Boolean Algebra & Logic Gates. By : Ali Mustafa

Size: px
Start display at page:

Download "Boolean Algebra & Logic Gates. By : Ali Mustafa"

Transcription

1 Boolean Algebra & Logic Gates By : Ali Mustafa

2 Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions are named: AND OR NOT (INVERTER)

3 AND Gate Represented by any of the following notations: X AND Y X. Y X Y Function definition: Z = 1 only if X=Y=1 0 otherwise

4 OR Gate Represented by any of the following notations: X OR Y X + Y Function definition: Z = 1 if X=1 or Y =1 or both X=Y=1 0 if X=Y=0

5 NOT (Inverter) Gate Represented by a bar over the variable Function definition: It is also called complement operation, as it changes 1 s to 0 s and 0 s to 1 s.

6 Logic gates timing Diagram Timing diagrams illustrate the response of any gate to all possible input signal combinations. The horizontal axis of the timing diagram represents time and the vertical axis represents the signal as it changes between the two possible voltage levels 1 or 0

7 Other Gates

8 Others Gates

9 Digital Logic How to describe Digital logic system? We have two methods: Truth Table Boolean Expression

10 TRUTH TABLE A Truth Table is a table of combinations of the binary variables showing the relationship between the different values that the input variables take and the result of the operation (output). The number of rows in the Truth Table is where n = number of input variables in the function. The binary combinations are obtained from the binary number by counting from 0 to Example: AND gate with 2 inputs n=2 The truth table has 2 rows = 4 The binary combinations is from 0 to (22-1=(3)) [00,01,10,11]

11 BOOLEAN EXPRESSIONS We can use these basic operations to form more complex expressions: Some terminology and notation: f is the name of the function. (x,y,z) are the input variables, each representing 1 or 0. Listing the inputs is optional, but sometimes helpful. A literalis any occurrence of an input variable or its complement. The function above has four literals: x, y, z, and x.

12 How to get the Boolean Expression from the truth table?

13 Boolean Expressions From Truth Tables Each 1 in the output of a truth table specifies one term in the corresponding boolean expression.

14 Example Find boolean expression?

15 Example Solution

16 Basic Logic gates We have defined three basic logic gates and operators Also, we could build any digital circuit from those basic logic gates. In digital Logic, we are not using normal mathematics we are using Boolean algebra So, we need to know the laws & rules of Boolean Algebra

17 Boolean Algebra What s the difference between the Boolean Algebra and arithmetic algebra? The First obvious difference that in Boolean algebra we have only (+) and (.) operators we don t have subtraction(-) or division(/) like math.

18 Binary Logic You should distinguish between binary logic and binary arithmetic. Arithmetic variables are numbers that consist of many digits. Arithmetic A binary logic variable is always either 1 or 0. Binary

19 Laws & Rules of Boolean Algebra The basic laws of Boolean algebra: The commutative law The associative law The distributive law

20 Commutative Law The commutative law of addition for two variables is written as: A+B = B+A The commutative law of multiplication for two variables is written as: AB = BA

21 Associative Law The associative law of addition for 3 variables is written as: A+(B+C) = (A+B)+C The associative law of multiplication for 3 variables is written as: A(BC) = (AB)C

22 Distributive Law The distributive law for multiplication as follows: A(B+C) = AB + AC The distributive law for addition is as follows A+(B.C) = (A+B)(A+C)

23 Basic Theorems of Boolean Algebra

24 Basic Theorems of Boolean Algebra OR Laws AND Laws Inversion Law

25 Theorem 1(a) Proof X + X = X Solution X + X = (X + X ). 1 X.1=1 =(X + X)(X + X ) X + X =1 =X + XX Dist X + YZ= (X+Y)(X+Z) =X + 0 X.X =0 =X

26 Theorem 1(b) Prove X.X = X Solution: X.X = XX + 0 X + 0 =X =XX + XX XX = 0 =X(X + X ) =X(1) X + X =1 =X

27 Self Tasks (Theorems) X + 1 = 1 X.0 = 0 X + XY= X

28 Duality Principle A Boolean equation remains valid if we take the dual of the expressions on both sides of the equals sign Dual of expression it means, Interchange 1 s with 0 s (and Vice-versa) Interchange AND (.) with OR (+) (and Vice-versa)

29 DeMorgan s Law Theorem 1: NAND = Bubbled OR Complement of product is equal to addition of the compliments. Theorem 2: NOR = Bubbled AND is equal to product of the compliments. Complement of sum

30 Truth Tables for DeMorgan s

31 Example Get the logic function from the following truth table and implement it using basic logic gates (AND, OR, NOT)

32 Simplification of the logic function A B + A B + AB Solution: A (B + B) + AB A (1) + AB (A + A)(A + B ) Hint A+AB Dist 1 (A + B ) (AB) DeMorgan Law(1 NAND Gate only) From 7 gates using simplification rule can be optimized to 1 gate

33 Home Task: Simplify the following Expressions Find the dual of the following expressions 1. A + AB = A 2. A + A B = A + B 3. A + A = 1 4. (A + B)(A + C) = A + BC

34 Home Task: Simplify the following Expressions Prove the following binary expressions ( Using Postulates) 1. A + AB = A 2. (A + B)(A + C) = A + BC 3. (A + B + AB)(A + B )(A B) = 0 4. AB + ABC + AB = A 5. AB + A B + A B

35 Fewer Gates F XY XZ

36 Algebraic Manipulation F XYZ XYZ XZ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ECE 238L Boolean Algebra - Part I

ECE 238L Boolean Algebra - Part I ECE 238L Boolean Algebra - Part I August 29, 2008 Typeset by FoilTEX Understand basic Boolean Algebra Boolean Algebra Objectives Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand

More 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

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

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

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

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

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

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

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

Combinational Logic. By : Ali Mustafa

Combinational Logic. By : Ali Mustafa Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

CHAPTER 3 BOOLEAN ALGEBRA

CHAPTER 3 BOOLEAN ALGEBRA CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 Exclusive-OR and Equivalence Operations 3.3 The Consensus

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

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

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

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

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

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

1 Boolean Algebra Simplification

1 Boolean Algebra Simplification cs281: Computer Organization Lab3 Prelab Our objective in this prelab is to lay the groundwork for simplifying boolean expressions in order to minimize the complexity of the resultant digital logic circuit.

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

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

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

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

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

If f = ABC + ABC + A B C then f = AB C + A BC + AB C + A BC + A B C

If f = ABC + ABC + A B C then f = AB C + A BC + AB C + A BC + A B C Examples: If f 5 = AB + AB then f 5 = A B + A B = f 10 If f = ABC + ABC + A B C then f = AB C + A BC + AB C + A BC + A B C In terms of a truth table, if f is the sum (OR) of all the minterms with a 1 in

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

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

Prof.Manoj Kavedia 2 Algebra

Prof.Manoj Kavedia 2 Algebra ` Logic Gates and Boolean 2 Algebra Chapter-2 (Hours:06 Marks:14 )( 12064 Digital Techniques) Logic Gates And Boolean Algebra 2.1 Logical symbol, logical expression and truth table of AND, OR, NOT, NAND,

More information

Chapter 3. Boolean Algebra. (continued)

Chapter 3. Boolean Algebra. (continued) Chapter 3. Boolean Algebra (continued) Algebraic structure consisting of: set of elements B binary operations {+, -} unary operation {'} Boolean Algebra such that the following axioms hold:. B contains

More 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

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

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

Digital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit

Digital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit Digital Techniques 1. Binary System The digital computer is the best example of a digital system. A main characteristic of digital system is its ability to manipulate discrete elements of information.

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

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

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

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

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

Review for Test 1 : Ch1 5

Review for Test 1 : Ch1 5 Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8

More 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

BOOLEAN LOGIC. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift

BOOLEAN LOGIC. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift BOOLEAN LOGIC Based on CBSE curriculum Class 11 By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region Neha Tyagi, KV 5 Jaipur II Shift Introduction Boolean Logic, also known as boolean algebra was

More 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

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

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

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

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

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

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

Chapter 7 Logic Circuits

Chapter 7 Logic Circuits Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary

More information

University of Technology

University of Technology University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year معالجات دقيقة المرحلة الرابعة ھندسة الليزر والبصريات االلكترونية Lecture 3 & 4 Boolean Algebra and Logic Gates

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

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.)

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.) DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.) 1 Learning Objectives 1. Apply the laws and theorems of Boolean algebra to to the manipulation of algebraic expressions to simplifying an expression, finding

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

Binary logic consists of binary variables and logical operations. The variables are

Binary logic consists of binary variables and logical operations. The variables are 1) Define binary logic? Binary logic consists of binary variables and logical operations. The variables are designated by the alphabets such as A, B, C, x, y, z, etc., with each variable having only two

More information

EE40 Lec 15. Logic Synthesis and Sequential Logic Circuits

EE40 Lec 15. Logic Synthesis and Sequential Logic Circuits EE40 Lec 15 Logic Synthesis and Sequential Logic Circuits Prof. Nathan Cheung 10/20/2009 Reading: Hambley Chapters 7.4-7.6 Karnaugh Maps: Read following before reading textbook http://www.facstaff.bucknell.edu/mastascu/elessonshtml/logic/logic3.html

More information

Lecture 3: Boolean Algebra

Lecture 3: Boolean Algebra Lecture 3: 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 Overview

More information

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION

More 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

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

CSC9R6 Computer Design. Practical Digital Logic

CSC9R6 Computer Design. Practical Digital Logic CSC9R6 Computer Design Practical Digital Logic 1 References (for this part of CSC9R6) Hamacher et al: Computer Organization App A. In library Floyd: Digital Fundamentals Ch 1, 3-6, 8-10 web page: www.prenhall.com/floyd/

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

LOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D.

LOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D. Basic Eperiment and Design of Electronics LOGIC GATES Ho Kyung Kim, Ph.D. hokyung@pusan.ac.kr School of Mechanical Engineering Pusan National University Outline Boolean algebra Logic gates Karnaugh maps

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

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

UNIT 3 BOOLEAN ALGEBRA (CONT D)

UNIT 3 BOOLEAN ALGEBRA (CONT D) UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic

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

7.1. Unit 7. Minterm and Canonical Sums 2- and 3-Variable Boolean Algebra Theorems DeMorgan's Theorem Simplification using Boolean Algebra

7.1. Unit 7. Minterm and Canonical Sums 2- and 3-Variable Boolean Algebra Theorems DeMorgan's Theorem Simplification using Boolean Algebra 7.1 Unit 7 Minterm and Canonical Sums 2- and 3-Variable Boolean Algebra Theorems DeMorgan's Theorem Simplification using Boolean Algebra CHECKERS / DECODERS 7.2 7.3 Gates Gates can have more than 2 inputs

More information