Chapter 2 Combinational Logic Circuits


 Clement Burke
 3 years ago
 Views:
Transcription
1 Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations
2 Chapter 2  Part 1 2
3 Chapter 2  Part 1 3
4 Chapter 2  Part 1 4
5 Chapter 2  Part 1 5
6 Y = A B z = x + y X = A Note: The statement: = 2 ( one plus one equals two ) is not the same as = 1 ( 1 or 1 equals 1 ). Chapter 2  Part 1 6
7 Operations are defined on the values "0" and "1" for each operator: AND 0 0 = = = = 1 OR = = = = 1 NOT 0 = 1 1 = 0 Chapter 2  Part 1 7
8 Chapter 2  Part 1 8
9 Switches in parallel => OR Switches in series => AND Normallyclosed switch => NOT C Chapter 2  Part 1 9
10 B C A D Chapter 2  Part 1 10
11 +V +V +V F G = X + Y X X. Y X X X Y Y (a) NOR (b) NAND (c) NOT Transistor of logic functions are called logic gates or just gates Transistor gate circuits can be modeled by switch circuits Chapter 2  Part 1 11
12 X Y X Z5 X Y Z 5 X 1 Y Y AND gate OR gate (a) Graphic symbols X X Z 5 X NOT gate or inverter Y (AND) X Y (OR) X1 Y (NOT) X (b) Timing diagram Chapter 2  Part 1 12
13 Truth Table = X + X Y Z Equation F = X + Y Z Logic Diagram F Chapter 2  Part 1 13
14 An algebraic structure defined on a set of at least two elements, B, together with three binary operators (denoted +, and ) that satisfies the following basic identities: 1. X + 0 = X 2. X. 1 = X 3. X + 1 = 1 4. X. 0 0 = 5. X + X = X 6. X. X = X 7. X + X = 1 8. X. X = 0 9. X = X X + Y = Y + X (X + Y) + Z = X + (Y + Z) X(Y + Z) = XY+ XZ X + Y = X. Y XY = YX (XY) Z = X(YZ) X+ YZ = (X + Y)(X + Z) X. Y = X + Y Commutative Associative Distributive DeMorgan s Chapter 2  Part 1 14
15 The identities above are organized into pairs. These pairs have names as follows: 14 Existence of 0 and Idempotence 78 Existence of complement 9 Involution Commutative Laws Associative Laws Distributive Laws DeMorgan s Laws The dual of an algebraic expression is obtained by interchanging + and and interchanging 0 s and 1 s. The identities appear in dual pairs. When there is only one identity on a line the identity is selfdual, i. e., the dual expression = the original expression. Chapter 2  Part 1 15
16 Chapter 2  Part 1 16
17 Chapter 2  Part 1 17
18 Chapter 2  Part 1 18
19 Chapter 2  Part 1 19
20 ( X + Y)Z + XY = Y(X + ( X + Y )Z + X Y Z) Chapter 2  Part 1 20
21 Chapter 2  Part 1 21 x y y ( )( ) n Minimizatio y y y x y y y x = + + = ( ) tion Simplifica y x y x y x y x = + + = + ( ) Absorption x y x x x y x x = + = + Consensus z y x z y z y x + = + + ( ) ( )( ) ( ) ( ) z y x z y z y x + + = DeMorgan's Laws x x = + + x x x x x x x x y x = +y
22 F1 = xyz F2 = x + yz F3 = xyz + x y z F4 = xy + x z + xy x y z F1 F2 F3 F Chapter 2  Part 1 22
23 A B + ACD + A BD + AC D + A BCD Chapter 2  Part 1 23
24 xyz + xyz Chapter 2  Part 1 24
25 Chapter 2  Part 1 25
26 Chapter 2  Part 1 26
27 XY X Y X Y X Y Chapter 2  Part 1 27
28 X + Y X + Y X + Y X + Y Chapter 2  Part 1 28
29 Chapter 2  Part 1 29
30 Chapter 2  Part 1 30
31 abcd abcd abcd abcd abcd abcd a + b + c + a + b + c + a + b + c + a + b + c + a + b + c + a + b + c + d d d d d d Chapter 2  Part 1 31
32 x y m 0 m 1 m 2 m x y M 0 M 1 M 2 M Chapter 2  Part 1 32
33 Chapter 2  Part 1 33
34 x y z index m 1 + m 4 + m 7 = F = = = = = = = = 1 Chapter 2  Part 1 34
35 Chapter 2  Part 1 35
36 F1 = (x + y + z) (x + y + z) (x + y + z) ( x + y + z) (x + y + z) x y z i M 0 M 2 M 3 M 5 M 6 = F = = = = = = = = 1 Chapter 2  Part 1 36
37 F(A,B,C,D) = M 3 M8 M11 M14 Chapter 2  Part 1 37
38 v + v f = x + x y f = x( y+ y ) + x y f = xy + xy + x y Chapter 2  Part 1 38
39 F = A + B C Chapter 2  Part 1 39
40 F = A + B C F(A, B,C) = Σ m (1,4,5,6,7) Chapter 2  Part 1 40
41 v v f ( x, y, z ) = x + x y x + x y = (x + x)(x + y) = 1(x + y) = x + y x + y + z z = (x + y + z) x+ y + z ( ) Chapter 2  Part 1 41
42 Convert to Product of Maxterms: f(a, B,C) = A C + BC + A B Use x + y z = (x+y) (x+z) with x = (A C + BC), y = A, and z = B to get: f = (A C + BC + A)(A C + BC + B) Then use x + x y = x + y to get: f = (C + BC + A)(AC + C + B) and a second time to get: f = (C + B + A)(A + C + B) Rearrange to standard order, f = (A + B + C)(A + B + C) to give f = M 5 M 2 Chapter 2  Part 1 42
43 F(x, y, z) = Σ m (1,3,5,7) F(x, y, z) F(x, y, z) = Σm(0,2,4,6) = ΠM(1,3,5,7) Chapter 2  Part 1 43
44 Chapter 2  Part 1 44
45 F(A, B,C) = Σm(1,4,5,6,7) Chapter 2  Part 1 45
46 Chapter 2  Part 1 46
47 Chapter 2  Part 1 47
48 Chapter 2  Part 1 48
49 A B C A B C A B C A B C A B C F A B C F Chapter 2  Part 1 49
50 Chapter 2  Part 1 50
Chapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active
More informationCHAPTER 2 BOOLEAN ALGEBRA
CHAPTER 2 BOOLEAN ALGEBRA This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operations 2.3 Boolean Expressions and Truth Tables 2.4 Basic Theorems 2.5 Commutative, Associative,
More informationChapter 2 Boolean Algebra and Logic Gates
Ch1: Digital Systems and Binary Numbers Ch2: Ch3: GateLevel Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan
More informationLogic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 1 Gate Circuits and Boolean Equations
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hperlinks are active in
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 1. Developed by George Boole in 1847 2. Applied to the Design of Switching Circuit by Claude Shannon in 1939 Department of Communication Engineering, NCTU 1 2.1 Basic Operations
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationLecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps
EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York
More informationChapter 2 Combinational
Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations HOANG Trang Reference: 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean
More informationChap 2. Combinational Logic Circuits
Overview 2 Chap 2. Combinational Logic Circuits Spring 24 Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization TwoLevel Optimization
More informationECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2
ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 8588224884 office, 8583530550 cell Office: 3802
More informationBoolean Algebra & Logic Gates. By : Ali Mustafa
Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions
More informationCHAPTER 3 BOOLEAN ALGEBRA
CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 ExclusiveOR and Equivalence Operations 3.3 The Consensus
More informationCS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman
CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions
More informationDiscrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST
Discrete Mathematics CS204: Spring, 2008 Jong C. Park Computer Science Department KAIST Today s Topics Combinatorial Circuits Properties of Combinatorial Circuits Boolean Algebras Boolean Functions and
More informationCombinational Logic. Review of Combinational Logic 1
Combinational Logic! Switches > Boolean algebra! Representation of Boolean functions! Logic circuit elements  logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational
More informationLogic Design. Chapter 2: Introduction to Logic Circuits
Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic
More informationIn Module 3, we have learned about Exclusive OR (XOR) gate. Boolean Expression AB + A B = Y also A B = Y. Logic Gate. Truth table
Module 8 In Module 3, we have learned about Exclusive OR (XOR) gate. Boolean Expression AB + A B = Y also A B = Y Logic Gate Truth table A B Y 0 0 0 0 1 1 1 0 1 1 1 0 In Module 3, we have learned about
More informationNumber System conversions
Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number
More informationMC9211 Computer Organization
MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (292/ODD) (29  / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra
More informationLecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions
EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The
More informationDigital Circuit And Logic Design I. Lecture 3
Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinationalcircuit analysis 4. Combinationalcircuit synthesis Panupong
More informationBinary Logic and Gates. Our objective is to learn how to design digital circuits.
Binary Logic and Gates Introduction Our objective is to learn how to design digital circuits. These circuits use binary systems. Signals in such binary systems may represent only one of 2 possible values
More informationChapter 2: Switching Algebra and Logic Circuits
Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used
More informationENG2410 Digital Design Combinational Logic Circuits
ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single
More informationChapter2 BOOLEAN ALGEBRA
Chapter2 BOOLEAN ALGEBRA Introduction: An algebra that deals with binary number system is called Boolean Algebra. It is very power in designing logic circuits used by the processor of computer system.
More informationLogic Gate Level. Part 2
Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively
More informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More informationGateLevel Minimization
GateLevel Minimization Dr. Bassem A. Abdullah Computer and Systems Department Lectures Prepared by Dr.Mona Safar, Edited and Lectured by Dr.Bassem A. Abdullah Outline 1. The Map Method 2. Fourvariable
More informationDigital Logic Design. Combinational Logic
Digital Logic Design Combinational Logic Minterms A product term is a term where literals are ANDed. Example: x y, xz, xyz, A minterm is a product term in which all variables appear exactly once, in normal
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. Overview Part 1 Gate
More informationLecture 2 Review on Digital Logic (Part 1)
Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Huntington Postulates 1. (a) Closure w.r.t. +. (b) Closure w.r.t.. 2. (a) Identity element 0 w.r.t. +. x + 0 = 0 + x = x. (b) Identity element 1 w.r.t.. x 1 =
More informationBoolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.
The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or
More informationEECS150  Digital Design Lecture 4  Boolean Algebra I (Representations of Combinational Logic Circuits)
EECS150  Digital Design Lecture 4  Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4bool1 Page 1, 9/5 9am Outline Review of
More informationCombinational Logic Design Principles
Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems
More informationStandard Expression Forms
ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms
More informationII. COMBINATIONAL LOGIC DESIGN.  algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):
ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra  developed by George Boole
More informationChapter 2: Boolean Algebra and Logic Gates
Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a
More informationEEA051  Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系. September 2004
EEA051  Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系 September 2004 Boolean Algebra (formulated by E.V. Huntington, 1904) A set of elements B={0,1} and two binary operators + and Huntington postulates 1. Closure
More informationOutline. EECS150  Digital Design Lecture 4  Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined
EECS150  Digital Design Lecture 4  Boolean Algebra I (Representations of Combinational Logic Circuits) January 30, 2003 John Wawrzynek Outline Review of three representations for combinational logic:
More informationChapter 2: Princess Sumaya Univ. Computer Engineering Dept.
hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev
E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean
More informationChapter 2 Boolean Algebra and Logic Gates
CSA051  Digital Systems 數位系統導論 Chapter 2 Boolean Algebra and Logic Gates 吳俊興國立高雄大學資訊工程學系 Chapter 2. Boolean Algebra and Logic Gates 21 Basic Definitions 22 Axiomatic Definition of Boolean Algebra 23
More informationBOOLEAN ALGEBRA TRUTH TABLE
BOOLEAN ALGEBRA TRUTH TABLE Truth table is a table which represents all the possible values of logical variables / statements along with all the possible results of the given combinations of values. Eg:
More informationChapter 2 : Boolean Algebra and Logic Gates
Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 14311432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean
More informationEx: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.
Boolean Expression Forms: Sumofproducts (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the
More information2 Application of Boolean Algebra Theorems (15 Points  graded for completion only)
CSE140 HW1 Solution (100 Points) 1 Introduction The purpose of this assignment is threefold. First, it aims to help you practice the application of Boolean Algebra theorems to transform and reduce Boolean
More informationEECS150  Digital Design Lecture 19  Combinational Logic Circuits : A Deep Dive
EECS150  Digital Design Lecture 19  Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150  Lec19cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationChapter 2. Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common
More informationUNIT 3 BOOLEAN ALGEBRA (CONT D)
UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions ExclusiveOR and ExclusiveNOR operations The consensus theorem Summary of algebraic
More informationBoolean Algebra, Gates and Circuits
Boolean Algebra, Gates and Circuits Kasper Brink November 21, 2017 (Images taken from Tanenbaum, Structured Computer Organization, Fifth Edition, (c) 2006 Pearson Education, Inc.) Outline Last week: Von
More informationCprE 281: Digital Logic
CprE 281: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Boolean Algebra CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
More informationAdministrative Notes. Chapter 2 <9>
Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2
More informationNumber System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary
Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION
More informationBoolean Algebra and Logic Gates Chapter 2. Topics. Boolean Algebra 9/21/10. EECE 256 Dr. Sidney Fels Steven Oldridge
Boolean Algebra and Logic Gates Chapter 2 EECE 256 Dr. Sidney Fels Steven Oldridge Topics DefiniGons of Boolean Algebra Axioms and Theorems of Boolean Algebra two valued Boolean Algebra Boolean FuncGons
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic
More informationGoals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations
Introduction to Electrical Engineering, II LETURE NOTES #2 Instructor: Email: Telephone: Office: ndrew. Kahng (lecture) abk@ucsd.edu 8588224884 office 3802 P&M lass Website: http://vlsicad.ucsd.edu/courses/ece20b/wi04/
More informationEvery time has a value associated with it, not just some times. A variable can take on any value within a range
Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»
More informationBinary Logic and Gates
1 COE 202 Digital Logic Binary Logic and Gates Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Introduction Boolean Algebra Elements of Boolean Algebra (Binary Logic) Logic Operations & Logic
More informationIntroduction to Digital Logic Missouri S&T University CPE 2210 Boolean Representations
Introduction to Digital Logic Missouri S&T University CPE 2210 Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology cetinkayae@mst.edu
More informationSignals and Systems Digital Logic System
Signals and Systems Digital Logic System Prof. Wonhee Kim Chapter 2 Design Process for Combinational Systems Step 1: Represent each of the inputs and outputs in binary Step 1.5: If necessary, break the
More information1. Expand each of the following functions into a canonical sumofproducts expression.
CHAPTER 4 PROLEMS 1. Expand each of the following functions into a canonical sumofproducts expression. (a) F(x, y, z) = xy + y z + x (b) F(w, x, y, z) = x y + wxy + w yz (c) F(A,,C,D) = AC + CD + C D
More informationBoolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE
Boolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE Today s Topics Boolean algebra applications in logic
More informationWeekI. Combinational Logic & Circuits
WeekI Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and
More informationECE 238L Boolean Algebra  Part I
ECE 238L Boolean Algebra  Part I August 29, 2008 Typeset by FoilTEX Understand basic Boolean Algebra Boolean Algebra Objectives Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand
More informationCHAPTER1: Digital Logic Circuits Combination Circuits
CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.
More informationECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering
ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of
More informationDIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA
DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS
More informationChapter 3. Boolean Algebra. (continued)
Chapter 3. Boolean Algebra (continued) Algebraic structure consisting of: set of elements B binary operations {+, } unary operation {'} Boolean Algebra such that the following axioms hold:. B contains
More informationINTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC110T
INTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC110T Credit : (2 + 1) / Week TEXT AND REF. BOOKS Text Book: Peter Norton (2011), Introduction to Computers, 7 /e, McGrawHill
More informationUnit 2 Session  6 Combinational Logic Circuits
Objectives Unit 2 Session  6 Combinational Logic Circuits Draw 3 variable and 4 variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the ProductofSums
More informationNAND, NOR and XOR functions properties
Laboratory NAND, NOR and XOR functions properties. Laboratory work goals Enumeration of NAND, NOR and XOR functions properties Presentation of NAND, NOR and XOR modules Realisation of circuits with gates
More informationMidterm1 Review. Jan 24 Armita
Midterm1 Review Jan 24 Armita Outline Boolean Algebra Axioms closure, Identity elements, complements, commutativity, distributivity theorems Associativity, Duality, De Morgan, Consensus theorem Shannon
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates
Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra
More informationComputer Organization I
Computer Organization I Lecture 6: Boolean Algebra /2/29 Wei Lu CS283 Overview Two Principles in Boolean Algebra () Duality Principle (2) Complement Principle Standard Form of Logic Expression () Sum of
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates The most common postulates used to formulate various algebraic structures are: 1. Closure. N={1,2,3,4 }, for any a,b N we obtain a unique c N by the operation
More informationCircuits & Boolean algebra.
Circuits & Boolean algebra http://xkcd.com/730/ CSCI 255: Introduction to Embedded Systems Keith Vertanen Copyright 2011 Digital circuits Overview How a switch works Building basic gates from switches
More informationEEE130 Digital Electronics I Lecture #4
EEE130 Digital Electronics I Lecture #4  Boolean Algebra and Logic Simplification  By Dr. Shahrel A. Suandi Topics to be discussed 41 Boolean Operations and Expressions 42 Laws and Rules of Boolean
More informationBOOLEAN ALGEBRA. Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra
BOOLEAN ALGEBRA Introduction 1854: Logical algebra was published by George Boole known today as Boolean Algebra It s a convenient way and systematic way of expressing and analyzing the operation of logic
More informationUNIVERSITI TENAGA NASIONAL. College of Information Technology
UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours
More informationEECS Variable Logic Functions
EECS150 Section 1 Introduction to Combinational Logic Fall 2001 2Variable Logic Functions There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs X
More informationCombinational Logic Fundamentals
Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic
More informationReview for Test 1 : Ch1 5
Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8
More informationBOOLEAN LOGIC. By Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift
BOOLEAN LOGIC Based on CBSE curriculum Class 11 By Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region Neha Tyagi, KV 5 Jaipur II Shift Introduction Boolean Logic, also known as boolean algebra was
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two and threevariable Fourvariable Determination of minimum expressions using essential prime implicants Fivevariable
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 4 Following the slides of Dr. Ahmed H. Madian محرم 439 ه Winter 28
More informationSwitches: basic element of physical implementations
Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by rewriting and by perfect induction Winter 200 CSE370  II  Boolean Algebra Switches: basic element
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 11 and 41 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationEC121 Digital Logic Design
EC121 Digital Logic Design Lecture 2 [Updated on 020418] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What
More informationCh 2. Combinational Logic. II  Combinational Logic Contemporary Logic Design 1
Ch 2. Combinational Logic II  Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:
More informationFunctions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:
Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)
More informationUC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits
2 Wawrzynek, Garcia 2004 c UCB UC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits 1 Introduction Original document by J. Wawrzynek (20031115) Revised
More informationChapter 2. Digital Logic Basics
Chapter 2 Digital Logic Basics 1 2 Chapter 2 2 1 Implementation using NND gates: We can write the XOR logical expression B + B using double negation as B+ B = B+B = B B From this logical expression, we
More informationCHAPTER 12 Boolean Algebra
318 Chapter 12 Boolean Algebra CHAPTER 12 Boolean Algebra SECTION 12.1 Boolean Functions 2. a) Since x 1 = x, the only solution is x = 0. b) Since 0 + 0 = 0 and 1 + 1 = 1, the only solution is x = 0. c)
More informationBoolean Algebra. Sungho Kang. Yonsei University
Boolean Algebra Sungho Kang Yonsei University Outline Set, Relations, and Functions Partial Orders Boolean Functions Don t Care Conditions Incomplete Specifications 2 Set Notation $09,3/#0,9 438 v V Element
More informationIntroduction to Digital Logic Missouri S&T University CPE 2210 Boolean Algebra
Introduction to Digital Logic Missouri S&T University CPE 2210 Boolean Algebra Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and
More informationSlide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary
Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016 SN s ENEL 353 Fall 2016 Slide Set 3 slide
More informationCombinational Logic. Fanin/ Fanout Timing. Copyright (c) 2012 Sean Key
Combinational Logic Fanin/ Fanout Timing Copyright (c) 2012 Sean Key Fanin & Fanout Fanin The number of inputs to a logic gate Higher fanin can lead to longer gate delays because of the higher input
More information