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

Size: px
Start display at page:

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

Transcription

1 CMSC 33 Lecture 6 Postulates & Theorems of oolean lgebra Semiconductors CMOS Logic Gates UMC, CMSC33, Richard Chang

2 Last Time Overview of second half of this course Logic gates & symbols Equivalence of oolean functions, truth tables, oolean formulas and combinational circuits Universality of NND gates UMC, CMSC33, Richard Chang

3 Postulates of oolean lgebra Commutative: =, + = + ssociative: ()C = (C), ( + ) + C = + ( + C) Distributive: ( + C) = + C, + C = ( + )( + C) Identity: there exists and such that for all, = and + =. Complement: for all, there exists such that = and + = where and are the identity elements.

4 Some Theorems of oolean lgebra Zero and One: =, + = Idempotence: =, + = Involution: = DeMorgan s: = +, + = bsorption: ( + ) =, + = Consensus: + C + C = + C, ( + )( + C)( + C) = ( + )( + C) 2

5 Theorems are derived from the postulates Idempotence: = identity, commutative = ( + ) complement = + distributive = + complement = commutative, identity = + identity = () + complement = ( + )( + ) distributive = ( + ) commutative, complement = + commutative, identity 3

6 Theorems are derived from the postulates Zero and One: = () complement = () commutative = () associative = idempotent = complement + = ( + ) + complement = + ( + ) commutative = ( + ) + associative = + idempotent = complement 4

7 Theorems are derived from the postulates bsorption: + = + identity, commutative = ( + ) distributive = one = commutative, identity ( + ) = + distributive = + idempotent = absorption 5

8 Proof by truth table? bsorption: + = + Proof by truth table only applies to - oolean lgebra. Proof by derivation from postulates holds for any oolean lgebra. 6

9 oolean lgebra with 8 elements Elements are the subsets of {a, b, c}:, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} Operations:, +. Union and intersection are commutative and associative. Union distributes over intersection: ( C) = ( ) ( C). Intersection distributes over union: ( C) = ( ) ( C). Identity: =, = {a, b, c}, {a, b, c} = and =. Complement: = {a, b, c}, = and = {a, b, c}. ll postulates hold. Therefore, all derived theorems also hold. 7

10 Simplifying MJ3 Simplify MJ3 using the postulates and theorems of oolean lgebra MJ3(,, C) = C + C + C + C SOP form = C + C + C + C + C + C idempotent = C + C + C + C + C + C commutative = ( + )C + ( + )C + (C + C) distributive = C + C + complement = C + C + identity Resulting circuit uses fewer gates. 8

11 -6 ppendix : Reduction of Digital Logic The lgebraic Method This majority circuit is functionally equivalent to the previous majority circuit, but this one is in its minimal two-level form: C F Principles of Computer rchitecture by M. Murdocca and V. Heuring 999 M. Murdocca and V. Heuring

12 -9 ppendix : Digital Logic ND-OR Implementation of Majority C Gate count is 8, gate input count is 9. C C F C C Principles of Computer rchitecture by M. Murdocca and V. Heuring 999 M. Murdocca and V. Heuring

13 How do we make gates??? UMC, CMSC33, Richard Chang

14 -5 ppendix : Digital Logic Truth Table Developed in 854 by George oole. Further developed by Claude Shannon (ell Labs). Outputs are computed for all possible input combinations (how many input combinations are there?) Consider a room with two light switches. How must they work? Inputs Output Hot Light Z Z Switch Switch Principles of Computer rchitecture by M. Murdocca and V. Heuring 999 M. Murdocca and V. Heuring

15 Electrically Operated Switch Example: a relay source: UMC, CMSC33, Richard Chang

16 Semiconductors Electrical properties of silicon Doping: adding impurities to silicon Diodes and the P-N junction Field-effect transistors UMC, CMSC33, Richard Chang

17 n Inverter using MOSFET CMOS = complementary metal oxide semiconductor P-type transistor conducts when gate is low N-type transistor conducts when gate is high p-type MOSFET n-type MOSFET UMC, CMSC33, Richard Chang

18 NND GTE

19 NOR GTE

20 CMOS Logic vs ipolar Logic MOSFET transistors are easier to miniaturie CMOS logic has lower current drain CMOS logic is easier to manufacture UMC, CMSC33, Richard Chang

21 CMSC 33, Computer Organiation & ssembly Language Programming, section Fall 23 Homework 3 Due: October 3, 23. (2 points) Draw schematics for the following functions using ND, OR and NOT gates. (Do not simplify the formulas.) (a) X(Y + Z) (b) X + Y Z (c) X(Y + Z) (d) W(X + Y Z) 2. ( points) Question.3, page 493, Murdocca & Heuring 3. ( points) Prove the Consensus Theorem + C + C = + C using the postulates and theorems of oolean algebra (except the Consensus Theorem itself) in Table - (p. 45). Hint: use absorption creatively. 4. (4 points) For each CMOS circuit below, (a) Provide a truth table for the circuit s function. (b) For diagram (a), write down the Sum-of-Products (SOP) oolean formula for the truth table. For diagram (b), wrtie down the Product-of-Sums (POS) oolean formula. (c) Simplify the SOP or POS formula using the postulates and theorems of oolean lgebra (p. 45). Show all work. (d) Draw the logic diagram of the simplified formula using ND, OR, NND, NOR and NOT gates. C D C D (a) (b)

22 Circuits for ddition Next time UMC, CMSC33, Richard Chang

CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates UMBC, CMSC313, Richard Chang Last Time Overview of second half of this course Logic gates &

More information

CMSC 313 Lecture 17. Focus Groups. Announcement: in-class lab Thu 10/30 Homework 3 Questions Circuits for Addition Midterm Exam returned

CMSC 313 Lecture 17. Focus Groups. Announcement: in-class lab Thu 10/30 Homework 3 Questions Circuits for Addition Midterm Exam returned Focus Groups CMSC 33 Lecture 7 Need good sample of all types of CS students Mon /7 & Thu /2, 2:3p-2:p & 6:p-7:3p Announcement: in-class lab Thu /3 Homework 3 Questions Circuits for Addition Midterm Exam

More information

CMSC 313 Lecture 15 Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo

CMSC 313 Lecture 15 Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo CMSC 33 Lecture 5 Good-bye ssembly Language Programming Overview of second half on Digital Logic DigSim Demo UMC, CMSC33, Richard Chang Good-bye ssembly Language What a pain! Understand

More information

CMSC 313 Lecture 16 Announcement: no office hours today. Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo

CMSC 313 Lecture 16 Announcement: no office hours today. Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo CMSC 33 Lecture 6 nnouncement: no office hours today. Good-bye ssembly Language Programming Overview of second half on Digital Logic DigSim Demo UMC, CMSC33, Richard Chang Good-bye ssembly

More information

Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring

Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring - Principles of Computer rchitecture Miles Murdocca and Vincent Heuring 999 M. Murdocca and V. Heuring -2 Chapter Contents. Introduction.2 Combinational Logic.3 Truth Tables.4 Logic Gates.5 Properties

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

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

CMSC 313 Lecture 19 Combinational Logic Components Programmable Logic Arrays Karnaugh Maps

CMSC 313 Lecture 19 Combinational Logic Components Programmable Logic Arrays Karnaugh Maps CMSC 33 Lecture 9 Combinational Logic Components Programmable Logic rrays Karnaugh Maps UMC, CMSC33, Richard Chang Last Time & efore Returned midterm exam Half adders & full adders Ripple

More information

Switches: basic element of physical implementations

Switches: basic element of physical implementations Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element

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

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

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

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

CMSC 313 Lecture 19 Homework 4 Questions Combinational Logic Components Programmable Logic Arrays Introduction to Circuit Simplification

CMSC 313 Lecture 19 Homework 4 Questions Combinational Logic Components Programmable Logic Arrays Introduction to Circuit Simplification CMSC 33 Lecture 9 Homework 4 Questions Combinational Logic Components Programmable Logic rrays Introduction to Circuit Simplification UMC, CMSC33, Richard Chang CMSC 33, Computer Organization

More information

CS 226: Digital Logic Design

CS 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 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

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

Boolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation.

Boolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation. Section 3 Boolean Algebra The Building Blocks of Digital Logic Design Section Overview Binary Operations (AND, OR, NOT), Basic laws, Proof by Perfect Induction, De Morgan s Theorem, Canonical and Standard

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 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

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

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

Digital Circuits. 1. Inputs & Outputs are quantized at two levels. 2. Binary arithmetic, only digits are 0 & 1. Position indicates power of 2.

Digital Circuits. 1. Inputs & Outputs are quantized at two levels. 2. Binary arithmetic, only digits are 0 & 1. Position indicates power of 2. Digital Circuits 1. Inputs & Outputs are quantized at two levels. 2. inary arithmetic, only digits are 0 & 1. Position indicates power of 2. 11001 = 2 4 + 2 3 + 0 + 0 +2 0 16 + 8 + 0 + 0 + 1 = 25 Digital

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

Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring

Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring - Principles of Computer rchitecture Miles Murdocca and Vincent Heuring 999 M. Murdocca and V. Heuring -2 Chapter Contents. Introduction.2 Combinational Logic.3 Truth Tables.4 Logic Gates.5 Properties

More information

Lecture 1. Notes. Notes. Notes. Introduction. Introduction digital logic February Bern University of Applied Sciences

Lecture 1. Notes. Notes. Notes. Introduction. Introduction digital logic February Bern University of Applied Sciences Output voltage Input voltage 3.3V Digital operation (Switch) Lecture digital logic February 26 ern University of pplied Sciences Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev.

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

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

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

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 CMSC 33 Lecture 8 Midterm Exam returned ssign Homework 3 Circuits for ddition Digital Logic Components Programmable Logic rrays UMC, CMSC33, Richard Chang Half dder Inputs: and Outputs:

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

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

Computer Organization: Boolean Logic

Computer Organization: Boolean Logic Computer Organization: Boolean Logic Representing and Manipulating Data Last Unit How to represent data as a sequence of bits How to interpret bit representations Use of levels of abstraction in representing

More information

Chapter 2: Boolean Algebra and Logic Gates

Chapter 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 information

COSC3330 Computer Architecture Lecture 2. Combinational Logic

COSC3330 Computer Architecture Lecture 2. Combinational Logic COSC333 Computer rchitecture Lecture 2. Combinational Logic Instructor: Weidong Shi (Larry), PhD Computer Science Department University of Houston Today Combinational Logic oolean lgebra Mux, DeMux, Decoder

More information

Part 5: Digital Circuits

Part 5: Digital Circuits Characteristics of any number system are: Part 5: Digital Circuits 5.: Number Systems & Code Conversions. ase or radix is equal to the number of possible symbols in the system 2. The largest value of digit

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

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

LOGIC GATES A Y=A+B. Logic symbol of OR gate B The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.

LOGIC GATES A Y=A+B. Logic symbol of OR gate B The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B. LOGIC GTS J-Physics INTRODUCTION : logic gate is a digital circuit which is based on certain logical relationship between the input and the output voltages of the circuit. The logic gates are built using

More information

Intro To Digital Logic

Intro To Digital Logic Intro To Digital Logic 1 Announcements... Project 2.2 out But delayed till after the midterm Midterm in a week Covers up to last lecture + next week's homework & lab Nick goes "H-Bomb of Justice" About

More information

Prove that if not fat and not triangle necessarily means not green then green must be fat or triangle (or both).

Prove that if not fat and not triangle necessarily means not green then green must be fat or triangle (or both). hapter : oolean lgebra.) Definition of oolean lgebra The oolean algebra is named after George ool who developed this algebra (854) in order to analyze logical problems. n example to such problem is: Prove

More information

Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits

Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University See: P&H ppendix C.2 and C.3 (lso, see C.0 and

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

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

EECS Variable Logic Functions

EECS Variable Logic Functions EECS150 Section 1 Introduction to Combinational Logic Fall 2001 2-Variable Logic Functions There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs X

More information

EECS150 - 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) EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am Outline Review of

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

Lecture 9: Digital Electronics

Lecture 9: Digital Electronics Introduction: We can classify the building blocks of a circuit or system as being either analog or digital in nature. If we focus on voltage as the circuit parameter of interest: nalog: The voltage can

More information

Outline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined

Outline. 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 information

Digital Circuit And Logic Design I. Lecture 3

Digital 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. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong

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

Unit 2 Boolean Algebra

Unit 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 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

Circuits & Boolean algebra.

Circuits & 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 information

Appendix A: Digital Logic. CPSC 352- Computer Organization

Appendix A: Digital Logic. CPSC 352- Computer Organization - CPSC 352- Computer Organization -2 Chapter Contents. Introduction.2 Combinational Logic.3 Truth Tables.4 Logic Gates.5 Properties of oolean lgebra.6 The Sum-of-Products Form, and Logic Diagrams.7 The

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

L2: Combinational Logic Design (Construction and Boolean Algebra)

L2: Combinational Logic Design (Construction and Boolean Algebra) L2: Combinational Logic Design (Construction and oolean lgebra) cknowledgements: Materials in this lecture are courtesy of the following people and used with permission. - Randy H. Katz (University of

More information

Boole Algebra and Logic Series

Boole Algebra and Logic Series S1 Teknik Telekomunikasi Fakultas Teknik Elektro oole lgebra and Logic Series 2016/2017 CLO1-Week2-asic Logic Operation and Logic Gate Outline Understand the basic theory of oolean Understand the basic

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

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

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

Gates. Quiz 1 will cover up to and including this lecture. The book says something about NAND... maybe an in-law. Is he talking about BILL??? 6.

Gates. Quiz 1 will cover up to and including this lecture. The book says something about NAND... maybe an in-law. Is he talking about BILL??? 6. Gates Is he talking about ILL??? The book says something about NND... maybe an in-law. WRD & HLSTED 6.4 NERD KIT Quiz will cover up to and including this lecture 6.4 - Fall 22 9/7/2 L4 - Gates Quick Review

More information

Learning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr.

Learning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr. /7/ CE 4 Digital ystem Design Dr. Arshad Aziz Fundamental of ogic Design earning Objectives Review the basic concepts of logic circuits Variables and functions Boolean algebra Minterms and materms ogic

More information

Lecture 2. Notes. Notes. Notes. Boolean algebra and optimizing logic functions. BTF Electronics Fundamentals August 2014

Lecture 2. Notes. Notes. Notes. Boolean algebra and optimizing logic functions. BTF Electronics Fundamentals August 2014 Lecture 2 Electronics ndreas Electronics oolean algebra and optimizing logic functions TF322 - Electronics Fundamentals ugust 24 Exercise ndreas ern University of pplied Sciences Rev. 946f32 2. of oolean

More information

Electronics. Overview. Introducction to Synthetic Biology

Electronics. Overview. Introducction to Synthetic Biology Electronics Introducction to Synthetic iology E Navarro Montagud P Fernandez de Cordoba JF Urchueguía Overview Introduction oolean algebras Logical gates Representation of boolean functions Karnaugh maps

More information

L2: Combinational Logic Design (Construction and Boolean Algebra)

L2: Combinational Logic Design (Construction and Boolean Algebra) L2: Combinational Logic Design (Construction and oolean lgebra) cknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. orriello, Contemporary Logic Design (second edition), Pearson Education,

More information

ELEC Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2)

ELEC Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2) ELEC 2200-002 Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn,

More information

2. Associative Law: A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C S.

2. Associative Law: A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C S. BOOLEAN ALGEBRA 2.1 Introduction Binary logic deals with variables that have two discrete values: 1 for TRUE and 0 for FALSE. A simple switching circuit containing active elements such as a diode and transistor

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

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

Chapter 2 Boolean Algebra and Logic Gates

Chapter 2 Boolean Algebra and Logic Gates Ch1: Digital Systems and Binary Numbers Ch2: Ch3: Gate-Level Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan

More information

BOOLEAN ALGEBRA INTRODUCTION SUBSETS

BOOLEAN ALGEBRA INTRODUCTION SUBSETS BOOLEAN ALGEBRA M. Ragheb 1/294/2018 INTRODUCTION Modern algebra is centered around the concept of an algebraic system: A, consisting of a set of elements: ai, i=1, 2,, which are combined by a set of operations

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

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

Digital electronic systems are designed to process voltage signals which change quickly between two levels. Low time.

Digital electronic systems are designed to process voltage signals which change quickly between two levels. Low time. DIGITL ELECTRONIC SYSTEMS Digital electronic systems are designed to process voltage signals which change quickly between two levels. High Voltage Low time Fig. 1 digital signal LOGIC GTES The TTL digital

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

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

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.

Boolean 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 information

Chapter 2 Part 7 Combinational Logic Circuits

Chapter 2 Part 7 Combinational Logic Circuits 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

More information

Standard Expression Forms

Standard 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 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

4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra

4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra 4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra To design a digital circuit that will perform a required function, it is necessary to manipulate and combine the various input signals in certain

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

3.4 Set Operations Given a set A, the complement (in the Universal set U) A c is the set of all elements of U that are not in A. So A c = {x x / A}.

3.4 Set Operations Given a set A, the complement (in the Universal set U) A c is the set of all elements of U that are not in A. So A c = {x x / A}. 3.4 Set Operations Given a set, the complement (in the niversal set ) c is the set of all elements of that are not in. So c = {x x /. (This type of picture is called a Venn diagram.) Example 39 Let = {1,

More information

CprE 281: Digital Logic

CprE 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 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

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. Overview Part 1 Gate

More information

Unit 8A Computer Organization. Boolean Logic and Gates

Unit 8A Computer Organization. Boolean Logic and Gates Unit 8A Computer Organization Boolean Logic and Gates Announcements Bring ear buds or headphones to lab! 15110 Principles of Computing, Carnegie Mellon University - CORTINA 2 Representing and Manipulating

More information

UC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits

UC 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 (2003-11-15) Revised

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

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

CS61c: Representations of Combinational Logic Circuits

CS61c: Representations of Combinational Logic Circuits CS61c: Representations of Combinational Logic Circuits J. Wawrzynek March 5, 2003 1 Introduction Recall that synchronous systems are composed of two basic types of circuits, combination logic circuits,

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

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture A Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Laws

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

ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2

ECE 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: 858-822-4884 office, 858-353-0550 cell Office: 3802

More information

Algebraic Methods for the Analysis and Synthesis

Algebraic Methods for the Analysis and Synthesis lgebraic ethods for the nalysis and Synthesis Fundaentals of oolean lgebra asic Postulates. oolean algebra is closed algebraic syste containing a set K of two or ore eleents and two operators and, ND and

More information

Floating Point Representation and Digital Logic. Lecture 11 CS301

Floating Point Representation and Digital Logic. Lecture 11 CS301 Floating Point Representation and Digital Logic Lecture 11 CS301 Administrative Daily Review of today s lecture w Due tomorrow (10/4) at 8am Lab #3 due Friday (9/7) 1:29pm HW #5 assigned w Due Monday 10/8

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