Digital Circuit And Logic Design I. Lecture 3

Size: px
Start display at page:

Download "Digital Circuit And Logic Design I. Lecture 3"

Transcription

1 Digital Circuit And Logic Design I Lecture 3

2 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong Sornkhom, 25/2 2

3 Combinational Logic Design Principles ()

4 . Introduction Logic circuits are classified into two types Combinational circuits outputs depend only on its current inputs Sequential circuits outputs depend not only on its current inputs but also on its current state. Combinational-circuit analysis Logic diagram formal description (truth table or logic expression) Combinational-circuit synthesis Formal description logic diagram Panupong Sornkhom, 25/2 4

5 . Introduction (cont.) Combinational-circuit design Requirements Informal description Informal description formal description Formal description logic diagram Panupong Sornkhom, 25/2 5

6 2. Switching algebra In 854, George Boole invented a two-value algebraic system, now called Boolean algebra In 938, Claude E. Shannon showed how to adapt Boolean algebra to analyze and describe the behavior of circuits built from relays Axioms (A) X = if X (A ) X = if X (A2) If X =, then X = (A2 ) If X =, then X = (A3) = (A3 ) + = (A4) = (A4 ) + = (A5) = = (A5 ) + = + = Panupong Sornkhom, 25/2 6

7 2. Switching algebra (cont.) Single-Variable Theorems (T) X+ = X (T ) X = X (Identities) (T2) X+ = (T2 ) X = (Null elements) (T3) X+X = X (T3 ) X X = X (Idempotency) (T4) (X ) = X (Involution) (T5) X+X = (T5 ) X X = (Complements) Two- and Three-Variable Theorems (T6) X+Y = Y+X (T6 ) X Y = Y X (Commutativity) (T7) (X+Y)+Z = X+(Y+Z) (T7 ) (X Y) Z = X (Y Z) (Associativity) (T8) X Y+X Z = X (Y+Z) (T8 ) (X+Y) (X+Z) = X+Y Z (Distributivity) Panupong Sornkhom, 25/2 7

8 2. Switching algebra (cont.) Two- and Three-Variable Theorems (cont.) (T9) X+X Y = X (T9 ) X (X+Y) = X (Covering) (T) X Y+ X Y = X (T ) (X+Y) (X+Y ) = X (Combining) (T) X Y+X Z+Y Z = X Y+X Z (T ) (X+Y) (X +Z) (Y+Z)= (X+Y) (X +Z) (Consensus) n-variable Theorems (T2) X+X+ +X = X (T2 ) X X X = X (Generalized idempotency) (T3) (X X 2 X n ) = X +X 2 + +X n (T3 ) (X +X 2 + +X n ) = X X 2 X n (DeMorgan s theorems) Panupong Sornkhom, 25/2 8

9 2. Switching algebra (cont.) Principle of Duality If a Boolean statement is proved true, the dual of statement is also true. The dual of expression is obtained by replacing each + in the expression by and vice versa, and replacing by and vice versa. Notice that we must preserving the existence of all parentheses, whether present or implied For example, given X Y+X Z = X (Y+Z) its dual is (X+Y) (X+Z) = X +(Y Z) Panupong Sornkhom, 25/2 9

10 2. Switching algebra (cont.) Standard representations of logic functions Truth table An algebraic sum of minterms, the canonical sum A minterm list using the Σ notation An algebraic product of maxterms, the canonical product A maxterm list using the Π notation Panupong Sornkhom, 25/2

11 Panupong Sornkhom, 25/2 2. Switching algebra (cont.) Standard representations of logic functions (cont.) F Z Y X Row Truth table for a particular 3-variable logic function, F(X,Y,Z)

12 2. Switching algebra (cont.) Standard representations of logic functions (cont.) A literal is a variable or the complement of a variable. A product term is a single literal or a logical product of two or more literals. A sum-of-products expression is a logical sum of product terms A sum term is a single literal or a logical sum of two or more literals A product-of-sums expression is a logical product of sum terms A normal term is a product or sum term in which no variable appears more than once. An n-variable minterm is a normal product term with n literals. An n-variable maxterm is a normal sum term with n literals Panupong Sornkhom, 25/2 2

13 2. Switching algebra (cont.) Standard representations of logic functions (cont.) There is a close correspondence between the truth table and minterms and maxterms. A minterm can be defined as a product term that is in exactly one row of the truth table. A maxterm can be defined as a sum term that is in exactly one row of the truth table. Picture from Textbook DDPP Panupong Sornkhom, 25/2 3

14 2. Switching algebra (cont.) Standard representations of logic functions (cont.) Based on the correspondence between the truth table and minterms, we can create an algebraic representation of a logic function from its truth table The canonical sum of a logic function is a sum of the minterms corresponding to truth-table rows (input combinations) for which the function produce a output. The cannonical product of a logic function is a product of the maxterms corresponding to input combinations for which the function produces a output. Panupong Sornkhom, 25/2 4

15 2. Switching algebra (cont.) Standard representations of logic functions (cont.) F=Σ X,Y,Z (,3,4,6,7) =X Y Z + X Y Z+ X Y Z + X Y Z +X Y Z Panupong Sornkhom, 25/2 5

16 2. Switching algebra (cont.) Standard representations of logic functions (cont.) F=Π X,Y,Z (,2,5) =(X+Y+Z ) (X+Y +Z) (X +Y+Z ) Panupong Sornkhom, 25/2 6

17 2. Switching algebra (cont.) Standard representations of logic functions (cont.) Therefore, Σ X,Y,Z (,3,4,6,7) = Π X,Y,Z (,2,5) In fact, for any logic function F, minterm list is complement of maxterm list and vice versa Panupong Sornkhom, 25/2 7

18 Panupong Sornkhom, 25/ Combinational-Circuit Analysis F Z Y X Row Pictures from text book DDPP Exhaustive approach

19 3. Combinational-Circuit Analysis (cont.) Algebraic approach F = ((X+Y ) Z)+(X Y Z ) Pictures from text book DDPP Panupong Sornkhom, 25/2 9

20 3. Combinational-Circuit Analysis (cont.) From F = ((X+Y ) Z)+(X Y Z ) We can use theorems to transform this expression into another form F = X Z+Y Z+X Y Z (sum of product form) Or F = ((X+Y ) Z)+(X Y Z ) = ((X+Y )+X Y Z ) (Z+X Y Z ) = ((X+Y )+X ) ((X+Y )+Y) ((X+Y )+Z ) (Z+X ) (Z+Y) (Z+Z ) = (X+Y +Z ) (X +Z) (Y+Z) = (X+Y +Z ) (X +Z) (Y+Z) (product of sum form) Panupong Sornkhom, 25/2 2

21 4. Combinational-Circuit Synthesis Circuit description and designs Given a circuit description, we convert it to formal description From formal description, we can derive it to logic diagram For example The description of a 4-bit prime-number detector might be, Given a 4-bit input combination N = N 3 N 2 N N, this function produces a output for N =, 2, 3, 5, 7,, 3, and otherwise. A formal description (logic function) can be designed directly from the canonical sum of product expression F = Σ N3N2NN (, 2, 3, 5, 7,, 3) = N 3 N 2 N N + N 3 N 2 N N + N 3 N 2 N N + N 3 N 2 N N + N 3 N 2 N N + N 3 N 2 N N + N 3 N 2 N N Panupong Sornkhom, 25/2 2

22 4. Combinational-Circuit Synthesis (cont.) Canonical-sum design for 4-bit prime-number detector Picture from text book DDPP Panupong Sornkhom, 25/2 22

23 4. Combinational-Circuit Synthesis (cont.) Circuit manipulations Sometimes we would like to use NAND and NOR gates instead of use AND, OR, and NOT gates Because NANDs and NORs are faster than ANDs and ORs in most technologies We can translate any logic expression into an equivalent sum-ofproducts expression, such an expression may be realized directly with two-level AND-OR circuit or may be converted to two-level NAND-NAND circuit Similarly, we can translate any logic expression into an equivalent product-of-sums expression, such an expression may be realized directly with two-level OR-AND circuit or may be converted to twolevel NOR-NOR circuit Panupong Sornkhom, 25/2 23

24 4. Combinational-Circuit Synthesis (cont.) Pictures from text book DDPP Panupong Sornkhom, 25/2 24

25 4. Combinational-Circuit Synthesis (cont.) Pictures from text book DDPP Panupong Sornkhom, 25/2 25

26 4. Combinational-Circuit Synthesis (cont.) Pictures from text book DDPP Panupong Sornkhom, 25/2 26

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

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

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

CS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman

CS 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 information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 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 information

Digital Circuit And Logic Design I. Lecture 4

Digital Circuit And Logic Design I. Lecture 4 Digital Circuit And Logic Design I Lecture 4 Outline Combinational Logic Design Principles (2) 1. Combinational-circuit minimization 2. Karnaugh maps 3. Quine-McCluskey procedure Panupong Sornkhom, 2005/2

More information

Chapter 2. Boolean Algebra and Logic Gates

Chapter 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 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

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

Boolean Algebra and Logic Gates

Boolean 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 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

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

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

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

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

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

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

Functions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:

Functions. 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 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

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

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

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

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

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

ECEN 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 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 information

Week-I. Combinational Logic & Circuits

Week-I. Combinational Logic & Circuits Week-I 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 information

Digital Logic Design. Combinational Logic

Digital 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 information

Contents. Chapter 2 Digital Circuits Page 1 of 30

Contents. Chapter 2 Digital Circuits Page 1 of 30 Chapter 2 Digital Circuits Page 1 of 30 Contents Contents... 1 2 Digital Circuits... 2 2.1 Binary Numbers... 2 2.2 Binary Switch... 4 2.3 Basic Logic Operators and Logic Expressions... 5 2.4 Truth Tables...

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

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

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

control in out in out Figure 1. Binary switch: (a) opened or off; (b) closed or on.

control in out in out Figure 1. Binary switch: (a) opened or off; (b) closed or on. Chapter 2 Digital Circuits Page 1 of 18 2. Digital Circuits Our world is an analog world. Measurements that we make of the physical objects around us are never in discrete units but rather in a continuous

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

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

Unit 2 Boolean Algebra

Unit 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 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

Computer Organization I

Computer 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 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

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

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

Combinational Logic. Review of Combinational Logic 1

Combinational 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 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

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

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

Chapter 2 : Boolean Algebra and Logic Gates

Chapter 2 : Boolean Algebra and Logic Gates Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 1431-1432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean

More information

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC. Boolean Expression Forms: Sum-of-products (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 information

Administrative Notes. Chapter 2 <9>

Administrative 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 information

CHAPTER 12 Boolean Algebra

CHAPTER 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 information

Chapter-2 BOOLEAN ALGEBRA

Chapter-2 BOOLEAN ALGEBRA Chapter-2 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 information

Binary Logic and Gates

Binary 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 information

Boolean Algebra CHAPTER 15

Boolean Algebra CHAPTER 15 CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an

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

Slide 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 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 information

Boolean 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. 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 information

Fundamentals of Computer Systems

Fundamentals of Computer Systems Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Fall 2011 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables x, y,... represent

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: 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

Binary Logic and Gates. Our objective is to learn how to design digital circuits.

Binary 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 information

Midterm1 Review. Jan 24 Armita

Midterm1 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 information

Chapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms

Chapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms Chapter 2 (Lect 2) Canonical and Standard Forms Sum of Minterms Product of Maxterms Standard Form Sum of products Product of sums Other Logic Operators Logic Gates Basic and Multiple Inputs Positive and

More information

Digital Systems and Information Part II

Digital Systems and Information Part II Digital Systems and Information Part II Overview Arithmetic Operations General Remarks Unsigned and Signed Binary Operations Number representation using Decimal Codes BCD code and Seven-Segment Code Text

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

Chapter 2 Boolean Algebra and Logic Gates

Chapter 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 2-1 Basic Definitions 2-2 Axiomatic Definition of Boolean Algebra 2-3

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

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

EEA051 - 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 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 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

Slides for Lecture 10

Slides for Lecture 10 Slides for Lecture 10 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 30 September, 2013 ENEL 353

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

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

ECE380 Digital Logic. Axioms of Boolean algebra

ECE380 Digital Logic. Axioms of Boolean algebra ECE380 Digital Logic Introduction to Logic Circuits: Boolean algebra Dr. D. J. Jackson Lecture 3-1 Axioms of Boolean algebra Boolean algebra: based on a set of rules derived from a small number of basic

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

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

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

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

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

Digital Logic Design. Malik Najmus Siraj

Digital Logic Design. Malik Najmus Siraj Digital Logic Design Malik Najmus Siraj siraj@case.edu.pkedu LECTURE 4 Today s Agenda Recap 2 s complement Binary Logic Boolean algebra Recap Computer Arithmetic Signed numbers Radix and diminished radix

More information

Fundamentals of Computer Systems

Fundamentals of Computer Systems Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Summer 2015 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables X,,... represent

More information

1. Name the person who developed Boolean algebra

1. Name the person who developed Boolean algebra MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA

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

Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps

Lecture 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 information

14:332:231 DIGITAL LOGIC DESIGN. Combinational Circuit Synthesis

14:332:231 DIGITAL LOGIC DESIGN. Combinational Circuit Synthesis :: DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering all Lecture #: Combinational Circuit Synthesis I Combinational Circuit Synthesis Recall: Combinational circuit

More information

Boolean Algebra, Gates and Circuits

Boolean 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 information

Standard & Canonical Forms

Standard & Canonical Forms 1 COE 202- Digital Logic Standard & Canonical Forms Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Minterms and Maxterms From truth table to Boolean expression Sum of minterms Product of Maxterms

More information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 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 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

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

INTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC-110-T

INTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC-110-T INTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC-110-T Credit : (2 + 1) / Week TEXT AND REF. BOOKS Text Book: Peter Norton (2011), Introduction to Computers, 7 /e, McGraw-Hill

More information

3 Boolean Algebra 3.1 BOOLEAN ALGEBRA

3 Boolean Algebra 3.1 BOOLEAN ALGEBRA 3 Boolean Algebra 3.1 BOOLEAN ALGEBRA In 1854, George Boole introduced the following formalism which eventually became Boolean Algebra. Definition. An algebraic system consisting of a set B of elements

More information

CSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication

CSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm

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

Spiral 1 / Unit 3

Spiral 1 / Unit 3 -3. Spiral / Unit 3 Minterm and Maxterms Canonical Sums and Products 2- and 3-Variable Boolean Algebra Theorems DeMorgan's Theorem Function Synthesis use Canonical Sums/Products -3.2 Outcomes I know the

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

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

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

BOOLEAN 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 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 information

Every time has a value associated with it, not just some times. A variable can take on any value within a range

Every 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 information

CPE100: Digital Logic Design I

CPE100: Digital Logic Design I Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter

More information

COE 202: Digital Logic Design Combinational Logic Part 2. Dr. Ahmad Almulhem ahmadsm AT kfupm Phone: Office:

COE 202: Digital Logic Design Combinational Logic Part 2. Dr. Ahmad Almulhem   ahmadsm AT kfupm Phone: Office: COE 202: Digital Logic Design Combinational Logic Part 2 Dr. Ahmad Almulhem Email: ahmadsm AT kfupm Phone: 860-7554 Office: 22-324 Objectives Minterms and Maxterms From truth table to Boolean expression

More information

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

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 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 information