Lecture (04) Boolean Algebra and Logic Gates


 Silvester Willis
 3 years ago
 Views:
Transcription
1 Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Boolean algebra properties basic assumptions and properties: Closure law A set S is closed with respect to a binary operator, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S. ٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
2 Commutative law. A binary operator * on a set S is said to be commutative whenever ٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Associative law. A binary operator * on a set S is said to be associative whenever ٤ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
3 ٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Identity element. A set S is said to have an identity element with respect to a binary operation * on S if there exists an element e S with the property that Example: The element is an identity element with respect to the binary operator + on the set of integers I = {, 3, 2,,,, 2, 3, }, since x + = + x = x for any x I Example: The element is an identity element with respect to the binary operator. on the set of integers I = {, 3, 2,,,, 2, 3, }, since x. =. x = x for any x I
4 ٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Inverse. A set S having the identity element e with respect to a binary operator * is said to have an inverse whenever, for every x S, there exists an element y S such that Example: In the set of integers, I, and the operator +, with e =, the inverse of an element a is ( a), since a + ( a) =. ٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
5 ٩ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Distributive law If * and. are two binary operators on a set S, * is said to be distributive over. Whenever ١٠ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
6 ١١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Summery The binary operator + defines addition (OR). The additive identity is. The additive inverse defines subtraction. The binary operator. defines multiplication (AND). The multiplicative identity is. For a, the multiplicative inverse of a = /a defines division (i.e., a. /a = ). The only distributive law applicable is that of. over +: a. (b + c) = (a. b) + (a.. c) ١٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
7 Ordinary and Boolean Algebra George Boole developed an algebraic system now called Boolean algebra. Claude E. Shannon introduced a two valued Boolean algebra called switching algebra that represented the properties of bistable electrical switching circuits. ١٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design. (a) The structure is closed with respect to the operator +. (b) The structure is closed with respect to the operator.. 2. (a) The element is an identity element with respect to +; that is, x + = + x = x. (b) The element is an identity element with respect to. ; that is, x. =. x = x. 3. (a) The structure is commutative with respect to +; that is, x + y = y + x. (b) The structure is commutative with respect to. ; that is, x. y = y. x. ١٤ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
8 4. (a) The operator. is distributive over +; that is, x. (y + z) = (x. y) + (x. z). (b) The operator + is distributive over. ; that is, x + (y. z) = (x + y). (x + z). 5. For every element x B, there exists an element x B (called the complement of x) such that (a) x + x = and (b) x. x =, 6. There exist at least two elements x, y B such that x y. ١٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Comparing Boolean algebra with arithmetic and ordinary algebra: The distributive law of + over. (i.e., x + (y. z) = (x + y). (x + z) ) is valid for Boolean algebra, but not for ordinary algebra. Boolean algebra does not have additive or multiplicative inverses; therefore, there are no subtraction or division operations. defines an operator called the complement that is not available in ordinary algebra.
9 Ordinary algebra deals with the real numbers, which constitute an infinite set of elements. Boolean algebra deals with the as yet undefined set of elements, B, but in the two valued Boolean algebra defined next (and of interest in our subsequent use of that algebra), B is defined as a set with only two elements, and. ١٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Two Valued Boolean Algebra defined on a set of two elements, B = {, }, with rules for the two binary operators + and. Truth table rules are exactly the same as the AND, OR, and NOT operations ١٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
10 the structure is closed with respect to the two operators, each operation is either or and, B. From the tables, we see that (a) + = + = + = ; (b). =. =. =. This establishes the two identity elements, for + and for., The commutative laws are obvious from the symmetry of the binary operator table (A+B = B+A) & (A.B = B.A) ١٩ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design The distributive law x. (y + z) = (x. y) + (x. z) can be shown to hold from the operator tables The distributive law of + over. can be shown to hold by means of a truth table ٢٠ Truth table Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
11 the complement table (a) x + x =, since + = + = and + = + =. (b) x. x =, since. =. = and. =. =. ٢١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Postulates and Basic theorem of Boolean algebra Postulates and Theorems of Boolean Algebra ٢٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
12 Duality principle postulates were listed in pairs and designated by part (a) and part (b). One part may be obtained from the other if the binary operators and the identity elements are interchanged we simply interchange OR and AND operators and replace s by s and s by s. ٢٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Theorem 5.a Truth table ٢٤ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
13 Truth table ٢٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Operator Precedence () parentheses, (2) NOT, (3) AND, and (4) OR. expressions inside parentheses must be evaluated before all other operations. The next operation that holds precedence is the complement, and then follows the AND and, finally, the OR. example: demorgan s ٢٦ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
14 Boolean functions Boolean function described by an algebraic expression consists of binary variables A Boolean function can be represented in a truth table. The number of rows in the truth table is 2 n, where n is the number of variables in the function. The interconnection of gates will dictate the logic expression. ٢٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Logic diagram ٢٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
15 Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٢٩ F x y z Y' z y x it is sometimes possible to obtain a simpler expression for the same function and thus reduce the number of gates in the circuit and the number of inputs to the gate. Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٠
16 Logic diagram ٣١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Logic diagram ٣٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
17 Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٣ z y x f2 X z z X Xy Y X Truth table Algebraic Manipulation When a Boolean expression is implemented with logic gates, each term requires a gate and each variable within the term designates an input to the gate. We define a literal to be a single variable within a term, in complemented or un complemented form. By reducing the number of terms, the number of literals, or both in a Boolean expression, it is often possible to obtain a simpler circuit Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٤
18 Example ٣٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٦ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
19 Complement of a Function The complement of a function may be derived algebraically through DeMorgan s theorems DeMorgan s theorems can be extended to three or more variables. ٣٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
20 Example 2 ٣٩ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٤٠ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
21 Example 3 ٤١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٤٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
22 Thanks,.. See you next week (ISA), ٤٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design
Lecture (04) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee
Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee Boolean algebra properties basic assumptions and properties: Closure law A set S is closed with respect to a binary operator, for every
More informationSYNERGY INSTITUTE OF ENGINEERING & TECHNOLOGY,DHENKANAL LECTURE NOTES ON DIGITAL ELECTRONICS CIRCUIT(SUBJECT CODE:PCEC4202)
Lecture No:5 Boolean Expressions and Definitions Boolean Algebra Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates Binary logic is used in all of today's digital computers and devices Cost of the circuits is an important factor Finding simpler and cheaper but equivalent circuits can
More informationChapter 2. Boolean Algebra and Logic Gates
Chapter 2. Boolean Algebra and Logic Gates Tong In Oh 1 Basic Definitions 2 3 2.3 Axiomatic Definition of Boolean Algebra Boolean algebra: Algebraic structure defined by a set of elements, B, together
More informationChapter 2. Boolean Expressions:
Chapter 2 Boolean Expressions: A Boolean expression or a function is an expression which consists of binary variables joined by the Boolean connectives AND and OR along with NOT operation. Any Boolean
More informationBawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University
Logic Design First Stage Lecture No.5 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Boolean Operations Laws of Boolean Algebra Rules of Boolean Algebra
More informationENGIN 112 Intro to Electrical and Computer Engineering
ENGIN 2 Intro to Electrical and Computer Engineering Lecture 5 Boolean Algebra Overview Logic functions with s and s Building digital circuitry Truth tables Logic symbols and waveforms Boolean algebra
More informationLecture (05) Boolean Algebra and Logic Gates
Lecture (05) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Minterms and Maxterms consider two binary variables x and y combined with an AND operation. Since eachv ariable may appear in either
More informationUNIT 2 BOOLEAN ALGEBRA
UNIT 2 BOOLEN LGEBR Spring 2 2 Contents Introduction Basic operations Boolean expressions and truth tables Theorems and laws Basic theorems Commutative, associative, and distributive laws Simplification
More informationExperiment 4 Boolean Functions Implementation
Experiment 4 Boolean Functions Implementation Introduction: Generally you will find that the basic logic functions AND, OR, NAND, NOR, and NOT are not sufficient to implement complex digital logic functions.
More informationBoolean algebra. June 17, Howard Huang 1
Boolean algebra Yesterday we talked about how analog voltages can represent the logical values true and false. We introduced the basic Boolean operations AND, OR and NOT, which can be implemented in hardware
More informationLecture (03) Binary Codes Registers and Logic Gates
Lecture (03) Binary Codes Registers and Logic Gates By: Dr. Ahmed ElShafee Binary Codes Digital systems use signals that have two distinct values and circuit elements that have two stable states. binary
More informationBawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University
Logic Design First Stage Lecture No.6 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Outlines Boolean Operations Laws of Boolean Algebra Rules of
More informationCombinational Logic & Circuits
WeekI Combinational Logic & Circuits Spring' 232  Logic Design Page Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other
More informationBinary logic. Dr.AbuArqoub
Binary logic Binary logic deals with variables like (a, b, c,, x, y) that take on two discrete values (, ) and with operations that assume logic meaning ( AND, OR, NOT) Truth table is a table of all possible
More informationLecture 3: Binary Subtraction, Switching Algebra, Gates, and Algebraic Expressions
EE210: Switching Systems Lecture 3: Binary Subtraction, Switching Algebra, Gates, and Algebraic Expressions Prof. YingLi Tian Feb. 5/7, 2019 Department of Electrical Engineering The City College of New
More informationLecture (02) Operations on numbering systems
Lecture (02) Operations on numbering systems By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I Complements of a number Complements are used in digital computers to simplify
More informationLSN 4 Boolean Algebra & Logic Simplification. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology
LSN 4 Boolean Algebra & Logic Simplification Department of Engineering Technology LSN 4 Key Terms Variable: a symbol used to represent a logic quantity Compliment: the inverse of a variable Literal: a
More informationBoolean Algebra & Digital Logic
Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to
More informationIntroduction to Boolean Algebra
Introduction to Boolean Algebra Boolean algebra which deals with twovalued (true / false or and ) variables and functions find its use in modern digital computers since they too use twolevel systems
More informationLogic Design: Part 2
Orange Coast College Business Division Computer Science Department CS 6 Computer Architecture Logic Design: Part 2 Where are we? Number systems Decimal Binary (and related Octal and Hexadecimal) Binary
More informationIntroduction to Boolean Algebra
Introduction to Boolean Algebra Boolean algebra which deals with twovalued (true / false or and ) variables and functions find its use in modern digital computers since they too use twolevel systems
More informationBoolean Algebra A B A AND B = A*B A B A OR B = A+B
Boolean Algebra Algebra is the branch of mathematics that deals with variables. Variables represent unknown values and usually can stand for any real number. Because computers use only 2 numbers as we
More informationGet Free notes at ModuleI One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)
More informationBoolean Algebra. BME208 Logic Circuits Yalçın İŞLER
Boolean Algebra BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com 5 Boolean Algebra /2 A set of elements B There exist at least two elements x, y B s. t. x y Binary operators: +
More information4&5 Binary Operations and Relations. The Integers. (part I)
c Oksana Shatalov, Spring 2016 1 4&5 Binary Operations and Relations. The Integers. (part I) 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition,
More information2.6 BOOLEAN FUNCTIONS
2.6 BOOLEAN FUNCTIONS Binary variables have two values, either 0 or 1. A Boolean function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses
More informationChapter 2 Boolean algebra and Logic Gates
Chapter 2 Boolean algebra and Logic Gates 2. Introduction In working with logic relations in digital form, we need a set of rules for symbolic manipulation which will enable us to simplify complex expressions
More informationAssignment (36) Boolean Algebra and Logic Simplification  General Questions
Assignment (36) Boolean Algebra and Logic Simplification  General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make
More informationCircuit analysis summary
Boolean Algebra Circuit analysis summary After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian محرم 1439 ه Winter
More informationIT 201 Digital System Design Module II Notes
IT 201 Digital System Design Module II Notes BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More informationENGIN 112. Intro to Electrical and Computer Engineering
ENIN 2 Intro to Electrical and Computer Engineering Lecture 6 More Boolean Algebra ENIN2 L6: More Boolean Algebra September 5, 23 A B Overview Epressing Boolean functions Relationships between algebraic
More informationVariable, Complement, and Literal are terms used in Boolean Algebra.
We have met gate logic and combination of gates. Another way of representing gate logic is through Boolean algebra, a way of algebraically representing logic gates. You should have already covered the
More information[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics
400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {
More informationUnitIV Boolean Algebra
UnitIV Boolean Algebra Boolean Algebra Chapter: 08 Truth table: Truth table is a table, which represents all the possible values of logical variables/statements along with all the possible results of
More informationCGF Lecture 2 Numbers
CGF Lecture 2 Numbers Numbers A number is an abstract entity used originally to describe quantity. i.e. 80 Students etc The most familiar numbers are the natural numbers {0, 1, 2,...} or {1, 2, 3,...},
More informationPropositional Calculus. Math Foundations of Computer Science
Propositional Calculus Math Foundations of Computer Science Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus so that they can use it to
More information6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )
6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2) Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate logical values from place to place.
More informationPropositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus: Boolean Algebra and Simplification CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Topics Motivation: Simplifying Conditional Expressions
More informationIntroduction to Computer Architecture
Boolean Operators The Boolean operators AND and OR are binary infix operators (that is, they take two arguments, and the operator appears between them.) A AND B D OR E We will form Boolean Functions of
More informationMenu. Algebraic Simplification  Boolean Algebra EEL3701 EEL3701. MSOP, MPOS, Simplification
Menu Minterms & Maxterms SOP & POS MSOP & MPOS Simplification using the theorems/laws/axioms Look into my... 1 Definitions (Review) Algebraic Simplification  Boolean Algebra Minterms (written as m i ):
More informationCombinational Logic Circuits
Chapter 2 Combinational Logic Circuits J.J. Shann (Slightly trimmed by C.P. Chung) Chapter Overview 21 Binary Logic and Gates 22 Boolean Algebra 23 Standard Forms 24 TwoLevel Circuit Optimization
More informationDIGITAL SYSTEM DESIGN
DIGITAL SYSTEM DESIGN UNIT I: Introduction to Number Systems and Boolean Algebra Digital and Analog Basic Concepts, Some history of Digital SystemsIntroduction to number systems, Binary numbers, Number
More information6. Combinational Circuits. Building Blocks. Digital Circuits. Wires. Q. What is a digital system? A. Digital: signals are 0 or 1.
Digital Circuits 6 Combinational Circuits Q What is a digital system? A Digital: signals are or analog: signals vary continuously Q Why digital systems? A Accurate, reliable, fast, cheap Basic abstractions
More informationInformation Science 1
Information Science Boolean Expressions Week College of Information Science and Engineering Ritsumeikan University Topics covered l Terms and concepts from Week 9 l Binary (Boolean) logic History Boolean
More informationCombinational Devices and Boolean Algebra
Combinational Devices and Boolean Algebra Silvina Hanono Wachman M.I.T. L021 6004.mit.edu Home: Announcements, course staff Course information: Lecture and recitation times and locations Course materials
More informationSummary. Boolean Addition
Summary Boolean Addition In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of or 0. The complement represents the inverse
More informationBoolean Analysis of Logic Circuits
Course: B.Sc. Applied Physical Science (Computer Science) Year & Sem.: IInd Year, Sem  IIIrd Subject: Computer Science Paper No.: IX Paper Title: Computer System Architecture Lecture No.: 7 Lecture Title:
More informationCDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012
CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6311 and Excess3? Basic Operations of Boolean Algebra Examples
More informationComputer Science. Unit4: Introduction to Boolean Algebra
Unit4: Introduction to Boolean Algebra Learning Objective At the end of the chapter students will: Learn Fundamental concepts and basic laws of Boolean algebra. Learn about Boolean expression and will
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian ذو الحجة 1438 ه Winter
More informationObjectives: 1 Bolean Algebra. Eng. Ayman Metwali
Objectives: Chapter 3 : 1 Boolean Algebra Boolean Expressions Boolean Identities Simplification of Boolean Expressions Complements Representing Boolean Functions 2 Logic gates 3 Digital Components 4
More informationDefinitions. 03 Logic networks Boolean algebra. Boolean set: B 0,
3. Boolean algebra 3 Logic networks 3. Boolean algebra Definitions Boolean functions Properties Canonical forms Synthesis and minimization alessandro bogliolo isti information science and technology institute
More informationDesigning Computer Systems Boolean Algebra
Designing Computer Systems Boolean Algebra 08:34:45 PM 4 June 2013 BA1 Scott & Linda Wills Designing Computer Systems Boolean Algebra Programmable computers can exhibit amazing complexity and generality.
More informationChapter 3. Boolean Algebra and Digital Logic
Chapter 3 Boolean Algebra and Digital Logic Chapter 3 Objectives Understand the relationship between Boolean logic and digital computer circuits. Learn how to design simple logic circuits. Understand how
More informationCSCI2467: Systems Programming Concepts
CSCI2467: Systems Programming Concepts Slideset 2: Information as Data (CS:APP Chap. 2) Instructor: M. Toups Spring 2018 Course updates datalab out today!  due after Mardi gras...  do not wait until
More information2. BOOLEAN ALGEBRA 2.1 INTRODUCTION
2. BOOLEAN ALGEBRA 2.1 INTRODUCTION In the previous chapter, we introduced binary numbers and binary arithmetic. As you saw in binary arithmetic and in the handling of floatingpoint numbers, there is
More informationDr. Chuck Cartledge. 10 June 2015
Miscellanea Exam #1 Break Exam review 2.1 2.2 2.3 2.4 Break 3 4 Conclusion References CSC205 Computer Organization Lecture #003 Chapter 2, Sections 2.1 through 4 Dr. Chuck Cartledge 10 June 2015 1/30
More informationCombinational Circuits Digital Logic (Materials taken primarily from:
Combinational Circuits Digital Logic (Materials taken primarily from: http://www.facstaff.bucknell.edu/mastascu/elessonshtml/eeindex.html http://www.cs.princeton.edu/~cos126 ) Digital Systems What is a
More informationSoftware Engineering 2DA4. Slides 2: Introduction to Logic Circuits
Software Engineering 2DA4 Slides 2: Introduction to Logic Circuits Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on S. Brown and Z. Vranesic, Fundamentals of Digital
More information1. Mark the correct statement(s)
1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another
More information2.1 Binary Logic and Gates
1 EED2003 Digital Design Presentation 2: Boolean Algebra Asst. Prof.Dr. Ahmet ÖZKURT Asst. Prof.Dr Hakkı T. YALAZAN Based on the Lecture Notes by Jaeyoung Choi choi@comp.ssu.ac.kr Fall 2000 2.1 Binary
More informationGate Level Minimization Map Method
Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically
More informationPropositional Calculus. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus
More informationStandard Forms of Expression. Minterms and Maxterms
Standard Forms of Expression Minterms and Maxterms Standard forms of expressions We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains:
More informationLECTURE 4. Logic Design
LECTURE 4 Logic Design LOGIC DESIGN The language of the machine is binary that is, sequences of 1 s and 0 s. But why? At the hardware level, computers are streams of signals. These signals only have two
More informationAlgebra 1 Review. Properties of Real Numbers. Algebraic Expressions
Algebra 1 Review Properties of Real Numbers Algebraic Expressions Real Numbers Natural Numbers: 1, 2, 3, 4,.. Numbers used for counting Whole Numbers: 0, 1, 2, 3, 4,.. Natural Numbers and 0 Integers:,
More informationQUESTION BANK FOR TEST
CSCI 2121 Computer Organization and Assembly Language PRACTICE QUESTION BANK FOR TEST 1 Note: This represents a sample set. Please study all the topics from the lecture notes. Question 1. Multiple Choice
More informationBoolean Algebra. P1. The OR operation is closed for all x, y B x + y B
Boolean Algebra A Boolean Algebra is a mathematical system consisting of a set of elements B, two binary operations OR (+) and AND ( ), a unary operation NOT ('), an equality sign (=) to indicate equivalence
More informationBoolean Logic CS.352.F12
Boolean Logic CS.352.F12 Boolean Algebra Boolean Algebra Mathematical system used to manipulate logic equations. Boolean: deals with binary values (True/False, yes/no, on/off, 1/0) Algebra: set of operations
More informationRead this before starting!
Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Spring Semester,
More information6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )
6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2) Signals and Wires Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate digital signals
More informationBOOLEAN ALGEBRA AND CIRCUITS
UNIT 3 Structure BOOLEAN ALGEBRA AND CIRCUITS Boolean Algebra and 3. Introduction 3. Objectives 3.2 Boolean Algebras 3.3 Logic 3.4 Boolean Functions 3.5 Summary 3.6 Solutions/ Answers 3. INTRODUCTION This
More informationLecture 5. Chapter 2: Sections 47
Lecture 5 Chapter 2: Sections 47 Outline Boolean Functions What are Canonical Forms? Minterms and Maxterms Index Representation of Minterms and Maxterms SumofMinterm (SOM) Representations ProductofMaxterm
More informationReview. EECS Components and Design Techniques for Digital Systems. Lec 05 Boolean Logic 9/404. Seq. Circuit Behavior. Outline.
Review EECS 150  Components and Design Techniques for Digital Systems Lec 05 Boolean Logic 9404 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley Design flow
More information(Refer Slide Time 6:48)
Digital Circuits and Systems Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology Madras Lecture  8 Karnaugh Map Minimization using Maxterms We have been taking about
More informationChap2 Boolean Algebra
Chap2 Boolean Algebra Contents: My name Outline: My position, contact Basic information theorem and postulate of Boolean Algebra. or project description Boolean Algebra. Canonical and Standard form. Digital
More informationDSAS Laboratory no 4. Laboratory 4. Logic forms
Laboratory 4 Logic forms 4.1 Laboratory work goals Going from Boolean functions to Boolean forms. Logic forms equivalence. Boolean forms simplification. Shannon s theorems. Representation in NAND and NOR
More informationSoftware and Hardware
Software and Hardware Numbers At the most fundamental level, a computer manipulates electricity according to specific rules To make those rules produce something useful, we need to associate the electrical
More informationUNIT4 BOOLEAN LOGIC. NOT Operator Operates on single variable. It gives the complement value of variable.
UNIT4 BOOLEAN LOGIC Boolean algebra is an algebra that deals with Boolean values((true and FALSE). Everyday we have to make logic decisions: Should I carry the book or not?, Should I watch TV or not?
More informationExperiment 3: Logic Simplification
Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed ElSaied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions
More information01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008
Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 28 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom
More information8.3 Common Loop Patterns
Lecture 17 Topics: Chapter 8. Loop Structures and Booleans 8.3 (Continues) nested loops 8.4. Computing with booleans 8.5 Other common structures: posttest, loop and half. 1 8.3 Common Loop Patterns Nested
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:0011:15 SEB 1242 Lecture 22 121115 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Binary Number Representation Binary Arithmetic Combinatorial Logic
More informationSWITCHING THEORY AND LOGIC CIRCUITS
SWITCHING THEORY AND LOGIC CIRCUITS COURSE OBJECTIVES. To understand the concepts and techniques associated with the number systems and codes 2. To understand the simplification methods (Boolean algebra
More informationCombinational Circuits
Combinational Circuits Q. What is a combinational circuit? A. Digital: signals are or. A. No feedback: no loops. analog circuits: signals vary continuously sequential circuits: loops allowed (stay tuned)
More informationNAND. Grade (10) Instructor. Logic Design 1 / 13
Logic Design I Laboratory 02 NAND NOR XOR # Student ID 1 Student Name Grade (10) Instructor signature 2 3 Delivery Date 1 / 13 Objective To find the basic NAND & NOR & XOR gates concept and study on multiple
More informationHenry Lin, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 7 (Digital Logic) July 24 th, 2012
Henry Lin, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 7 (Digital Logic) July 24 th, 2012 1 Digital vs Analog Digital signals are binary; analog
More informationRead this before starting!
Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Spring Semester,
More informationSE311: Design of Digital Systems
SE311: Design of Digital Systems Lecture 3: Complements and Binary arithmetic Dr. Samir AlAmer (Term 041) SE311_Lec3 (c) 2004 ALAMER ١ Outlines Complements Signed Numbers Representations Arithmetic Binary
More informationComputer Architecture
Computer Architecture Lecture 1: Digital logic circuits The digital computer is a digital system that performs various computational tasks. Digital computers use the binary number system, which has two
More informationModule 7. Karnaugh Maps
1 Module 7 Karnaugh Maps 1. Introduction 2. Canonical and Standard forms 2.1 Minterms 2.2 Maxterms 2.3 Canonical Sum of Product or SumofMinterms (SOM) 2.4 Canonical product of sum or ProductofMaxterms(POM)
More informationWhy Don t Computers Use Base 10? Lecture 2 Bits and Bytes. Binary Representations. ByteOriented Memory Organization. Base 10 Number Representation
Lecture 2 Bits and Bytes Topics! Why bits?! Representing information as bits " Binary/Hexadecimal " Byte representations» numbers» characters and strings» Instructions! Bitlevel manipulations " Boolean
More informationLecture 10: Combinational Circuits
Computer Architecture Lecture : Combinational Circuits Previous two lectures.! TOY machine. Net two lectures.! Digital circuits. George Boole (85 864) Claude Shannon (96 2) Culminating lecture.! Putting
More informationOutline. What Digit? => Number System. Decimal (base 10) Significant Digits. Lect 03 Number System, Gates, Boolean Algebra. CS221: Digital Design
Lect 3 Number System, Gates, Boolean Algebra CS22: Digital Design Dr. A. Sahu Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati Outline Number System Decimal, Binary, Octal, Hex Conversions
More informationRead this before starting!
Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Spring Semester,
More informationEEE130 Digital Electronics I Lecture #4_1
EEE130 Digital Electronics I Lecture #4_1  Boolean Algebra and Logic Simplification  By Dr. Shahrel A. Suandi 46 Standard Forms of Boolean Expressions There are two standard forms: Sumofproducts form
More informationX Y Z F=X+Y+Z
This circuit is used to obtain the compliment of a value. If X = 0, then X = 1. The truth table for NOT gate is : X X 0 1 1 0 2. OR gate : The OR gate has two or more input signals but only one output
More informationRead this before starting!
Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Spring Semester,
More information