The document discusses combinational logic circuits and Boolean algebra. It contains 17 questions covering topics such as:
1. The three basic logic operations - AND, OR, NOT - and their truth tables and logic diagrams.
2. Boolean algebra, functions, truth tables, and logic diagrams.
3. Identities of Boolean algebra, dual expressions, and the consensus theorem.
4. Minimizing logic functions using Boolean algebra identities and Karnaugh maps.
5. Minterms, maxterms, and representing logic functions with sums of products and products of sums.
6. Demorgan's theorem and complementing logic functions.
7. Simplifying logic functions through
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.
This document provides an overview of functions and relations. It begins by defining the learning objectives and outcomes for understanding functions. It then discusses representing relations using arrow diagrams, ordered pairs, and graphs. It introduces the concepts of domain, codomain, object, image, and range for relations. Different types of relations like one-to-one, many-to-one, one-to-many, and many-to-many are classified. Functions are introduced as a special type of relation where each element in the domain maps to only one element in the codomain. Notation for expressing functions is explained along with determining the domain, object, image, and range of functions. Examples are provided to illustrate these concepts.
The document provides instructions for a mathematics scholarship test. It explains that the test has 3 sections (Algebra, Analysis, Geometry), with 10 questions each for a total of 30 questions. Candidates should answer each question in the provided answer booklet, not on the question paper. Calculators are not allowed. The instructions also define various mathematical terms and notations used in the questions.
The document provides properties of determinants and examples of their applications. It also gives tips for solving problems based on properties of determinants. Finally, it lists 20 assignment questions related to matrices and determinants, covering topics like solving systems of equations using matrices, finding the inverse of a matrix, and applying properties of determinants.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.
This document provides an overview of functions and relations. It begins by defining the learning objectives and outcomes for understanding functions. It then discusses representing relations using arrow diagrams, ordered pairs, and graphs. It introduces the concepts of domain, codomain, object, image, and range for relations. Different types of relations like one-to-one, many-to-one, one-to-many, and many-to-many are classified. Functions are introduced as a special type of relation where each element in the domain maps to only one element in the codomain. Notation for expressing functions is explained along with determining the domain, object, image, and range of functions. Examples are provided to illustrate these concepts.
The document provides instructions for a mathematics scholarship test. It explains that the test has 3 sections (Algebra, Analysis, Geometry), with 10 questions each for a total of 30 questions. Candidates should answer each question in the provided answer booklet, not on the question paper. Calculators are not allowed. The instructions also define various mathematical terms and notations used in the questions.
The document provides properties of determinants and examples of their applications. It also gives tips for solving problems based on properties of determinants. Finally, it lists 20 assignment questions related to matrices and determinants, covering topics like solving systems of equations using matrices, finding the inverse of a matrix, and applying properties of determinants.
1. understand the terms function, domain, range, one-one function,inverse function and composition of functions
2. identify the range of a given function in simple cases, and find the
composition of two given functions
3. determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
4. illustrate in graphical terms the relation between a one-one function and its inverse.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2009. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
This document discusses composite functions and the order of operations when combining functions.
It provides an example of a mother converting the temperature of her baby's bath water from Celsius to Fahrenheit using two separate functions. The first function converts the Celsius reading to Fahrenheit, and the second maps the Fahrenheit reading to whether the water is too cold, alright, or too hot. Together these functions form a composite function.
Algebraically, a composite function f∘g(x) is defined as applying the inner function f first to the input x, and then applying the outer function g to the output of f. The domain of the inner function must be contained within the range of the outer function. The order of
This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.
The document discusses functions and their properties. It defines a function as a rule that assigns exactly one output value to each input value in its domain. Functions can be represented graphically, numerically in a table, or with an algebraic rule. The domain of a function is the set of input values, while the range is the set of output values. Basic operations like addition, subtraction, multiplication and division can be performed on functions in the same way as real numbers. Composition of functions is defined as evaluating one function using the output of another as the input.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document provides 15 multi-part math word problems involving indices, logarithms, and coordinate geometry. The problems cover topics such as simplifying expressions with indices, solving logarithmic and exponential equations, finding equations of lines and loci, determining properties of geometric figures defined by coordinate points, and calculating areas. Students must use their understanding of indices, logarithms, coordinate geometry, and geometric relationships to solve the problems.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. Parabolas are the graphs of quadratic functions and have certain properties: they are symmetric about a center line, with the highest/lowest point (called the vertex) sitting on the center line. The vertex position can be found using the formula x = -b/2a. Examples are given of finding the vertex and graphing parabolas.
The question asks to solve a system of linear inequalities graphically. The system is: x + y ≤ 9, y > x, x ≥ 0. The solution region is the shaded area where all the individual inequality regions overlap, which is the triangular region bounded by the lines y = x, x = 0, and x + y = 9.
1. The document provides information on multiple integrals including double integrals, triple integrals, and integrals in spherical and cylindrical coordinates. It defines each type of integral and gives their general formulas.
2. Examples are provided for calculating double and triple integrals over different regions in rectangular, cylindrical, and spherical coordinate systems. The order of integration can be changed by considering strips or slices of the region.
3. Properties of the integrals include applying Fubini's theorem to change the order of integration, and relating the triple integral over a region to the double integral over the bounds and integrating over the third variable.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2006. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The document discusses the difference quotient formula for calculating the slope of a cord connecting two points (x, f(x)) and (x+h, f(x+h)) on a function graph. It defines the difference quotient as (f(x+h) - f(x))/h, which calculates the slope as the change in y-values (f(x+h) - f(x)) over the change in x-values (h). An example calculates the slope of the cord connecting the points (2, f(2)) and (2.2, f(2.2)) on the function f(x) = x^2 - 2x + 2.
This document provides a module on functions and simultaneous equations for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 15 problems on functions and 12 problems on simultaneous equations to help students prepare for their SPM examinations. The module is published by the Terengganu Education Department and involves several teachers from technical and science schools in the state.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2013. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
The document is a mathematics textbook for Additional Mathematics Form 4. It covers topics on functions, simultaneous equations, quadratic equations, and quadratic functions. It contains examples and practice questions for students to work through with answers. The questions range from simple calculations to solving equations and inequalities involving quadratic expressions.
Module 4 exponential and logarithmic functionsAya Chavez
This document provides an overview of logarithmic functions including:
1) Logarithmic functions are defined as the inverse of exponential functions.
2) The module will cover defining logarithmic functions, graphing them, applying logarithmic laws, and solving logarithmic equations.
3) Students are expected to learn to define logarithmic functions, draw their graphs, describe properties from graphs, apply laws of logarithms, and solve simple logarithmic equations.
This document is a resume for Martin K. Hastings that summarizes his education, employment experience, honors, and qualifications. Hastings graduated cum laude from Arizona State University with a Bachelor of Science in Marketing. He has marketing intern experience at Sedona Golf Resort where he coordinated promotions, cultivated business relationships, and executed social media campaigns. His honors include graduating cum laude and being named to the Dean's List for 6 semesters. Hastings' qualifications include strong communication, organizational, and leadership skills as well as experience in marketing, promotions, and business operations.
This document announces the birth of i.materialise, a new business built upon Materialise's core that aims to make 3D printing accessible to everyone. It will offer customizable and personalized products like lamps, interior items, jewelry, architectural models, bronzes, and gifts that are designed and printed using 3D printing technology. The document promotes i.materialise and provides contact information for anyone with questions or ideas.
1. understand the terms function, domain, range, one-one function,inverse function and composition of functions
2. identify the range of a given function in simple cases, and find the
composition of two given functions
3. determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
4. illustrate in graphical terms the relation between a one-one function and its inverse.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2009. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
This document discusses composite functions and the order of operations when combining functions.
It provides an example of a mother converting the temperature of her baby's bath water from Celsius to Fahrenheit using two separate functions. The first function converts the Celsius reading to Fahrenheit, and the second maps the Fahrenheit reading to whether the water is too cold, alright, or too hot. Together these functions form a composite function.
Algebraically, a composite function f∘g(x) is defined as applying the inner function f first to the input x, and then applying the outer function g to the output of f. The domain of the inner function must be contained within the range of the outer function. The order of
This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.
The document discusses functions and their properties. It defines a function as a rule that assigns exactly one output value to each input value in its domain. Functions can be represented graphically, numerically in a table, or with an algebraic rule. The domain of a function is the set of input values, while the range is the set of output values. Basic operations like addition, subtraction, multiplication and division can be performed on functions in the same way as real numbers. Composition of functions is defined as evaluating one function using the output of another as the input.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document provides 15 multi-part math word problems involving indices, logarithms, and coordinate geometry. The problems cover topics such as simplifying expressions with indices, solving logarithmic and exponential equations, finding equations of lines and loci, determining properties of geometric figures defined by coordinate points, and calculating areas. Students must use their understanding of indices, logarithms, coordinate geometry, and geometric relationships to solve the problems.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. Parabolas are the graphs of quadratic functions and have certain properties: they are symmetric about a center line, with the highest/lowest point (called the vertex) sitting on the center line. The vertex position can be found using the formula x = -b/2a. Examples are given of finding the vertex and graphing parabolas.
The question asks to solve a system of linear inequalities graphically. The system is: x + y ≤ 9, y > x, x ≥ 0. The solution region is the shaded area where all the individual inequality regions overlap, which is the triangular region bounded by the lines y = x, x = 0, and x + y = 9.
1. The document provides information on multiple integrals including double integrals, triple integrals, and integrals in spherical and cylindrical coordinates. It defines each type of integral and gives their general formulas.
2. Examples are provided for calculating double and triple integrals over different regions in rectangular, cylindrical, and spherical coordinate systems. The order of integration can be changed by considering strips or slices of the region.
3. Properties of the integrals include applying Fubini's theorem to change the order of integration, and relating the triple integral over a region to the double integral over the bounds and integrating over the third variable.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2006. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The document discusses the difference quotient formula for calculating the slope of a cord connecting two points (x, f(x)) and (x+h, f(x+h)) on a function graph. It defines the difference quotient as (f(x+h) - f(x))/h, which calculates the slope as the change in y-values (f(x+h) - f(x)) over the change in x-values (h). An example calculates the slope of the cord connecting the points (2, f(2)) and (2.2, f(2.2)) on the function f(x) = x^2 - 2x + 2.
This document provides a module on functions and simultaneous equations for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 15 problems on functions and 12 problems on simultaneous equations to help students prepare for their SPM examinations. The module is published by the Terengganu Education Department and involves several teachers from technical and science schools in the state.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2013. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
The document is a mathematics textbook for Additional Mathematics Form 4. It covers topics on functions, simultaneous equations, quadratic equations, and quadratic functions. It contains examples and practice questions for students to work through with answers. The questions range from simple calculations to solving equations and inequalities involving quadratic expressions.
Module 4 exponential and logarithmic functionsAya Chavez
This document provides an overview of logarithmic functions including:
1) Logarithmic functions are defined as the inverse of exponential functions.
2) The module will cover defining logarithmic functions, graphing them, applying logarithmic laws, and solving logarithmic equations.
3) Students are expected to learn to define logarithmic functions, draw their graphs, describe properties from graphs, apply laws of logarithms, and solve simple logarithmic equations.
This document is a resume for Martin K. Hastings that summarizes his education, employment experience, honors, and qualifications. Hastings graduated cum laude from Arizona State University with a Bachelor of Science in Marketing. He has marketing intern experience at Sedona Golf Resort where he coordinated promotions, cultivated business relationships, and executed social media campaigns. His honors include graduating cum laude and being named to the Dean's List for 6 semesters. Hastings' qualifications include strong communication, organizational, and leadership skills as well as experience in marketing, promotions, and business operations.
This document announces the birth of i.materialise, a new business built upon Materialise's core that aims to make 3D printing accessible to everyone. It will offer customizable and personalized products like lamps, interior items, jewelry, architectural models, bronzes, and gifts that are designed and printed using 3D printing technology. The document promotes i.materialise and provides contact information for anyone with questions or ideas.
This document provides a summary and analysis of the 2016 US presidential election as of March 15, 2016. On the Republican side, Donald Trump has won the most primary contests but Ted Cruz is not far behind in the delegate count. The Republican contests on March 15th, especially in Florida, Ohio, and Illinois, will be crucial in determining whether Trump can build an insurmountable lead. For the Democrats, Hillary Clinton maintains a sizable delegate lead over Bernie Sanders despite Sanders winning almost as many contests, with the March 15th primaries in key states also important for the trajectory of the race.
An overview of online writing best practices. Topics include conciseness, active voice, finding your voice, scannable text, calls to action, and design.
Matthew Pizzullo has over 15 years of experience in customer service roles. He currently works as an Uber driver in Phoenix, AZ where he maintains a 4.80 star rating. Previously, he worked at WageWorks Inc. for nearly a decade, ultimately becoming a supervisor managing teams of 20-33 agents. In this role, he assisted with training and improving processes. He also has experience handling high volumes of phone calls in call center roles. Pizzullo has a high school diploma and taken various college courses.
This document is a resume for Kenneth R. Smith that outlines his professional experience and qualifications. It summarizes his background in systems security, troubleshooting, LAN/WAN technical management, and technical specialties. His experience includes roles such as systems analyst, network administration, project management, and operations management. It also lists his education credentials and provides details on previous positions held at companies like Flextronics, PSR Professional, Affiliated Computer Services, and Outback Steakhouse between 2003-2013.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document discusses key themes in student recruitment: attraction, engagement, and conversion. It provides insights on when prospective students typically perform activities like course research, open day registration, and requesting prospectuses at different stages. The rest of the document offers practical actions institutions can take to improve in these areas, such as tracking marketing spending and results, keeping leads engaged through personalized communications, and testing approaches to optimize conversion rates throughout the student lifecycle.
Se muestra un estudio a un portafolio de acciones del sector tecnologico, mediante alteryx se hace un analisis para dar conclusion si se recomienda o no invertir en las acciones que forman parte del portafolio
Katie Allison has over 13 years of experience in customer service, administrative, and teaching roles. She has a Master's degree in English from Kansas State University and excels at communicating, problem solving, and managing multiple projects simultaneously. Currently she is a Lead Employment Assistant at H&R Block, where she leads a team, develops processes, and handles a high volume of communications.
Cubrir Riesgo Cambiario de ABC y Evaluar Flujo de Caja DescontadoOmar Castellanos R.
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This document appears to be a presentation by Tony D'Onofrio about his personal branding journey and expertise in social media and the retail technology industry. Over the past 10 years, D'Onofrio has built a personal brand on social media focused on retail innovation, reaching over 5,000 connections and 26,000 followers. The presentation outlines D'Onofrio's evolution in social media and personal branding, and how he has become a thought leader through his writing on emerging retail technology trends.
Aportes que TRANSPARENCIA envió al Jurado Nacional de Elecciones para la reglamentación de propaganda electoral, publicidad y neutralidad estatal para las Elecciones Generales 2016.
El documento presenta una introducción a la contabilidad computarizada, discutiendo su definición, ventajas y desventajas. También describe los sistemas de información contable y los modelos de procesamiento de información, incluidos los sistemas manuales y computarizados. Finalmente, cubre los tipos de sistemas de contabilidad computarizada e instrucciones para instalar un sistema en una empresa nueva o existente.
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 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.
This document discusses four ways to represent functions: verbally, numerically, visually through graphs, and algebraically through explicit formulas. It provides examples of each type of representation and discusses key properties of functions like domain, range, and the vertical line test. The document also covers piecewise-defined functions and functions with symmetry properties like even functions whose graphs are symmetric about the y-axis.
DLD Lecture No 15 Prime and Essential Implicants, Five Variable Map.pptxSaveraAyub2
Digital Logic and Design Lecture 15 discusses prime and essential prime implicants using Karnaugh maps. Examples are provided to show how to:
1) Identify minterms and maxterms on a Karnaugh map and group adjacent squares to find prime implicants in sum of products (SOP) or product of sums (POS) form.
2) Simplify Boolean functions defined by minterms using 4-variable and 5-variable Karnaugh maps to find essential prime implicants in SOP or POS form.
3) Implement simplified Boolean functions using logic gates in both SOP and POS forms to compare implementations.
MAT-121 COLLEGE ALGEBRAWritten Assignment 32 points eac.docxjessiehampson
MAT-121: COLLEGE ALGEBRA
Written Assignment 3
2 points each except for 5, 6, 9, 15, 16, which are 4 points each as indicated.
SECTION 3.1
Algebraic
For the following exercise, determine whether the relationship represents y as a function of x. If the relationship represents a function then write the relationship as a function of
x
using
f
as the function.
x+y2=5
Consider the relationship 7n-5m=4.
Write the relationship as a function
n
=
k
(
m
).
Evaluate
k
(
5
).
Solve for
k
(
m
) = 7.
Graphical
Given the following graph
Evaluate
f
(4)
Solve for
f
(x) = 4
Numeric
For the following exercise, determine whether the relationship represents a function.
{(0, 5), (-5, 8), (0, -8)}
For the following exercise, use the function
f
represented in table below. (4 points)
x
-18
-12
-6
0
6
12
18
f(x)
24
17
10
3
-4
-11
-18
Answer the following:
Evaluate
f
(-6).
Solve
f
(
x
) = -11
Evaluate
f
(12)
Solve
f
(
x
) = -18
For the following exercise, evaluate the expressions, given functions
f
,
g
, and
h
:
f(x)=4x+2
; g(x)=7-6x; h(x)=7x2-3x+6
f(-1)g(1)h(0) (4 points)
Real-world applications
The number of cubic yards of compost,
C
, needed to cover a garden with an area of
A
square feet is given by
C
=
h
(
A
).
A garden with an area of 5,000 ft2 requires 25 yd3 of compost. Express this information in terms of the function
h
.
Explain the meaning of the statement
h
(2500) = 12.5.
SECTION 3.2
Algebraic
For the following exercise, find the domain and range of each function and state it using interval notation.
f(x)=9-2x5x+13
Numeric
For the following exercise, given each function
f
, evaluate
f
(3),
f
(-2),
f
(1), and f (0). (4 points)
Real-World Applications
The height,
h,
of a projectile is a function of the time,
t,
it is in the air. The height in meters for
t
seconds is given by the function h(t)= -9.8t2+19.6t. What is the domain of the function? What does the domain mean in the context of the problem?
SECTION 3.3
Algebraic
For the following exercise, find the average rate of change of each function on the interval specified in simplest form.
k(x)=23x+1
on [2, 2+h]
Graphical
For the following exercise, use the graph of each function to
estimate
the intervals on which the function is increasing or decreasing.
For the following exercise, find the average rate of change of each function on the interval specified.
g(x)=3x2-23x3 on [1, 3]
Real-World Applications
Near the surface of the moon, the distance that an object falls is a function of time. It is given by d(t)=1.6t2, where
t
is in seconds and d(t) is in meters. If an object is dropped from a certain height, find the average velocity of the object from t = 2 to t = 5.
SECTION 3.4
Algebraic
For the following exercise, determine the domain for each function in interval notation. (4 points)
f(x)=2x+5 and g(x)=4x+9, find f-g, f+g, fg, and fg
For.
18 pc09 1.2_ digital logic gates _ boolean algebra_basic theoremsarunachalamr16
Digital logic gates are basic building blocks of digital circuits that make logical decisions based on input combinations. There are three basic logic gates: OR, AND, and NOT. Other common gates such as NAND, NOR, XOR, and XNOR are derived from these. Boolean algebra uses variables that can be 1 or 0, and logical operators like AND, OR, and NOT to represent logic functions. Logic functions can be expressed in canonical forms such as sum of minterms or product of maxterms. Standard forms like SOP and POS are also used. Conversions between these forms allow simplifying logic functions.
This document provides an overview of various mathematical models and functions, including:
1) Linear, polynomial, power, rational, algebraic, trigonometric, exponential, and logarithmic functions.
2) It describes the key properties of each type of function, such as their domains, ranges, and graph shapes.
3) It also discusses how functions can be transformed through stretching, shrinking, shifting, and reflection.
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.
1) The document outlines a course curriculum that covers functions, rational functions, one-to-one functions, exponential functions, and logarithmic functions over 32 hours spread across 8 weeks.
2) It provides the chapter titles and learning objectives for each chapter, along with the topics and hours allocated to each lesson.
3) Key concepts covered include functions as models of real-life situations, representing functions as sets of ordered pairs, tables, graphs, and piecewise functions.
This document provides an overview of various types of functions and their graphs. It begins with linear functions of the form y=mx+c and discusses how shifting these functions along the x- or y-axis changes their graphs. It then covers quadratic, square root, cube, reciprocal, constant, identity and absolute value functions. Piecewise, polynomial, algebraic, and transcendental functions are also defined. The document discusses bounded vs unbounded functions and concludes by examining circular and hyperbolic functions and their graphs.
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
This document provides definitions and examples of various types of numbers and functions. It discusses:
- Number sets including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- Types of intervals such as closed, open, and semi-open/semi-closed.
- Definitions of a function, including domain, co-domain, and range. Methods of representing functions include mapping, algebraic, and ordered pairs.
- Classification of functions as algebraic vs. transcendental, even vs. odd, explicit vs. implicit, continuous vs. discontinuous, and increasing vs. decreasing.
- Properties of even and odd functions are also discussed.
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 module introduces quadratic functions. It discusses identifying quadratic functions as those with the highest exponent of 2, rewriting quadratic functions in general form f(x) = ax^2 + bx + c and standard form f(x) = a(x-h)^2 + k, and the key properties of quadratic graphs including the vertex and axis of symmetry. The module provides examples of identifying quadratic functions from equations and ordered pairs/tables and rewriting quadratic functions between general and standard form using completing the square.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
This document summarizes three applications of linear algebra:
1) Fast integer multiplication, which can be done in O(n log n) time using linear algebra and Fourier transforms to represent integers as polynomials and multiply the polynomials.
2) Data structures like databases and graphs can be represented using matrices and vectors from linear algebra.
3) Multimedia like images, sound, and video can be stored as vectors and matrices, with images as pixel arrays, sound as amplitude arrays, and video as arrays of images.
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While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
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What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
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What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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My slides at Nordic Testing Days 6.6.2024
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1. SUMMARY COMBINATIONAL LOGIC CIRCUITS LECT1
Q1-What are the three basic Binary Logic operatios , their truth
tables , and their logic diagrames
three basic logical operations called AND, OR, and NOT:
when (X,Y,Z are binary variables equal 1 or 0)
AND Z = X • Y is called Z equal X AND Y
OR. Z = X + Y is called Z equal X OR Y
−
NOT. Z= X is called Z equal NOT X
The truth tables for the operations are shown in following Table
Logic Gat diagrames:
Q2- what is the Boolean algebra with example
1
2. The Boolean algebra is an algebra dealing with binary variables and logic
operations as example the function F is afunction in logic variables
X,Y,Z.
F ( X ,Y , Z ) = X + YZ
A Boolean function can be represented by a truth table.
The truth table for the logic operation is number of rows in a truth table is
2n, where n is the number of variables in the function. The binary
combinations for the truth table are the n-bit binary numbers that
correspond to counting in decimal from 0 through 2n – 1, the logic
diagrame for the function as following.
SUMMARY COMBINATIONAL LOGIC CIRCUITS LECT2
Q3- what are Basic Identities of Boolean algebra
2
3. Basic Identities of Boolean algebra are:
Examples:
1. X+XY=X(1+Y)=X
2. XY + XY = X (Y + Y ) = X
3. X + XY = ( X + X )( X + Y ) = X + Y
more examples:
4. X ( X + Y ) = X + XY = X
5. ( X + Y )( X + Y ) = X + YY = X
6. X ( X + Y ) = XX + XY = XY
Q4- what are meant by Dual, and its dual Boolean expressions?
The dual expression is change the OR to AND and AND to OR
XY to X+Y
X+Y to XY
The consensus theorem
XY + XZ + YZ = XY + XZ
3
4. The theorem shows that the third terrm, YZ, is redundant and can be
eliminated. Note that Y and Z are associated with X and X in the first
two terms and appear together in the term that is eliminated.
The dual of the consensus theorem is
( X + Y )( X + Z )(Y + Z ) = ( X + Y )( X + Z )
Q5- draw the logic diagram for the following function , and using the
Boolean algebra to simplify it , then draw the simplified logic
diagram for the simplified function with truth table check for
two function
F = XYZ + XYZ + XZ
The three terms in the expression are implemented with three AND gates,
,one OR, and two NOT.
Now consider a simplification of the expression for F by
applying Boolean algebra:
F = XYZ + XYZ + XZ
F = XY ( Z + Z ) + XZ by identity 14
F = XY .1 + XZ by identity7
Truth table for function check a, and b:
4
5. Q6- what is the Demorgan’s theorem, using it to complement
the Functions:
The the Demorgan’s theorem for complement the expressions are:
X + Y = X ⋅Y
X .Y = X + Y
X =X
Q7- Find the complements of thru function in example 1-1 by
taking the duals of the equations and complementing each
literal.
5
6. SUMMARY COMBINATIONAL LOGIC CIRCUITS LECT3
Q8- what are the Minterms and Maxterms
a- Minterms: A product term in which all the variables appear exactly once,
either complemented or uncomplemented, is called a minterm.
b- Maxterms: A sum term that contains all the variables in
complemented or uncomplemented form is called a maxterm.
6
7. Note from Tables a minterm and maxterm with the same subscript are the
complements of each other; that is, Mi = mi .
Q9- represents the following logic function with minterm and maxterm from
a given truth table:
By examining the truth tables for these minterms it is evident that the function F
can be expressed algebraically as the logical sum of the stated minterms:
F = XY Z + XYZ + XY Z + XYZ
Boolean Function of Three Variables can be further abbreviated by listing only
the decimal subscripts of the minterms:
F ( X , Y , Z ) = ∑ m(0,2,5,7)
7
8. Note that if the function contains all minterms, it is equal to 1.
F by maxterm:
This shows the procedure for expressing a Boolean function as a
product of maxterms. The abbreviated form for this product is
F ( X , Y , Z ) = Π M (1,3,4,6)
Q10- draw the logic diagram for the following functions
using Sum of Products, and Product of sum:
a-Sum of Products
An example of logical a Boolean function expressed as a sum of
products is
F = Y + XYZ + XY
The AND gates followed by the OR gate form a circuit configuration
referred to as a two-level implementation .
If the expression not in the standard form of sum of product, the
expression can be converted to a sum of products by applying the
appropriate distributive law as follows:
F = AB + C(D + E) = AB + CD +CE
The function F is implemented in a nonstandard form in Figure 1-6(a).
This requires two AND gates and two OR gates. There are three levels of
gating in the circuit. F is implemented in sum-of-products form in Figure
1-6(b).
8
9. b- Product of Sums
F = X (Y + Z )( X + Y + Z )
This expression has sum terms of one, two, and three literals. The sum terms
perform an OR operation, and the product is an AND operation.
SUMMARY COMBINATIONAL LOGIC CIRCUITS LECT4,5
Q11-draw the Karnaug map (K-map) for two, three, and four
variables for Simplifying a Boolean Function :
a-Two-Variable Map
There are four minterms
9
10. Figure 1-9(b) shows the map for the logical sum of three minterms:
m1 + m2 + m3 = XY + XY + XY = X + Y
The optimized expression X + Y is determined from the two-square area
for the variable X in the second row and the two-square area for Y in the
second column.
b- Three-Variable Map
The characteristic of the listed sequence is that only one bit changes in
value from one adjacent column to the next, which corresponds to the
Gray code.
In general, Three-variable maps exhibit the following characteristics:
One square represents a minterm of three literals.
A rectangle of two squares represents a product term of two literals.
A rectangle of four squares represents a product term of one literal.
10
11. A rectangle of eight squares produces a function that is always equal to
logic 1 as in the following figures:
c- Four-Variable Map
There are 16 minterms for four binary variables, and therefore, a four-
variable map consists of 16 squares, as shown :
The combinations of squares that can be chosen during the optimization
process in the four-variable map are as follows:
One square represents a minterm of four literals.
A rectangle of 2 squares represents a product term of three literals.
A rectangle of 4 squares represents a product term of two literals.
A rectangle of 8 squares represents a product term of one literal.
A rectangle of 16 squares produces a function that is always equal to logic 1.
Q12- Simplifying a Boolean Function Using a K-Map
F ( X , Y , Z ) = ∑ m(2,3,4,5)
1-Fill K- map with minterms
11
12. 2- The optimized expression for F:
F = XY + XY
Q13- Simplifying a Boolean Function Using a K-Map
F1(X,Y.Z) = Sm(3.4,6,7)
F2 (X,Y,Z) = Sw(0,2,4,5.6)
The map for F1 is shown in Figure 1-13(a). There are four squares marked
with 1's, one for each minterm of the function. Two adjacent squares are
combined in the third column to give a two-literal term YZ. The
remaining two squares with 1's are also adjacent by the cylinder-based
definition and are shown in the diagram with their values enclosed in half
rectangles. When combined, these two squares give the two-literal term
XZ .The optimized function thus becomes
F 1= YZ + XZ
12
13. And with the same procedures
F2 = Z + XY
Q14- Simplify the following two Boolean functions
F1(X,Y.Z) = ∑ m(1,3.4,5,6)
F2 (X,Y,Z) = ∑ m(1,2,3,5.7)
As in the fig1.14a the minterms 1, and 3 give the term XZ ,
minterms 4, and 6 give term XZ . The minterm 5 can be
combined with the minterm 4 to give XY or minterm 1 to give
Y Z so :
F1(X,Y.Z) = ∑ m(1,3.4,5,6) = XZ + XZ + Y Z
For the F2 as in fig.2.14b the minterms 1,3,4,7 combined and give the
term Z, the minterm 2 combined with minterm 3 to give the term XY so :
F2 (X,Y,Z) = ∑ m(1,2,3,5.7) = Z + XY
Q15- Simplifying a 4-Variable Function with a Map
F(W,X,Y,Z) = ∑ m(0,l,2,4,5,6,8,9,10,12,13)
13
14. As in the fig. minterms 0, 1,4, 5, 8, 9, 12, 13 the eight minterms
combined to give the term Y .The remaining term 2,6,14. The 6, 14,
combined with 4, 12 to give X Z , and 2,6 combined with 0,4 to give term
W Z so F simplified as:
F= Y + X Z + W Z
Q16- Simplifying a 4- Variable Function with a Map
F = ∑ m(0,1,2,6,12,13,14)
As in the fig. minterms 0, 1,8,9 the four minterms combined to give the
term B C .The remaining term 2,6,10. The 2,10 combined with 0,8 to give
B D , and 6 combined with 2 to give term A CD so F simplified as:
F= B C + B D + A CD
Q17- Simplifying a 4- Variable Function with a Map using
don’t care condition:
14
15. Consider the following incompletely specified function F that has three
don't-care minterms d:
F(A,B,C,D) = ∑ m(l,3,7,11,15)
d(A,B,C,D) = ∑ m(0,2,5)
As in the next fig.a the don't care is signed by x sign. The minterms 3, 7,
11, 15 are combined to give term CD the remaining minterm 1, 2 is
combined with don't care minterm 0,3 to give term A B
So the simplified F function is as:
F= CD+ A B
As in fig.b minterms 3, 7, 11, 15 combine to give CD, minterms 1,3,7
combined with don't care minterm 5 to give A D
Q18- explain exclusive-or, and exclusive-NOR with gates
The exclusive-OR (XOR), denoted by ⊕ , is a logical operation that
performs the function
X ⊕ Y = XY + XY
It is equal to 1 if exactly one input variable is equal to l.
The exclusive-NOR, also known as the equivalence, is the complement of
the exclusive-OR and is expressed by the function
15
16. X ⊕ Y = XY + XY
It is equal to 1 if both X and Y are equal to 1 or if both are equal to 0.
for more than two variables
The exclusive-OR is replaced by the odd function; there is no symbol for
exclusive-OR for more than two inputs.
By duality, the exclusive-NOR is replaced by the even function and has
no symbol for more than two inputs.
Q18- explain Odd Function, and even function
Odd Function
The exclusive-OR operation with three or more variables can be
converted into an ordinary Boolean function by replacing the ⊕ symbol
with its equivalent Boolean expression as follows:
X ⊕ Y ⊕ Z = ( X ⊕ Y ) Z + Z ( X ⊕ Y ) = ( XY + XY ) Z + Z ( XY + XY )
= XY Z + XYZ + XY Z + XYZ
The Boolean expression clearly indicates that the three-variable
exclusive-OR is equal to 1 if only one variable is equal to 1 or if all three
variables are equal to 1. Hence, whereas in the two-variable function
only one variable need be equal to 1, with three or more variables an odd
number of variables must be equal to 1.
16
17. Next Figure(a) shows the map for the three-variable odd function. The
four-variable case is shown in Figure (b).
The odd function is implemented by means of two-input exclusive-OR
gates, as shown in the next fig.
b-even function
It should be mentioned that the minterms not marked with 1's in the map
have an even number of 1's and constitute the complement of the odd
function, called the even function.
The even function is obtained by replacing the output gate with an
exclusive-NOR gate.
17
18. Q19-draw The symbols, truth tables, and function for different types of
gates
First year computer science
Introduction to Computer Organization
CHAPTER TWO
18
19. ARITHMETIC FUNCTIONS AND CIRCUITS
LEC. 6,7
2-1 Binary adders
An arithmetic circuit is a combinational circuit that performs arithmetic
operation? Such as addition, subtraction, multiplication, and division with
binary numbers or with decimal numbers in a binary code. We will
develop arithmetic circuits by means of hierarchical, iterative design. We
begin at the lowest level by finding a circuit that performs the addition of
two binary digits. This simple addition consists of four possible
elementary 0 +0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 10. The first three
operations produce a sum requiring only one bit to represent it, but
when both the augend and addend are equal to 1, the binary sum
requires two bits. Because of this case, the result is always
represented by two bits, the carry and the sum. The carry
obtained from the addition of two bits is added to the next
higher order pair of significant bits.
A combinational circuit that performs the addition of two bits is
called a half adder.
One that performs the addition of three bits (two significant bits
and a previous carry) is called a full adder.
The names of the circuits stem from the fact that two half
adders can be employed to implement a full adder. The half
adder and the full adder are basic arithmetic blocks with which
other arithmetic circuits are designed.
Half Adder
19
20. A half adder is an arithmetic circuit that generates the sum of two
binary digits. The circuit has two inputs and two outputs. The input
variables are the augend and addend bits to be added, and the
output variables produce the sum and carry. We assign the symbols
X and Y to the two inputs and S(for "sum") and C (for "carry") to
the outputs. The truth table for the half adder is listed in Table 3-1.
The C output is 1 only when both inputs are l. The S output
represents the least significant bit of the sum.
table 2.1
The Boolean functions for the two outputs, easily obtained from
the truth table, are:
S = XY + XY = X ⊕ Y
C = XY
20
21. The half adder can be implemented with one exclusive-OR gate
and one AND gate, as shown in fig.2.1
Figure 2-1 Logic Diagram of Half Adder
Full Adder
A full adder is a combinational circuit that forms arithmetic sum
off three input bits. Besides the three inputs it has two ouputs.
Two of the inputs are denoted by X and Y, represent the two
significant bits to be added . the third input Z represents the
carry from the previous lower significant position.the two
output are designated by symbols S for sum, and C for carry as
in table 3.2.the S is equal to 1 when only one input equal to one
or when all three inputs are equal to one.the carry is 1 if two
input equal 1 or three inputs equal 1.
Truth table 2.2
21
22. The simplified sum of of product functions for the two output are:
The Boolean function for full adder in terms of exlusive OR
operation can be expressed as:
The maps as in fig.2.2 , and the logic diagram as in fig.2.3.
22
23. Fig. 2.2 maps for full adder
Fig. 2.3 logic diagram full adder
Binary Ripple Carry Adder
A parallel binary adder is a digital circuit that produces the arithmetic
sum of two binary numbers using only combinational logic. The parallel
adder uses n full adders in parallel, with all input bits applied
simultaneously to produce the sum. The full adders are connected in
cascade, with the carry output from one full adder connected to the carry
input of the next full adder Since a 1 carry may appear near the least
significant bit of the adder and yet propagate through many full adders to
the most significant bit, just as a wave ripples outward from a pebble
dropped in a pond, the parallel adder is referred to as a ripple carry adder
23
24. Figure 2-4 shows the interconnection of four full-adder blocks to form a
4-bit ripple carry adder. The augend bits of A and the addend bits of B are
designated by subscripts in increasing order from right to left, with
subscript 0 denoting the least significant bit. The carries are connected in
a chain through the full adders. The input carry to the parallel adder is C0,
and the output carry is C4. An n-bit ripple carry adder requires n full
adders, with each output carry connected to the input carry of the next-
higher-order full adder.
FIGURF 2-4 4bit Ripple carry Adder
For example, consider the two binary numbers A = 1011 and B = 0011.
Their sum, S = 1110, is formed with a 4-bit ripple carry adder as follows:
24
25. 2-2 Binary subtraction
2-2-1 binary adder-subtractors
The circuit for subtracting A - B consists of a parallel adder as
shown in Figure 2-4, with inverters placed between each B
terminal and the corresponding full-adder input. The input carry
C0 must be equal to l.The operation that is performed becomes A
plus the 1's complement of B plus 1. This is equal to A plus the
2's complement of B. For unsigned numbers, it gives A - B if A
>B or the 2's complement of (B – A) if A< B.
The addition and subtraction operations can be combined into
one circuit with one common binary adder. This is done by
including an exclusive-OR gate with each full adder. A 4-bit
adder-subtractor circuit is shown in Figure 2-5. Input S controls
the operation. When S=0 the circuit is an adder, and when S = 1
the circuit becomes a subtracter. Each exclusive-OR gate
receives input S and one of the inputs of B, Bi . When S = 0, we
have Bi ⊕ 0. If the full adders receive the value of B. and the
25
26. input carry is 0, the circuit performs A plus B. When S = 1, we
have Bi ⊕ 1 = Bi and C0 = 1. In this case, the circuit performs the
operation A plus the 2's complement of B.
Fig.2.5 adder subtractor circuit
Signed Binary Numbers
The convention is to make the sign bit 0 for positive numbers, 1
for negative numbers. If the binary number is signed, then the
leftmost represents the sign and the rest of the bits represent the
number. If the binary number is assumed to be unsigned, then
the leftmost bit is the most significant bit of number. Table 2.3
show the signed binary numbers
26
27. Table 2-3
Signed Binary Addition and Subtraction
If tne number is negative we replaced by the 2's complement,
and then add, if the most significant bit is zero the result value is
positive valu. If the most significant bit is 1 the result is negative
ant is in the 2's complement form, to get the exact value we get
its complement.
Example:
Note that in the case of unsigned number the negative number replaced
by its complement, and then add, if there is carry it's ignored and the
27
28. result is positive. if there is no carry the result is negative , and the result
is in complemt value , if we need the exact value replace the result by its
complement valus.
Overflow
To obtain a correct answer when adding and subtracting, we must ensure
that the result has a sufficient number of bits to accommodate the sum. If
we start with two n-bit numbers, and the sum occupies n + 1 bits, we say
that an overflow occurs.
Example:
Note that the 8-bit result that should have been positive has a
negative sign bit and that the 8-bit result that should have been
negative has a positive sign bit. If, however, the carry out of the
sign bit position is taken as the sign bit of the result, then the 9-
bit answer so obtained will be correct. But since there is no
position in ilic result for the 9th bit, we say that an overflow has
occurred.
2-3 BINARY MULTIPLICATION
Multiplication of binary numbers is performed in the same way
as with decimal numbers. The multiplicand is multiplied by each
bit of the multiplier, starting from the least significant bit. Each
28
29. such multiplication forms a partial product. Successive partial
products are shifted one bit to the left. The final product is
obtained from the sum of the partial products.
To see how a binary multiplier can be implemented with a
combinational circuit, consider the multiplication of two 2-bit
numbers, as shown in Figure 2-6. The multiplicand bits are B0
and B1, the multiplier bits are A1 and A0, and the product is
C3C2C1C0 as in figure 2-6.
Fig.2.6 2 bit binary multiplier
Example
Multiply binary number 1 0 1, and 010
01
x 10
00
01
00
0010
29
30. CHAPTER Three
SEQUENTIAL
CIRCUITS
LEC. 8,9
3.1 sequential circuits
To this point, we have studied only combinational logic. In order to
perform useful or flexible sequences of operations, we need to be able to
construct circuits that can store information between the operations.
Such circuits are called sequential circuits.
A block diagram of a sequential circuit is shown in Figure 3-1. A
combinational circuit and storage elements are interconnected to form
the sequential circuit. The sequential circuit receives binary information
from its environment via the inputs. These inputs, together with the
present state of the storage elements to determine the binary value of
the outputs and the storage element next state.
Figure 3.1 block diagram of a sequential circuit
The storage elements used in clocked sequential circuits that called
flip –flops. A flip –flop is a binary storage device capable of storing one
bit of information and having timing characteristics. The block diagram
of a synchronous clocked sequential circuit is shown in Figure 3-2.
30
31. The flip – flops receive their inputs from the combinational circuit and
also from a clock signal with pulses that occur at fixed intervals of time.
Figure 3.2 synchronous clocked circuit
3.2 Latches
A storage element can maintain a binary state indefinitely until
.directed by an input signal to switch states
SR Latches 3.2.1
The latch has two inputs, labeled S for set and R for reset, and two
useful states When output Q=1 =1and Q =0, the latch is said to be in
_
the set state. When Q =0 and Q=1, it is in the reset state. Outputs Q
_
and Q are normally the complements of each other. When both
inputs are equal to 1 at the same time, an undefined state with both
outputs equal to 0 occurs as in figure 3.3.
31
32. Figure 3.3 SR Latch with NOR gates
In normal operation these problems area avoided by making sure that
1's are not applied to both inputs simultaneously. The behavior of the
SR latch described in the preceding paragraph is illustrated by the
logic simulator waveforms shown in Figure 3.4 Initially.
Figure 3-4 logic simulation of SR latch behaviour
The SR latch with two cross –coupled NAND gates is shown in Figure
3.5. It operates with both inputs normally at 1, unless the state of the latch
has to be changed. The application of a 0 to the S input causes output Q
to go to1.putting the latch in the set state. When the S input goes back to
1, the circuit remains in the set state with both inputs at 1. The state if the
32
33. latch is changed by placing a 0 on the R input. This causes the circuit to
go to the reset state and stay there even after both inputs return to 1. The
condition that is undefined for this NAND latch is when both inputs are
equal to 0 at the same time, an input combination that should be avoided.
Figure 3.5 Latch with nand gates
The operation of the basic NOR and NAND latches can be modified
by providing an additional control input that determines when the state of
the latch can be changed. An SR latch with a control input is shown in
Figure 3.6. It consists of the
Figure 3.6 SR Latch with control
The D latch in VLSI circuits is often constructed with transmission gates
(TGs). As shown in Figure -3.8. The C input controls two TGs. When
C=1, the TG connected to input D conducts, and the TG connected to
output Q disconnects .this produces a path from input D through two
33
34. Figure ١.٨D latch with transmition gates
Figure 3.8D latch with transmition gates
Inverters to output Q, thus, the output follows the data input as long as c
remains active (1). When C changes to 0, the first TG disconnects input D
from the circuit and the second TG connects the two inverters at the
output into a loop. Hence, the value that was present at input D at the
time that C went from 1 to 0 is retained at the Q output by the loop.
Flip –Flops 3-3
Any changes in the data input will change the state of the latch. In
this sense the latch is transparent, since its input value can be seen
from the outputs. The key to the proper operation is to prevent them
from being transparent. In a flip-flop, before an output can change, the
path from its inputs to its outputs is broken.
There are two ways that latches are combined to form a flip-flop.
One way is to combine two latches such that (1) the inputs presented
to the flip-flop when a clock pulse is present control its state and (2)
the state of the flip-flop changes only when a clock pulse is not present.
Such a circuit is called a master-slave flip-flop. Another way is to
34
35. produce a flip-flop that triggers only during a signal transition from 0
to 1 (or from 1 to 0) on the clock pulse. Such a circuit is said to be an
edge triggered flip-flop.
3.3.1 Master-slave Flip-Flops
The master-slave SR flip-flop, consisting of two latches and an
inverter, is shown in Figure 3.9. When the clock input C is 0, the
output of the inverter is 1. The slave latch is then enabled, and its
output Q is equal to the master output Y. the master latch is disabled,
because C is 0. When logic 1 clock pulse is applied, the values on S and
R control the value stored in the master latch Y. The slave, however, is
disabled as long as the pulse remains at the 1 level, because its C input is
equal to 0. Any changes in the external S and R inputs change the master
output Y,
Figure 3.9 SR master slaves Flip-Flop
but cannot affect the slave output 0. When the pulse returns to 0, the
master is disabled and is isolated from the S and R inputs. At the same
time, the slave is enabled, and the current value of Y is transferred to the
output of the flip-flop at Q.
35
36. 3.3.2D flip-flop A master-slave D flip-flop can be constructed from the
SR master-slave flip-flop by simply replacing the master SR latch with a
master D latch. The resulting circuit is shown in Figure 3.10. The
resulting circuit changes its value on the negative edge of the clock
Figure 3.10 Negative edge-triggered D Flip-Flop
For the clock
input equal to 0,
the master latch
is enabled and
transparent and
Figure 3.11 positive edge-triggered D Flip-Flop
follows the D
input value. The slave latch is disabled and holds the state of the flip-flop
fixed. When the positive edge occurs, the clock input changes to 1. This
disables the master latch so that its value is fixed and enables the slave
latch so that it copies the state of the master latch. The state of the master
latch to be copied is the state that is present at the positive edge of the
clock. Thus .the behavior appears to be edge triggered with the clock
input equal to 1.the master latch is disabled and cannot change so the
state of both the master and the slave remain unchanged finally. When the
clock input changes from 1 to O. the master is enabled before any change
36
37. in the master can reach it. Thus the value stored in the slave remains
unchanged during this transition.
3.3.3 JK and T filp-flop
The JK flip-flop was a modified version of the master-slave SR. when SR
produces undefined output for S=1, R=1, the JK cause the output to
complement its current value.
_ _
Q(t + 1) = J (t ) Q(t ) + k (t )Q(t )
The T (toggle) flip-flop is equivalent to JK with J and K tied together so
that J=K=1are applied. If we take the characteristic equation for the JK
and make this connection the equation becomes:
_ _
T exclusive-ORQ
Q(t + 1) =T Q + TQ = T ⊕Q
Standard graphics symbols
The stander graphics symbols for the different types of latches and flip-
flops are shown in Figure 3-12.Aflip –flop or latch is designated by a
rectangular block with inputs on the left and outputs on the right. One
output designates the normal state of the flip-flop, and the other, with a
bubble, designates the complement output.
37
38. Figure 3.13 flip-flop logic characteristic tables
, Characteristic equations, and excitation table
See fig.11
Figure 3.12 standard graphics symbols for latches and flip-flops
See fig.9
38
39. Characteristic table: the table defines the next state Q(t+1) as a function
of the present state Q(t) , and inputs of D, or S,R, or J,K, or T. note that
the pulse at C is not listed, it is assumed to occur between time t, and t+1.
Characteristic equation: it defines the next state after the clock pulse as
a function of present state, and Flip-flop inputs.
Excitation table: these tables define the Flip-flops input value as a
function of next state value after the clock pulse, given the present state
values.
Sequential circuit analysis: we are given the flip-flop inputs, and asked
to give the corresponding output. To do that we must knew the
characteristic table of the given flip-flop.
39
40. Sequential circuit design: the inverse of analysis, given present state,
and next state, required designing the input of the flip-flop; we must
know the excitation table of the given flip-flop.
e.g for D type
analysis
Q(t), Q(t+1) design
D
40