Chapter 2 Part 7 Combinational Logic Circuits

Size: px
Start display at page:

Download "Chapter 2 Part 7 Combinational Logic Circuits"

Transcription

1 University of Wisconsin - Madison EE/omp Sci 352 Digital Systems Fundamentals Kewal K. Saluja and u Hen Hu Spring 2002 hapter 2 Part 7 ombinational Logic ircuits Originals by: harles R. Kime and Tom Kamisnski Modified for course use by: Kewal K. Saluja and u Hen Hu NND and NOR Implementation We found that we could implement general oolean equations with these three primitives: ND OR NOT In this section we will find that either of two gates, the NND gate or the NOR gate can be used to implement arbitrary logic functions. We use the Positive Logic onvention (where all signals are active high) and a small circle to on a symbol to represent NOT or invert. hapter 2-7 2

2 NND Gates The basic positive logic NND gate is denoted by the following symbol: ND-Invert (NND) NND comes from NOT ND, I. e., the ND function with a NOT applied. We call this symbol for a NND gate an ND-Invert. The small circle represents the invert function. If we apply DeMorgan's Law we get: = + + F (,, ) = hapter NND Gates (ont.) pplying DeMorgan's Law gives: Invert-OR (NND) We call this symbol for a NND gate the Invert- OR since all inputs are inverted, followed by the OR function. oth symbols represent the NND gate - it is sometimes more logically descriptive to use one form over the other. NND gate with one input degenerates to an inverter. F (,, ) = + + hapter

3 NND Function Implementation NND gates can implement a simplified Sum-of- Products form. onstructing two level NND-NND gate circuit: D G (,,, D) = + The first level is two 2-input NND gates using ND- Invert. The second level is one 2-input NND gate using Invert-OR. Using the NND relationship, we have: D G (,,, D) = D = + D = + D hapter NND Implementation (ont.) In the implementation, the bubbles are on opposite ends of the same line. y =, they can be combined and deleted: D G(,,,D) sum-of-products (SOP) form results To implement an equation like: F(,,) = +, the NND for degenerates to a NOT since there is only one input hapter

4 Degenerate ND Term The degenerate NND becomes an inverter: F(,,) To implement the complement of F using NND gates, add an inverter to the output: F'(,,) hapter NND-NND Example w Implement: F (w,x,y, z) = yz + wx + xy + wz y z F(w,x,y,z) x w z F (w,x,y,z) y 0 x hapter

5 Summary: Two-Level NND ircuits Find minimum literal SOP form for F and F Select SOP form with smallest literal count onvert selected form to NND circuit using ND-invert (inverters for single literal ND terms) and invert-or symbols If SOP form for F used, add inverter to circuit output. hapter NOR Gates The basic positive logic NOR gate (Not-OR) is denoted by the following symbol: OR-Invert (NOR) F (,, ) = + + This is called the OR-Invert, since it is logically an OR function followed by an invert. y DeMorgan's Law we have the following Invert -ND symbol for a NOR gate: Invert-ND single-input NOR gate is an inverter, too. hapter

6 NOR Gates The basic positive logic NOR gate is denoted by the following symbol: OR-Invert (NOR) NOR comes from NOT OR, I. e., the OR function with a NOT applied. We call this symbol for a NOR gate an OR-Invert. The small circle represents the invert function. If we apply DeMorgan's Law we get: + + = F (,, ) = + + hapter 2-7 NOR Gates (ont.) pplying DeMorgan's Law gives: Invert-ND (NOR) We call this symbol for a NOR gate the Invert- ND since all inputs are inverted, followed by the ND function. oth symbols represent the NOR gate - it is sometimes more logically descriptive to use one form over the other. NOR gate with one input degenerates to an inverter. F (,, ) = hapter

7 NOR Function Implementation NND gates can implement a simplified Sum-of- Products form. onstructing two-level NOR-NOR circuit: The first level is two 2-input NOR gates using OR- Invert. The second level is one 2-input NOR gate using Invert-ND. Using the NOR relationship, we have: G (,,, D) = D ( + ) + (+ D) ( + ) (+ D) = = ( + )(+ D) ( ) ( ) G (,,, D) = + + D hapter Useful Transformations From Involution (i.e. (')' = ) and DeMorgan's Law, we get the following useful equivalences: ( ) = (( )')' (+) = ((+)')' ( )' (+)' ('+')' (' ')' ('+') (' ') These simple transformations can be used to manipulate a two level network. hapter

8 Graphical Transformations The relations from the previous slide lead to the following transformations: ( ) = (( )')' (+) = ((+)')' ('+')' (' ')' ( )' ('+') (+)' (' ') Recall that two bubbles in series can be removed from the circuit hapter General Two-level Implementations We need to consider whether the form of a two-level implementation is to be:. SOP (ND-OR) or 2. POS (OR-ND). omplemented output functions (i.e. ND-NOR or OR- NND) can be handled by complementing the function. Given a function F expressed as a Karnaugh Map, we can use the same general procedures we have used before to minimize the function and express it in SOP or POS form. hapter

9 General Implementations (ont.) Given a two level implementation desired, use the previous transfromations to get it into one of the below forms. Then follow the steps to transform the function to the desired form: For Type: ND-OR (SOP Form) ND-NOR (SOP complemented) OR-ND (POS Form) OR-NND (POS complemented) Use: ircle 's in the K-Map and minimize (lso use for NND-NND) ircle 0's in the K-Map and minimize ircle 0's in the K-Map and minimize SOP. Use DeMorgan's to transform to POS. (lso use for NOR-NOR) ircle 's in the K-Map and minimize SOP. Use DeMorgan's to transform to POS. hapter Implementation Example 0 Implement the function in NOR-OR. We can remove the "Inverter" and replace it with the complement of the input variable hapter

10 Implementation Example 2 0 Implement the function in ND-NOR. hapter Multi-level NND Implementations dd inverters in two-level implementation into the cost picture ttempt to combine inverters to reduce the term count ttempt to reduce literal + term count by factoring expression into POSOP or SOPOS hapter

11 Multi-level NND Example F = = = ( + + ) + ( + + ) 5 inputs and 8 gates* 7 inputs and 4 gates F * ounting inverters (NOTS) as input and gate hapter Multilevel NND Example 2 F = + D + + D hapter

Chapter 2: Princess Sumaya Univ. Computer Engineering Dept.

Chapter 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 information

Digital Fundamentals

Digital Fundamentals Digital Fundamentals Tenth Edition Floyd hapter 5 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, 10 th 2008 Pearson Education ENE, KMUTT ed 2009 2009 Pearson Education, Upper Saddle

More information

ECE/Comp Sci 352 Digital System Fundamentals Quiz # 1 Solutions

ECE/Comp Sci 352 Digital System Fundamentals Quiz # 1 Solutions Last (Family) Name: KIME First (Given) Name: Student I: epartment of Electrical and omputer Engineering University of Wisconsin - Madison EE/omp Sci 352 igital System Fundamentals Quiz # Solutions October

More information

12/31/2010. Overview. 05-Boolean Algebra Part 3 Text: Unit 3, 7. DeMorgan s Law. Example. Example. DeMorgan s Law

12/31/2010. Overview. 05-Boolean Algebra Part 3 Text: Unit 3, 7. DeMorgan s Law. Example. Example. DeMorgan s Law Overview 05-oolean lgebra Part 3 Text: Unit 3, 7 EEGR/ISS 201 Digital Operations and omputations Winter 2011 DeMorgan s Laws lgebraic Simplifications Exclusive-OR and Equivalence Functionally omplete NND-NOR

More information

EE 110 Practice Problems for Exam 1: Solutions, Fall 2008

EE 110 Practice Problems for Exam 1: Solutions, Fall 2008 EE Practice Problems for Exam : Solutions, Fall 28. ircle T (true) or F (false) for each of these oolean equations. (a). T FO + = (b). T FO + = ( + )( + ) (c). TO F = (d). TO F () = () (e). TO F + + =

More information

Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions

Lecture 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 information

Learning Objectives. Boolean Algebra. In this chapter you will learn about:

Learning Objectives. Boolean Algebra. In this chapter you will learn about: Ref. Page Slide /78 Learning Objectives In this chapter you will learn about: oolean algebra Fundamental concepts and basic laws of oolean algebra oolean function and minimization Logic gates Logic circuits

More information

Textbook: Digital Design, 3 rd. Edition M. Morris Mano

Textbook: Digital Design, 3 rd. Edition M. Morris Mano : 25/5/ P-/70 Tetbook: Digital Design, 3 rd. Edition M. Morris Mano Prentice-Hall, Inc. : INSTRUCTOR : CHING-LUNG SU E-mail: kevinsu@yuntech.edu.tw Chapter 3 25/5/ P-2/70 Chapter 3 Gate-Level Minimization

More information

Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 2 Circuit Optimization

Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 2 Circuit Optimization Logic and omputer Design Fundamentals hapter 2 ombinational Logic ircuits Part 2 ircuit Optimization harles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in View Show mode)

More information

CHAPTER 7 MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES

CHAPTER 7 MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES CHAPTER 7 MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES This chapter in the book includes: Objectives Study Guide 7.1 Multi-Level Gate Circuits 7.2 NAND and NOR Gates 7.3 Design of Two-Level Circuits Using

More information

12/31/2010. Overview. 10-Combinational Circuit Design Text: Unit 8. Limited Fan-in. Limited Fan-in. Limited Fan-in. Limited Fan-in

12/31/2010. Overview. 10-Combinational Circuit Design Text: Unit 8. Limited Fan-in. Limited Fan-in. Limited Fan-in. Limited Fan-in Overview 10-ombinational ircuit esign Text: Unit 8 Gates with elays and Timing Other Hazards GR/ISS 201 igital Operations and omputations Winter 2011 r. Louie 2 Practical logic gates are limited by the

More information

Chapter 4 Part 2 Sequential Circuits

Chapter 4 Part 2 Sequential Circuits University of Wisconsin - Madison ECE/Comp Sci 352 Digital Systems Fundamentals Kewal K. Saluja and Yu Hen Hu Spring 2002 Chapter 4 Part 2 Sequential Circuits Originals by: Charles R. Kime and Tom Kamisnski

More information

ﻮﻧﺭﺎﮐ ﺔﺸﻘﻧ ﺎﺑ ﻱﺯﺎﺳ ﻪﻨﻴﻬﺑ

ﻮﻧﺭﺎﮐ ﺔﺸﻘﻧ ﺎﺑ ﻱﺯﺎﺳ ﻪﻨﻴﻬﺑ بهينه سازي با نقشة کارنو Karnaugh Map Karnaugh Map Method of graphically representing the truth table that helps visualize adjacencies 2-variable K-map 3-variable K-map 2 3 2 3 6 7 4 5 D 3 2 4 5 7 6 2

More information

This form sometimes used in logic circuit, example:

This form sometimes used in logic circuit, example: Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from

More information

Boolean Algebra and Logic Simplification

Boolean Algebra and Logic Simplification S302 Digital Logic Design Boolean Algebra and Logic Simplification Boolean Analysis of Logic ircuits, evaluating of Boolean expressions, representing the operation of Logic circuits and Boolean expressions

More information

Goals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations

Goals 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 858-822-4884 office 3802 P&M lass Website: http://vlsicad.ucsd.edu/courses/ece20b/wi04/

More information

ENGR 303 Introduction to Logic Design Lecture 3. Dr. Chuck Brown Engineering and Computer Information Science Folsom Lake College

ENGR 303 Introduction to Logic Design Lecture 3. Dr. Chuck Brown Engineering and Computer Information Science Folsom Lake College Introduction to Logic Design Lecture 3 Dr. Chuck rown Engineering and Computer Information Science Folsom Lake College Outline for Todays Lecture Logic Circuits SOP / POS oolean Theorems DeMorgan s Theorem

More information

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA

DIGITAL 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 information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal

More information

Lecture 3. Title goes here 1. level Networks. Boolean Algebra and Multi-level. level. level. level. level

Lecture 3. Title goes here 1. level Networks. Boolean Algebra and Multi-level. level. level. level. level Lecture 3 Dr Richard Reilly Dept. of Electronic & Electrical Engineering Room 53, Engineering uilding oolean lgebra and Multi- oolean algebra George oole, little formal education yet was a brilliant scholar.

More information

EEE130 Digital Electronics I Lecture #4

EEE130 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 4-1 Boolean Operations and Expressions 4-2 Laws and Rules of Boolean

More information

II. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):

II. 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 information

14:332:231 DIGITAL LOGIC DESIGN

14:332:231 DIGITAL LOGIC DESIGN :: DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall Lecture #: Combinational Circuit Synthesis II hat if we have input variables? V = V = of Example with variables

More information

ELC224C. Karnaugh Maps

ELC224C. Karnaugh Maps KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to K-maps How to use K-maps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on

More information

Chap 2. Combinational Logic Circuits

Chap 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 Two-Level Optimization

More information

Boolean Algebra. Boolean Variables, Functions. NOT operation. AND operation. AND operation (cont). OR operation

Boolean Algebra. Boolean Variables, Functions. NOT operation. AND operation. AND operation (cont). OR operation oolean lgebra asic mathematics for the study of logic design is oolean lgebra asic laws of oolean lgebra will be implemented as switching devices called logic gates. Networks of Logic gates allow us to

More information

Chapter 7 Combinational Logic Networks

Chapter 7 Combinational Logic Networks Overview Design Example Design Example 2 Universal Gates NND-NND Networks NND Chips Chapter 7 Combinational Logic Networks SKEE223 Digital Electronics Mun im/rif/izam KE, Universiti Teknologi Malaysia

More information

CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates CMSC 33 Lecture 6 Postulates & Theorems of oolean lgebra Semiconductors CMOS Logic Gates UMC, CMSC33, Richard Chang Last Time Overview of second half of this course Logic gates & symbols

More information

Combinational Logic Design Principles

Combinational 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 information

Logic Design. Chapter 2: Introduction to Logic Circuits

Logic 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 information

Logic Gates and Boolean Algebra

Logic Gates and Boolean Algebra Logic Gates and oolean lgebra The ridge etween Symbolic Logic nd Electronic Digital Computing Compiled y: Muzammil hmad Khan mukhan@ssuet.edu.pk asic Logic Functions and or nand nor xor xnor not 2 Logic

More information

MC9211 Computer Organization

MC9211 Computer Organization MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra

More information

BOOLEAN ALGEBRA TRUTH TABLE

BOOLEAN 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 information

UNIT 5 KARNAUGH MAPS Spring 2011

UNIT 5 KARNAUGH MAPS Spring 2011 UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two- and three-variable Four-variable Determination of minimum expressions using essential prime implicants Five-variable

More information

Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra

Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra oolean lgebra Introduction 1854: Logical algebra was published by George oole known today as oolean lgebra It s a convenient way and systematic way of expressing and analyzing the operation of logic circuits.

More information

Signals and Systems Digital Logic System

Signals 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 information

Boolean Algebra & Logic Gates. By : Ali Mustafa

Boolean 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 information

ELEC Digital Logic Circuits Fall 2014 Logic Minimization (Chapter 3)

ELEC Digital Logic Circuits Fall 2014 Logic Minimization (Chapter 3) ELE 2200-002 Digital Logic ircuits Fall 204 Logic Minimization (hapter 3) Vishwani D. grawal James J. Danaher Professor Department of Electrical and omputer Engineering uburn University, uburn, L 36849

More information

Chapter 2 Combinational Logic Circuits

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 information

Logic. Basic Logic Functions. Switches in series (AND) Truth Tables. Switches in Parallel (OR) Alternative view for OR

Logic. Basic Logic Functions. Switches in series (AND) Truth Tables. Switches in Parallel (OR) Alternative view for OR TOPIS: Logic Logic Expressions Logic Gates Simplifying Logic Expressions Sequential Logic (Logic with a Memory) George oole (85-864), English mathematician, oolean logic used in digital computers since

More information

Combinatorial Logic Design Principles

Combinatorial Logic Design Principles Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 4-1 Boolean algebra a.k.a. switching algebra deals with boolean values -- 0, 1 Positive-logic convention analog voltages

More information

LOGIC GATES (PRACTICE PROBLEMS)

LOGIC GATES (PRACTICE PROBLEMS) LOGIC GTES (PRCTICE PROLEMS) Key points and summary First set of problems from Q. Nos. 1 to 9 are based on the logic gates like ND, OR, NOT, NND & NOR etc. First four problems are basic in nature. Problems

More information

Department of Electrical and Computer Engineering University of Wisconsin - Madison. ECE/CS 352 Digital System Fundamentals.

Department of Electrical and Computer Engineering University of Wisconsin - Madison. ECE/CS 352 Digital System Fundamentals. Last (family) name: First (given) name: Student I.D. #: Circle section: Lipasti Kim Department of Electrical and Computer Engineering University of Wisconsin - Madison ECE/CS 352 Digital System Fundamentals

More information

Chapter 2. Introduction. Chapter 2 :: Topics. Circuits. Nodes. Circuit elements. Introduction

Chapter 2. Introduction. Chapter 2 :: Topics. Circuits. Nodes. Circuit elements. Introduction hapter 2 Introduction igital esign and omputer rchitecture, 2 nd Edition avid Money Harris and Sarah L. Harris logic circuit is composed of: Inputs Outputs Functional specification Timing specification

More information

CHAPTER1: Digital Logic Circuits Combination Circuits

CHAPTER1: 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 information

Why digital? Overview. Number Systems. Binary to Decimal conversion

Why digital? Overview. Number Systems. Binary to Decimal conversion Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not

More information

Discrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST

Discrete 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 information

Digital Design 2. Logic Gates and Boolean Algebra

Digital Design 2. Logic Gates and Boolean Algebra Digital Design 2. Logic Gates and oolean lgebra József Sütő ssistant Lecturer References: [1] D.M. Harris, S.L. Harris, Digital Design and Computer rchitecture, 2nd ed., Elsevier, 213. [2] T.L. Floyd,

More information

Chapter 2: Switching Algebra and Logic Circuits

Chapter 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 information

1. Expand each of the following functions into a canonical sum-of-products expression.

1. Expand each of the following functions into a canonical sum-of-products expression. CHAPTER 4 PROLEMS 1. Expand each of the following functions into a canonical sum-of-products 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 information

211: Computer Architecture Summer 2016

211: Computer Architecture Summer 2016 211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Review: cache hit rate - Project3 - Digital Logic: - truth table => SOP - simplification: Boolean

More information

Digital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1

Digital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1 Lecture 5 - Chapter 2 Outline Other Logic Gates and their uses Other Logic Operations CS 2420 Husain Gholoom - lecturer Page 1 Digital logic gates CS 2420 Husain Gholoom - lecturer Page 2 Buffer A buffer

More information

CHAPTER III BOOLEAN ALGEBRA

CHAPTER III BOOLEAN ALGEBRA CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.

More information

Number System conversions

Number 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 information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2 - Part 1 2 Chapter 2 - Part 1 3 Chapter 2 - Part 1 4 Chapter 2 - Part

More information

E&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates

E&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 information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in

More information

Lecture (02) NAND and NOR Gates

Lecture (02) NAND and NOR Gates Lecture (02) NAND and NOR Gates By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 2018, CSE303 Logic design II NAND gates and NOR gates In this section we will define NAND and NOR gates. Logic

More information

Theorem/Law/Axioms Over (.) Over (+)

Theorem/Law/Axioms Over (.) Over (+) material prepared by: MUKESH OHR Follow me on F : http://www.facebook.com/mukesh.sirji4u OOLEN LGER oolean lgebra is a set of rules, laws and theorems by which logical operations can be mathematically

More information

Midterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes

Midterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name

More information

ELEC Digital Logic Circuits Fall 2015 Logic Minimization (Chapter 3)

ELEC Digital Logic Circuits Fall 2015 Logic Minimization (Chapter 3) ELE 2200-002 igital Logic ircuits Fall 205 Logic Minimization (hapter 3) Vishwani. grawal James J. anaher Professor epartment of Electrical and omputer Engineering uburn University, uburn, L 36849 http://www.eng.auburn.edu/~vagrawal

More information

CHAPTER III BOOLEAN ALGEBRA

CHAPTER III BOOLEAN ALGEBRA CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.

More information

Gate-Level Minimization

Gate-Level Minimization Gate-Level 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. Four-variable

More information

Ch 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1

Ch 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 information

ENG2410 Digital Design Combinational Logic Circuits

ENG2410 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 information

Possible logic functions of two variables

Possible logic functions of two variables ombinational logic asic logic oolean algebra, proofs by re-writing, proofs by perfect induction logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,..., minimal set Logic realization

More information

Combinational Logic (mostly review!)

Combinational Logic (mostly review!) ombinational Logic (mostly review!)! Logic functions, truth tables, and switches " NOT, N, OR, NN, NOR, OR,... " Minimal set! xioms and theorems of oolean algebra " Proofs by re-writing " Proofs by perfect

More information

Show that the dual of the exclusive-or is equal to its compliment. 7

Show that the dual of the exclusive-or is equal to its compliment. 7 Darshan Institute of ngineering and Technology, Rajkot, Subject: Digital lectronics (2300) GTU Question ank Unit Group Questions Do as directed : I. Given that (6)0 = (00)x, find the value of x. II. dd

More information

The Karnaugh Map COE 202. Digital Logic Design. Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals

The Karnaugh Map COE 202. Digital Logic Design. Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals The Karnaugh Map COE 202 Digital Logic Design Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals Presentation Outline Boolean Function Minimization The Karnaugh Map (K-Map) Two, Three,

More information

Logic Design Combinational Circuits. Digital Computer Design

Logic Design Combinational Circuits. Digital Computer Design Logic Design Combinational Circuits Digital Computer Design Topics Combinational Logic Karnaugh Maps Combinational uilding locks Timing 2 Logic Circuit logic circuit is composed of: Inputs Outputs Functional

More information

Combinational Logic Fundamentals

Combinational 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 information

Karnaugh Maps Objectives

Karnaugh Maps Objectives Karnaugh Maps Objectives For Karnaugh Maps of up to 5 variables Plot a function from algebraic, minterm or maxterm form Obtain minimum Sum of Products and Product of Sums Understand the relationship between

More information

Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University

Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Aby K George, ECE Department, Wayne State University Contents The Map method Two variable

More information

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions.

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions. Combinational logic Possible logic functions of two variables Logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,... Minimal set xioms and theorems of oolean algebra Proofs by re-writing

More information

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA CHPTER 3 LOGIC GTES & OOLEN LGER C H P T E R O U T C O M E S Upon completion of this chapter, student should be able to: 1. Describe the basic logic gates operation 2. Construct the truth table for basic

More information

EC-121 Digital Logic Design

EC-121 Digital Logic Design EC-121 Digital Logic Design Lecture 2 [Updated on 02-04-18] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What

More information

Lecture 2 Review on Digital Logic (Part 1)

Lecture 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 information

UNIVERSITI TENAGA NASIONAL. College of Information Technology

UNIVERSITI 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 information

EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive

EECS150 - 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 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational

More information

CMSC 313 Lecture 18 Midterm Exam returned Assign Homework 3 Circuits for Addition Digital Logic Components Programmable Logic Arrays

CMSC 313 Lecture 18 Midterm Exam returned Assign Homework 3 Circuits for Addition Digital Logic Components Programmable Logic Arrays MS 33 Lecture 8 Midterm Exam returned Assign Homework 3 ircuits for Addition Digital Logic omponents Programmable Logic Arrays UMB, MS33, Richard hang MS 33, omputer Organization & Assembly

More information

Lecture 4: More Boolean Algebra

Lecture 4: More Boolean Algebra Lecture 4: More Boolean Algebra Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University ENGIN2

More information

Basic Gate Repertoire

Basic Gate Repertoire asic Gate Repertoire re we sure we have all the gates we need? Just how many two-input gates are there? ND OR NND NOR SURGE Hmmmm all of these have 2-inputs (no surprise) each with 4 combinations, giving

More information

E&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 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 information

Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps

Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology

More information

CHAPTER * 2-2.* Pearson Education Limited Problem Solutions, Global Edition Chapter 2. Verification of DeMorgan s Theorem

CHAPTER * 2-2.* Pearson Education Limited Problem Solutions, Global Edition Chapter 2. Verification of DeMorgan s Theorem HPTER 2 2-.* a) XYZ = X + Y + Z Verification of DeMorgan s Theorem Pearson Education Limited 206. X Y Z XYZ XYZ X + Y + Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b) X + YZ = ( X + Y) ( X + Z) The Second

More information

COMP2611: Computer Organization. Introduction to Digital Logic

COMP2611: Computer Organization. Introduction to Digital Logic 1 OMP2611: omputer Organization ombinational Logic OMP2611 Fall 2015 asics of Logic ircuits 2 its are the basis for binary number representation in digital computers ombining bits into patterns following

More information

Name. ECE-200 Intelligent Systems

Name. ECE-200 Intelligent Systems Name Spring 2003 EE-200 Intelligent Systems Pracice Final Solution ll problems have the same weight Problem 1. We are working with a multiplexor that is to switch between four sources (inputs), each one

More information

Digital Logic Design ABC. Representing Logic Operations. Dr. Kenneth Wong. Determining output level from a diagram. Laws of Boolean Algebra

Digital Logic Design ABC. Representing Logic Operations. Dr. Kenneth Wong. Determining output level from a diagram. Laws of Boolean Algebra Digital Logic Design ENGG1015 1 st Semester, 2011 Representing Logic Operations Each function can be represented equivalently in 3 ways: Truth table Boolean logic expression Schematics Truth Table Dr.

More information

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA

DIGITAL 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 information

Combinational Logic Circuits Part II -Theoretical Foundations

Combinational Logic Circuits Part II -Theoretical Foundations Combinational Logic Circuits Part II -Theoretical Foundations Overview Boolean Algebra Basic Logic Operations Basic Identities Basic Principles, Properties, and Theorems Boolean Function and Representations

More information

WEEK 2.1 BOOLEAN ALGEBRA

WEEK 2.1 BOOLEAN ALGEBRA WEEK 2.1 BOOLEAN ALGEBRA 1 Boolean Algebra Boolean algebra was introduced in 1854 by George Boole and in 1938 was shown by C. E. Shannon to be useful for manipulating Boolean logic functions. The postulates

More information

EEL 3701 Fall Multiply out the following logic expressions to obtain the Sum of Products.

EEL 3701 Fall Multiply out the following logic expressions to obtain the Sum of Products. Page 1/5 Version: 7/30/17 1. Multiply out the following logic expressions to obtain the Sum of Products.. (/W /X Y) (W /Z) (/W X /Z) (W X) (W Y /Z). (/W /Y Z) (/W Y) (X /Y /Z) (/W X Y) (Y Z) 2. Using oolean

More information

CHAPTER 2 BOOLEAN ALGEBRA

CHAPTER 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 information

New Students Day Activity

New Students Day Activity Course: S ELECTRONICS New Students Day ctivity Introduction: In S Level Electronics you need to gain an understanding of the electronic circuits so that you can then start to design your own circuits like

More information

Digital- or Logic Circuits. Outline Logic Circuits. Logic Voltage Levels. Binary Representation

Digital- or Logic Circuits. Outline Logic Circuits. Logic Voltage Levels. Binary Representation Outline Logic ircuits Introduction Logic Systems TTL MOS Logic Gates NOT, OR, N NOR, NN, XOR Implementation oolean lgebra ombinatorial ircuits Multipleer emultipleer rithmetic ircuits Simplifying Logic

More information

Chapter 2. Digital Logic Basics

Chapter 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 information

Logic Gate Level. Part 2

Logic 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 information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Overview Part 1 Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra

More information

3. PRINCIPLES OF COMBINATIONAL LOGIC

3. PRINCIPLES OF COMBINATIONAL LOGIC Principle of ombinational Logic -. PRINIPLES OF OMINTIONL LOGI Objectives. Understand the design & analysis procedure of combinational logic.. Understand the optimization of combinational logic.. efinitions

More information

2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS

2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital

More information