where each minter form which the
function is 1 is included in at least one prime of the subset
Known as a “cover”
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10. Constraint Matrix
Example 5
F(x, y, z) = yz + x’y + y’z’ + xyz + x’z’
P1 P2 P3 P4
Rows are minterms:
x’y x’z’ y’z’ yz
yz xyz, x’yz
(m0) x’y’z’ 0 1 1 0
x’y x’yz, x’yz’ (m2) x’yz’ 1 1 0 0
y’z’ xy’z’, x’y’z’ (m3) x’yz 1 0 0 1
(m7) xyz 0 0 0 1
xyz xyz (same)
(m4) xy’z’ 0 0 1 0
x’z’ x’yz’, x’y’z’
Cols are prime implicants: (get these from ex3)
yz + x’y + y’z’ + x’z’
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Constraint Matrix
Example 5
F(x, y, z) = yz + x’y + y’z’ + xyz + x’z’
P1 P2 P3 P4 Now we look for essential prime
x’y x’z’ y’z’ yz implicants
(m0) x’y’z’ 0 1 1 0
(m2) x’yz’ 1 1 0 0
(m3) x’yz 1 0 0 1
(m7) xyz 0 0 0 1 Singleton row (only one way to cover this minterm)
(m4) xy’z’ 0 0 1 0 Must include these primes (essential) in the cover
P1 P2 Remove P3 and P4 to simplify constraint matrix
x’y x’z’ Remove any minterm covered by these primes (m0, m3,
m7, m4)
(m2) x’yz’ 1 1
Easy to see how to cover remaining minterms (QM step 3)
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Constraint Matrix
Example 5
F(x, y, z) = yz + x’y + y’z’ + xyz + x’z’ From previous slides …
P1 P2 P3 P4
x’y x’z’ y’z’ yz
Solution 1 Solution 2
= P3 + P4 + P1 = P3 + P4 + P2
= y’z’ + yz + x’y = y’z’ + yz + x’z’
What happens when the solution is not
so obvious?
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11. Quine-McCluskey Overview
How is each step Are there alternatives?
Quine-McCluskey Algorithm currently done?
(1) Find all prime implicants Tabular Iterated Consensus to find
Minimization complete sum
(2) Find all essential prime Prime Implicant Chart Constraint Matrix
implicants (row with single “X”) (basically same thing
except axis switched)
(3) Select a minimal set of remaining Petrick’s Method Row/Column
prime implicants that covers the Dominance
on-set of the function
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Row Dominance (Constraint)
If a row ri in a constraint matrix has all the ones of another row rj, we say ri
dominates rj
ri is unneeded and all dominating row can be removed
Absorption property x (x + y) = x
P1 P2 P3
m1 1 1 0
m2 dominates m1
m2 1 1 1 remove m2
P1 P2 P3
m1 1 1 0
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Column Dominance (Variable)
If column Pi has all the ones of another column Pj, and the cost of Pi is not greater
than Pj, we say Pi dominates Pj
The dominated column can be removed
P1 P2 P3 P2
m1 1 1 0 m1 1 Assumes P2 does not cost more than P1
m2 0 1 1 m2 1 or P3
What is the cost of a column?
Each prime implicant (col) corresponds to
P2 dominates P1
one AND gate
P2 dominates P3
Remove P1 and P3 Our Choices
We could say each column = 1 gate and
everyone’s the same
We could include number of literal, then a
prime with 5 literals cost more than 3
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12. Reduction Techniques Using Row/Col Dominance
1. Remove rows covered by “essential columns” (i.e. essential prime implicants)
2. Remove rows through row dominance (dominating row removed)
3. Remove columns through column dominance (dominated column removed)
Re-iterate 1-3 until no further simplification is possible
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Reduction Techniques Using Row/Col Dominance
Example 6
P1 P2 P3 P4 P5 P6
(1A) No essential columns to remove
m1 1 1 1
m2 1 1
m3 1 1
m4 1 1 1
m5 1 1
m6 1 1 1
(2A) Row dominance
P1 P2 P3 P4 P5 P6
m1 dominates m2
m2 1 1
m4 dominates m3
m3 1 1 m6 dominates m5
m5 1 1
Remove the dominating rows
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Reduction Techniques Using Row/Col Dominance
Example 6
Note: P6 tell us nothing, you can remove
to simplify if you want
P1 P2 P3 P4 P5 P6
m2 1 1
(3A) Column dominance
m3 1 1
P2 dominates P1
m5 1 1
P2 dominates P3
P4 dominates P5, vice versa
Remove the dominated cols
P2 P4 (1A) Essential Columns
m2 1
m3 1
m5 1
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13. Reduction Techniques Using Row/Col Dominance
Example 6
P2 P4 (1A) Essential Columns
m2 1 P4 only column to cover m5
m3 1 essential prime implicant = {P4}
m5 1
P2 only column to cover m2, m3
essential prime implicant = {P4, P2}
Simplify matrix
Matrix empty – no further simplification possible
Cover = P2 + P4
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Reduction Techniques Using Row/Col Dominance
What happens when matrix cannot be simplified?
No rows left
We have a terminal case and solved the problem
Rows left
Problem is cyclic
Alternative techniques such as divide-and-conquer or branch-and-bound are needed
(Or guess, or use Petrick's)
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Conclusion
Quine-McCluskey with Don’t Cares
Alternative methods to perform Quine-McCluskey algorithm
Iterated consensus (iterative and recursive)
Generate a complete sum
Row/Column Dominance
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