Computer Organization I. Lecture 7: Boolean Algebra
|
|
- Reynard Riley
- 7 years ago
- Views:
Transcription
1 Computer Organization I Lecture 7: Boolean Algebra
2 Overview Standard Form of Logic Expression (1) Implementing Logic Function in Sum of Minterm (2) Implementing Logic Function in Product of Maxterm Alternative Standard Form of Logic Expression (1) Sum of Product (SOP) (2) Product of Sum (POS)
3 Objectives Know how to implement a logic function in SOM and POM given any logic expression
4 Standard Forms of Logic Expression - Sum of Minterms Any Boolean function can be expressed as a Sum of Minterms. For the function table, the minterms used are the terms corresponding to the 1's For expressions, expand all terms first to explicitly list all minterms. Do this by ANDing any term missing a variable x with a term ( x + x ). Example: Implement F = X + X Y as a sum of minterms. First expand terms: F = X (Y + Y) + X Y Then distribute terms: F = X Y + X Y + X Y Express as sum of minterms: F = m 3 + m 2 + m 0
5 Standard Forms of Logic Expression - Another Example about Sum of Minterms Example: Implement F = A + B C There are three variables, A, B, and C which we take to be the standard order. Expanding the terms with missing variables Collect terms (i.e. removing the duplicate terms) Express as SOM F = A (B + B) (C + C) + (A + A) B C = (A B + A B ) (C + C) + A B C + A B C = (A B C + A B C + A B C + A B C) + A B C + A B C = m 7 + m 6 + m 5 + m 4 + m 1 = m 1 + m 4 + m 5 + m 6 + m 7
6 Standard Forms of Logic Expression - Shorthand SOM Form Start with F = A + B C and end up with F = m 1 + m 4 + m 5 + m 6 + m 7 F can be denoted in the formal shorthand: F = Ʃm (1,4,5,6,7), Ʃ is the logical sum of minterms Note that we explicitly show the standard variables in order and drop the m designators.
7 Standard Forms of Logic Expression - Product of Maxterms Any Boolean function can be expressed as a Product of Maxterms. For the function table, the maxterms used are the terms corresponding to the 0's For expressions, expand all terms first to explicitly list all maxterms. Do this by first applying the distributive law, ORing terms missing variable x with a term equal to x x and then applying the distributive law again. Example: Convert F (X,Y,Z) = X + X Y as a product of maxterms. First apply distributive law: F = (X + X) (X + Y) = 1 (X + Y) = X + Y Then add missing variable: F = X + Y + Z Z = (X + Y + Z) (X+Y+Z) Express as sum of maxterms: F = M 2 M 3
8 Standard Forms of Logic Expression - Another Example about POM Example: Convert F(A,B,C) = A C + B C + A B Apply the distributive law repeatedly Collect terms F = (A + B + C) (A + B + C) Express as POM F = M 2 M 5
9 Standard Forms of Logic Expression - Shorthand POM Form Start with F = A C + B C + A B and end up with F = M 2 M 5 F can be denoted in the formal shorthand: F = M(2,5), is the logical consecutive product maxterms Note that we explicitly show the standard variables in order and drop the M designators.
10 A B C m 0 ABC Standard Forms of Logic Expression - Complement Function by SOM m 1 ABC m 2 ABC m 3 ABC m 4 ABC m 5 ABC m 6 ABC m 7 ABC The complement of a function expressed as a sum of minterms is constructed by selecting the minterms missing in the sum-of-minterms forms. F (A,B,C) = m 0 + m 2 + m 5 + m 7 = Ʃm (0,2,5,7) F (A,B,C) = m 1 + m 3 + m 4 + m 6 = Ʃm (1,3,4,6) F F
11 Standard Forms of Logic Expression - Complement Function by POM the complement of a function expressed by a Sum of Minterms form is simply the Product of Maxterms with the same indices according to DeMorgan Theorem. Example: given F (A,B,C) = Ʃm (0,2,5,7) we have F in maxterms: F (A,B,C) = M (0,2,5,7) we have F in minterms: F (A,B,C) = Ʃm (1,3,4,6) we have F in maxterms: F (A,B,C) = M (1,3,4,6)
12 Standard Forms of Logic Expression - Conversion between SOM and POM To convert between sum-of-minterms and product-of-maxterms form (or vice-versa) we follow these steps: Find the function complement by swapping terms in the list with terms not in the list. Change from products to sums, or vice versa. Example: Given F as: F (A,B,C) = Ʃm (1,3,5,7) Form the Complement: F (A,B,C) = Ʃm (0,2,4,6) Then use the other form with the same indices this forms the complement again, giving the other form of the original function: F (A,B,C) = M (0,2,4,6)
13 Alternative Standard Form of Logic Expression So far we have known that any logic function with a given truth table can be expressed as a standard form using sum of minterms or product of maxterms. This can be considered as the first step of logic circuit design. Once the standard form is obtained according to truth table, the second step is to try to simplify expression to see if the number of product or the number of literals can be reduced in order to simplify circuit. The result might be in a sum of product (SOP) form or a product of sum (POS) form, and we call the SOP and POS form as the alternative standard form of logic expression
14 Alternative Standard Form of Logic Expression Sum-of-Product (SOP) form: logic expressions are written as an OR of AND terms Product-of-Sums (POS) form: logic expressions are written as an AND of OR terms Examples: SOP: A B C + A B C + B POS: (A + B) (A + B + C) C These mixed forms are neither SOP nor POS (A B + C) (A + C) A B C + AC (A + B)
15 Summary Implementing any logic function in SOM and POM based on Truth Table or Boolean Algebraic Manipulation Introduction to SOP and POS
16 Thank you Q & A
BOOLEAN ALGEBRA & LOGIC GATES
BOOLEAN ALGEBRA & LOGIC GATES Logic gates are electronic circuits that can be used to implement the most elementary logic expressions, also known as Boolean expressions. The logic gate is the most basic
More informationCSEE 3827: Fundamentals of Computer Systems. Standard Forms and Simplification with Karnaugh Maps
CSEE 3827: Fundamentals of Computer Systems Standard Forms and Simplification with Karnaugh Maps Agenda (M&K 2.3-2.5) Standard Forms Product-of-Sums (PoS) Sum-of-Products (SoP) converting between Min-terms
More informationKarnaugh Maps. Circuit-wise, this leads to a minimal two-level implementation
Karnaugh Maps Applications of Boolean logic to circuit design The basic Boolean operations are AND, OR and NOT These operations can be combined to form complex expressions, which can also be directly translated
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationKarnaugh Maps & Combinational Logic Design. ECE 152A Winter 2012
Karnaugh Maps & Combinational Logic Design ECE 52A Winter 22 Reading Assignment Brown and Vranesic 4 Optimized Implementation of Logic Functions 4. Karnaugh Map 4.2 Strategy for Minimization 4.2. Terminology
More informationCSE140: Midterm 1 Solution and Rubric
CSE140: Midterm 1 Solution and Rubric April 23, 2014 1 Short Answers 1.1 True or (6pts) 1. A maxterm must include all input variables (1pt) True 2. A canonical product of sums is a product of minterms
More informationChapter 2: Boolean Algebra and Logic Gates. Boolean Algebra
The Universit Of Alabama in Huntsville Computer Science Chapter 2: Boolean Algebra and Logic Gates The Universit Of Alabama in Huntsville Computer Science Boolean Algebra The algebraic sstem usuall used
More informationBoolean Algebra. Boolean Algebra. Boolean Algebra. Boolean Algebra
2 Ver..4 George Boole was an English mathematician of XIX century can operate on logic (or Boolean) variables that can assume just 2 values: /, true/false, on/off, closed/open Usually value is associated
More informationSimplifying Logic Circuits with Karnaugh Maps
Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified
More informationUnited States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1
United States Naval Academy Electrical and Computer Engineering Department EC262 Exam 29 September 2. Do a page check now. You should have pages (cover & questions). 2. Read all problems in their entirety.
More information2.0 Chapter Overview. 2.1 Boolean Algebra
Thi d t t d ith F M k 4 0 2 Boolean Algebra Chapter Two Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital
More information1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.
File: chap04, Chapter 04 1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. 2. True or False? A gate is a device that accepts a single input signal and produces one
More informationCDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012
CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Multi-Level Gate Circuits NAND and NOR Gates Design of Two-Level Circuits Using NAND and NOR Gates
More informationCSE140: Components and Design Techniques for Digital Systems
CSE4: Components and Design Techniques for Digital Systems Tajana Simunic Rosing What we covered thus far: Number representations Logic gates Boolean algebra Introduction to CMOS HW#2 due, HW#3 assigned
More informationBasic Logic Gates Richard E. Haskell
BASIC LOGIC GATES 1 E Basic Logic Gates Richard E. Haskell All digital systems are made from a few basic digital circuits that we call logic gates. These circuits perform the basic logic functions that
More informationDigital circuits make up all computers and computer systems. The operation of digital circuits is based on
Digital Logic Circuits Digital circuits make up all computers and computer systems. The operation of digital circuits is based on Boolean algebra, the mathematics of binary numbers. Boolean algebra is
More informationElementary Logic Gates
Elementary Logic Gates Name Symbol Inverter (NOT Gate) ND Gate OR Gate Truth Table Logic Equation = = = = = + C. E. Stroud Combinational Logic Design (/6) Other Elementary Logic Gates NND Gate NOR Gate
More informationLogic Reference Guide
Logic eference Guide Advanced Micro evices INTOUCTION Throughout this data book and design guide we have assumed that you have a good working knowledge of logic. Unfortunately, there always comes a time
More informationBoolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris Hui-Ru Jiang Spring 2010
Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic simplification Proving validity of an
More informationGates, Circuits, and Boolean Algebra
Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks
More informationCH3 Boolean Algebra (cont d)
CH3 Boolean Algebra (cont d) Lecturer: 吳 安 宇 Date:2005/10/7 ACCESS IC LAB v Today, you ll know: Introduction 1. Guidelines for multiplying out/factoring expressions 2. Exclusive-OR and Equivalence operations
More informationUnit 3 Boolean Algebra (Continued)
Unit 3 Boolean Algebra (Continued) 1. Exclusive-OR Operation 2. Consensus Theorem Department of Communication Engineering, NCTU 1 3.1 Multiplying Out and Factoring Expressions Department of Communication
More informationKarnaugh Maps (K-map) Alternate representation of a truth table
Karnaugh Maps (K-map) lternate representation of a truth table Red decimal = minterm value Note that is the MS for this minterm numbering djacent squares have distance = 1 Valuable tool for logic minimization
More informationC H A P T E R. Logic Circuits
C H A P T E R Logic Circuits Many important functions are naturally computed with straight-line programs, programs without loops or branches. Such computations are conveniently described with circuits,
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More informationPhiladelphia University Faculty of Information Technology Department of Computer Science ----- Semester, 2007/2008.
Philadelphia University Faculty of Information Technology Department of Computer Science ----- Semester, 2007/2008 Course Syllabus Course Title: Computer Logic Design Course Level: 1 Lecture Time: Course
More informationLogic in Computer Science: Logic Gates
Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers
More informationSwitching Algebra and Logic Gates
Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design
More informationEE360: Digital Design I Course Syllabus
: Course Syllabus Dr. Mohammad H. Awedh Fall 2008 Course Description This course introduces students to the basic concepts of digital systems, including analysis and design. Both combinational and sequential
More informationTwo-level logic using NAND gates
CSE140: Components and Design Techniques for Digital Systems Two and Multilevel logic implementation Tajana Simunic Rosing 1 Two-level logic using NND gates Replace minterm ND gates with NND gates Place
More informationDigital Electronics Part I Combinational and Sequential Logic. Dr. I. J. Wassell
Digital Electronics Part I Combinational and Sequential Logic Dr. I. J. Wassell Introduction Aims To familiarise students with Combinational logic circuits Sequential logic circuits How digital logic gates
More informationGates & Boolean Algebra. Boolean Operators. Combinational Logic. Introduction
Introduction Gates & Boolean lgebra Boolean algebra: named after mathematician George Boole (85 864). 2-valued algebra. digital circuit can have one of 2 values. Signal between and volt =, between 4 and
More informationCHAPTER 3 Boolean Algebra and Digital Logic
CHAPTER 3 Boolean Algebra and Digital Logic 3.1 Introduction 121 3.2 Boolean Algebra 122 3.2.1 Boolean Expressions 123 3.2.2 Boolean Identities 124 3.2.3 Simplification of Boolean Expressions 126 3.2.4
More informationFORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. The Binary Adder
FORDHAM UNIVERITY CIC 3593 Fordham College Lincoln Center Computer Organization Dept. of Computer and Info. cience pring, 2011 1 Introduction The Binar Adder The binar adder circuit is an important building
More informationA single register, called the accumulator, stores the. operand before the operation, and stores the result. Add y # add y from memory to the acc
Other architectures Example. Accumulator-based machines A single register, called the accumulator, stores the operand before the operation, and stores the result after the operation. Load x # into acc
More informationUnderstanding Logic Design
Understanding Logic Design ppendix of your Textbook does not have the needed background information. This document supplements it. When you write add DD R0, R1, R2, you imagine something like this: R1
More informationA Course Material on DIGITAL PRINCIPLES AND SYSTEM DESIGN
A Course Material on By MS.G.MANJULA ASSISTANT PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM 638 56 QUALITY CERTIFICATE This is to certify
More informationDESIGN OF GATE NETWORKS
DESIGN OF GATE NETWORKS DESIGN OF TWO-LEVEL NETWORKS: and-or and or-and NETWORKS MINIMAL TWO-LEVEL NETWORKS KARNAUGH MAPS MINIMIZATION PROCEDURE AND TOOLS LIMITATIONS OF TWO-LEVEL NETWORKS DESIGN OF TWO-LEVEL
More informationCS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011
CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationCourse Requirements & Evaluation Methods
Course Title: Logic Circuits Course Prefix: ELEG Course No.: 3063 Sections: 01 & 02 Department of Electrical and Computer Engineering College of Engineering Instructor Name: Justin Foreman Office Location:
More informationCombinational circuits
Combinational circuits Combinational circuits are stateless The outputs are functions only of the inputs Inputs Combinational circuit Outputs 3 Thursday, September 2, 3 Enabler Circuit (High-level view)
More informationDigital Logic Design
Digital Logic Design Version 4.6 printed on February 2016 First published on August 2006 Background and Acknowledgements This material has been developed for the first course in Digital Logic Design. The
More informationSECTION C [short essay] [Not to exceed 120 words, Answer any SIX questions. Each question carries FOUR marks] 6 x 4=24 marks
UNIVERSITY OF KERALA First Degree Programme in Computer Applications Model Question Paper Semester I Course Code- CP 1121 Introduction to Computer Science TIME : 3 hrs Maximum Mark: 80 SECTION A [Very
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationScilab Textbook Companion for Digital Electronics: An Introduction To Theory And Practice by W. H. Gothmann 1
Scilab Textbook Companion for Digital Electronics: An Introduction To Theory And Practice by W. H. Gothmann 1 Created by Aritra Ray B.Tech Electronics Engineering NIT-DURGAPUR College Teacher Prof. Sabyasachi
More informationDigital Logic Design. Basics Combinational Circuits Sequential Circuits. Pu-Jen Cheng
Digital Logic Design Basics Combinational Circuits Sequential Circuits Pu-Jen Cheng Adapted from the slides prepared by S. Dandamudi for the book, Fundamentals of Computer Organization and Design. Introduction
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationMEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationDigital Electronics Detailed Outline
Digital Electronics Detailed Outline Unit 1: Fundamentals of Analog and Digital Electronics (32 Total Days) Lesson 1.1: Foundations and the Board Game Counter (9 days) 1. Safety is an important concept
More informationSet operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE
Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More informationELEC 2210 - EXPERIMENT 1 Basic Digital Logic Circuits
Objectives ELEC - EXPERIMENT Basic Digital Logic Circuits The experiments in this laboratory exercise will provide an introduction to digital electronic circuits. You will learn how to use the IDL-00 Bit
More informationChapter 1. Computation theory
Chapter 1. Computation theory In this chapter we will describe computation logic for the machines. This topic is a wide interdisciplinary field, so that the students can work in an interdisciplinary context.
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationOct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also
More information3.Basic Gate Combinations
3.Basic Gate Combinations 3.1 TTL NAND Gate In logic circuits transistors play the role of switches. For those in the TTL gate the conducting state (on) occurs when the baseemmiter signal is high, and
More informationComputer Engineering 290. Digital Design: I. Lecture Notes Summer 2002
Computer Engineering 290 Digital Design: I Lecture Notes Summer 2002 W.D. Little Dept. of Electrical and Computer Engineering University of Victoria 1 Preface These lecture notes complement the material
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More information6. BOOLEAN LOGIC DESIGN
6. OOLEN LOGI DESIGN 89 Topics: oolean algebra onverting between oolean algebra and logic gates and ladder logic Logic examples Objectives: e able to simplify designs with oolean algebra 6. INTRODUTION
More informationIntroduction. The Quine-McCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick
CSEE E6861y Prof. Steven Nowick The Quine-McCluskey Method Handout 5 January 21, 2016 Introduction The Quine-McCluskey method is an exact algorithm which finds a minimum-cost sum-of-products implementation
More informationNAND and NOR Implementation
University of Wisconsin - Madison EE/omp ci 352 Digital ystems Fundamentals harles R. Kime ection 2 Fall 200 hapter 2 ombinational Logic ircuits Part 7 harles Kime & Thomas Kaminski NND and NOR Implementation
More informationLecture 5: Gate Logic Logic Optimization
Lecture 5: Gate Logic Logic Optimization MAH, AEN EE271 Lecture 5 1 Overview Reading McCluskey, Logic Design Principles- or any text in boolean algebra Introduction We could design at the level of irsim
More informationLecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot- Nots
Lecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot- Nots Registers As you probably know (if you don t then you should consider changing your course), data processing is usually
More informationCloud Computing is NP-Complete
Working Paper, February 2, 20 Joe Weinman Permalink: http://www.joeweinman.com/resources/joe_weinman_cloud_computing_is_np-complete.pdf Abstract Cloud computing is a rapidly emerging paradigm for computing,
More informationThe string of digits 101101 in the binary number system represents the quantity
Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for
More informationBinary full adder. 2-bit ripple-carry adder. CSE 370 Spring 2006 Introduction to Digital Design Lecture 12: Adders
SE 370 Spring 2006 Introduction to Digital Design Lecture 12: dders Last Lecture Ls and Ls Today dders inary full 1-bit full omputes sum, carry-out arry-in allows cascaded s = xor xor = + + 32 ND2 11 ND2
More informationCombinational Logic Design Process
Combinational Logic Design Process Create truth table from specification Generate K-maps & obtain logic equations Draw logic diagram (sharing common gates) Simulate circuit for design verification Debug
More informationComputer Programming Lecturer: Dr. Laith Abdullah Mohammed
Algorithm: A step-by-step procedure for solving a problem in a finite amount of time. Algorithms can be represented using Flow Charts. CHARACTERISTICS OF AN ALGORITHM: Computer Programming Lecturer: Dr.
More informationENGI 241 Experiment 5 Basic Logic Gates
ENGI 24 Experiment 5 Basic Logic Gates OBJECTIVE This experiment will examine the operation of the AND, NAND, OR, and NOR logic gates and compare the expected outputs to the truth tables for these devices.
More informationEG1108: Electrical Engineering
EG1108: Electrical Engineering Part 2: Application Examples Ben M. Chen Professor of Electrical and Computer Engineering National University of Singapore Office: E4 06 08 Phone: 6516 2289 Email: bmchen@nus.edu.sg
More informationChapter 1: Digital Systems and Binary Numbers
Chapter 1: Digital Systems and Binary Numbers Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching
More informationANALOG & DIGITAL ELECTRONICS
ANALOG & DIGITAL ELECTRONICS Course Instructor: Course No: PH-218 3-1-0-8 Dr. A.P. Vajpeyi E-mail: apvajpeyi@iitg.ernet.in Room No: #305 Department of Physics, Indian Institute of Technology Guwahati,
More informationA Little Set Theory (Never Hurt Anybody)
A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra
More informationBoolean Design of Patterns
123 Boolean Design of Patterns Basic weave structures interlacement patterns can be described in many ways, but they all come down to representing the crossings of warp and weft threads. One or the other
More informationGrade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationThe GMAT Guru. Prime Factorization: Theory and Practice
. Prime Factorization: Theory and Practice The following is an ecerpt from The GMAT Guru Guide, available eclusively to clients of The GMAT Guru. If you would like more information about GMAT Guru services,
More informationLevent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC
Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n
More informationTheory of Logic Circuits. Laboratory manual. Exercise 3
Zakład Mikroinformatyki i Teorii Automatów yfrowych Theory of Logic ircuits Laboratory manual Exercise 3 Bistable devices 2008 Krzysztof yran, Piotr zekalski (edt.) 1. lassification of bistable devices
More informationCombinational Logic Design
Chapter 4 Combinational Logic Design The foundations for the design of digital logic circuits were established in the preceding chapters. The elements of Boolean algebra (two-element switching algebra
More informationSum-of-Products and Product-of-Sums expressions
Sum-of-Products and Product-of-Sums expressions This worksheet and all related files are licensed under the reative ommons ttribution License, version.. To view a copy of this license, visit http://creativecommons.org/licenses/by/./,
More informationexclusive-or and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576
exclusive-or and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576 Outline exclusive OR gate (XOR) Definition Properties Examples of Applications Odd Function Parity Generation and Checking
More informationDesign and Development of Virtual Instrument (VI) Modules for an Introductory Digital Logic Course
Session ENG 206-6 Design and Development of Virtual Instrument (VI) Modules for an Introductory Digital Logic Course Nikunja Swain, Ph.D., PE South Carolina State University swain@scsu.edu Raghu Korrapati,
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationChapter 7 Memory and Programmable Logic
NCNU_2013_DD_7_1 Chapter 7 Memory and Programmable Logic 71I 7.1 Introduction ti 7.2 Random Access Memory 7.3 Memory Decoding 7.5 Read Only Memory 7.6 Programmable Logic Array 77P 7.7 Programmable Array
More informationNUMBER SYSTEMS. 1.1 Introduction
NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationCOMBINATIONAL CIRCUITS
COMBINATIONAL CIRCUITS http://www.tutorialspoint.com/computer_logical_organization/combinational_circuits.htm Copyright tutorialspoint.com Combinational circuit is a circuit in which we combine the different
More informationRead-only memory Implementing logic with ROM Programmable logic devices Implementing logic with PLDs Static hazards
Points ddressed in this Lecture Lecture 8: ROM Programmable Logic Devices Professor Peter Cheung Department of EEE, Imperial College London Read-only memory Implementing logic with ROM Programmable logic
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationLecture 1: Course overview, circuits, and formulas
Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik
More informationResolution. Informatics 1 School of Informatics, University of Edinburgh
Resolution In this lecture you will see how to convert the natural proof system of previous lectures into one with fewer operators and only one proof rule. You will see how this proof system can be used
More informationCourse Manual Automata & Complexity 2015
Course Manual Automata & Complexity 2015 Course code: Course homepage: Coordinator: Teachers lectures: Teacher exercise classes: Credits: X_401049 http://www.cs.vu.nl/~tcs/ac prof. dr. W.J. Fokkink home:
More informationStandards for Mathematical Practice: Commentary and Elaborations for 6 8
Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:
More informationTake-Home Exercise. z y x. Erik Jonsson School of Engineering and Computer Science. The University of Texas at Dallas
Take-Home Exercise Assume you want the counter below to count mod-6 backward. That is, it would count 0-5-4-3-2-1-0, etc. Assume it is reset on startup, and design the wiring to make the counter count
More informationAdder.PPT(10/1/2009) 5.1. Lecture 13. Adder Circuits
Adder.T(//29) 5. Lecture 3 Adder ircuits Objectives Understand how to add both signed and unsigned numbers Appreciate how the delay of an adder circuit depends on the data values that are being added together
More informationLecture Notes in Discrete Mathematics. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Lecture Notes in Discrete Mathematics Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior
More information