Chapter 2 : Boolean Algebra and Logic Gates
|
|
- Edwin Francis
- 6 years ago
- Views:
Transcription
1 Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University Basic Definitions 2.2. Basic Theorems and Properties of Boolean Algebra 2.3. Boolean Function Examples 2.4. Canonical and Standard Forms 2.5. Example of three variables functions 2.6. SOP and POS Conversion 2.7. SOP and POS Forms Chapter 2 page: 1
2 Basic Definitions The Boolean Algebra may be defined with a set of elements, a set of operators and a number of postulates to deduce rules, theorems and properties of the system. The most common postulates used to formulate various algebraic structure are: o Closure: Given a binary operators * and a set of elements S. S is closed to the binary operator (*) if for any a, b S we obtain a unique c by the operation a*b =c o Associative law: A binary operator * on a set S is said to be associative whenever : (x*y)*z =x*(y*z) for all x,y,z S o Commutative law: A binary operator * on a set S is said to be commutative whenever : x*y = y*z for all x,y S o Identity Element: A set S is said to have an identity element with respect to a binary operation * if there exists an element e S with the property: e*x = x*e = x for every S o Inverse: A set S having the identity element e with respect to a binary operator * is said to have an inverse, for every S, there exists an element y S such that x*y = e o Distributive law: If * and : are two binary operators on a set S, * is said to be distributive over whenever : x*(y z) = (x*y) (x*z) Chapter 2 page: 2
3 Basic Definitions Boolean Algebra is an algebraic structure defined by a set of elements B, with two operators + and, provided that the following postulates are satisfied: o Closure with respect to the operator + Closure with respect to the operator o An identity element with respect to + designated by 0 An identity element with respect to designated by 1 o Commutative with respect to + : x+y=y+x Commutative with respect to : x y=y x o Distributive over + : x (y+z) = (x y)+(x z) o + Distributive over : x + (y z) = (x + y) (x + z) o For every element x B, there exists an element x B called the complement of x such that x+x =1 and x x =0 Chapter 2 page: 3
4 Basic Theorems and Properties of Boolean Algebra Note that every law has two expressions, (a) and (b). This is known as duality. These are obtained by changing every AND(.) to OR(+), every OR(+) to AND(.) and all 1's to 0's and vice-versa. It has become conventional to drop the. (AND symbol) i.e. A.B is written as AB. o Identity Law 1(a): X+X =X 1(b): X.X = X o Commutative Law 2(a): 2(b): o Associative Law 3(a): 3(b): X+Y= Y+X X.Y =Y.X X+(Y+Z) = (X+Y)+Z X.(Y.Z) =(X.Y).Z o Distributive Law 4(a): X.(Y+Z) = XY +XZ 4(b): X+(Y.Z) =(X+Y).(X+Z) o Rule 4(a): (X ) =X Chapter 2 page: 4
5 Basic Theorems and Properties of Boolean Algebra Theorems o Theorem 1(a): X+1=1 1(b): X.0=0 By duality o Theorem 2(a): (X+Y) = X Y DeMorgan 2(b): (X.Y) =X +Y o Theorem 3(a): X+XY = X Absorption 3(b): X.(X+Y) =X o Theorem 4(a): 4(b): X+X Y = X+Y X.(X +Y) = XY Chapter 2 page: 5
6 Boolean Functions Examples 1. Prove the Rule Rx Algebraically: A + A B = A+B A+ A B = A.1 + A B = A.(1 + B) +A B = A.1+ AB + A B = A. + B(A + A ) = A + B 2. Simplify the function: F(A,B,C) = (A + B + C )(A + B C) using only rules and Theorems F(A,B,C) = (A + B + C )(A + B C) = AA + AB C + B A + B B C + C A + C B C = A(1+ B C + B + C ) + B C + B CC = A + B C Chapter 2 page: 6
7 Boolean Functions Examples A Boolean function described by an algebraic expression consists of binary variables, the constants 0 and 1, and the logic operation symbols. F 1 =x+x y F 2 =x y z + x yz + xy Simplify the Boolean functions: 1. x(x +y) 2. x+x y 3. (x+y)(x+y ) 4. xy+ x z +yz Find the complement of the functions F 1 and F 2 : 1. F 1 = (x yz +x y z) 2. F 2 = x(y z +yz) Chapter 2 page: 7
8 Canonical and Standard Forms Minterms and Maxterms Definition: a minterm of n variables is a product of the variables in which each appears exactly once in true or complemented form. e.g.: minterms of 3 variables: Each minterm = 1 for only one combination of values of the variables, = 0 otherwise Definition: a maxterm of n variables is a sum of the variables in which each appears exactly once in true or complemented form. e.g.: maxterm of 3 variables: Each maxterm = 0 for only one combination of values of the variables, = 1 otherwise Chapter 2 page: 8
9 Canonical and Standard Forms Minterms and Maxterms Consider 2 binary variables, there are 4 possible configurations: x y, x y, xy and xy Each of these four AND terms is called minterm or standard product. F(a,b,c) = m i Where m i represent all minterms for which F(a,bc,)=1 In a similar manner, n variables can be combined to form 2 n minterms represented by a symbol m j Similarly, n variables forming 2 n possible combinations of an OR terms called Maxterms or standard sums represented by M j F(a,b,c) = M j Where M j represent all minterms for which F(a,bc,)=0 Chapter 2 page: 9
10 Canonical and Standard Forms SOP and POS Forms All possible minterms and maxterms are obtained from the truth table: Every function can be written as a sum of minterms, which is a special kind of Sum Of Products form (SOP). The sum of minterms form for any function is unique A Boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces 1 in the function, and then taking the OR of all those terms. Every function can be written as a unique product of maxterms (POS). If you have a truth table for a function, you can write a product of maxterms (POS) expression by picking out the rows of the table where the function output is 0. (Be careful if you re writing the actual literals!) Chapter 2 page: 10
11 Canonical and Standard Forms Minterms and Maxterms for Three variables Minterms Maxterms x y z Term Design. Term Desig x y z m 0 x+y+z M x y z m 1 x+y+z M x yz m 2 x+y +z M x yz m 3 x+y +z M xy z m 4 x +y+z M xy z m 5 x +y+z M xyz m 6 x +y +z M xyz m 7 x +y +z M 7 Chapter 2 page: 11
12 Canonical and Standard Forms If you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. Express F 1 and F 2 as a SOP form and its equivalent form using Maxterms. x y z F1 F F 1 =m 1 +m 4 +m 7 = m(1,4,7) F 2 =m 3 +m 5 +m 6 +m 7 = m(3,5,6,7) F 1 =M 0.M 2.M 3.M 5.M 6 F 2 =M 0.M 1.M 2.M 4 Chapter 2 page: 12
13 Example of three variables functions Express F and F as a SOP form. x y z F F F=m 0 +m 1 +m 2 +m 3 +m 6 = m(0,1,2,3,6) F =m 4 +m 5 +m 7 = m(4,5,7) F contains all the minterms not in F. Chapter 2 page: 13
14 Example of three variables functions Express F and F as a POS form (using Maxterms). If you have a truth table for a function, you can write a product of maxterms (POS) expression by picking out the rows of the table where the function output is 0. (Be careful if you re writing the actual literals!) x y z F F F = M 4 +M 5 +M 7 = M(4,5,7) F = M 0 +M 1 +M 2 +M 3 +M 6 = M (0,1,2,3,6) F contains all the maxterms not in F. Chapter 2 page: 14
15 SOP and POS Conversion We can convert a sum of minterms to a product of maxterms Consider F(A,B,C)= m(1,3,5,7) And F (A,B,C) = m(0,2,4,6) = m 0 + m 2 +m 4 +m 6 Complementing (F ) = (m 0 + m 2 +m 4 +m 6 ) So F = m 0. m 2. m 4. m 6 = M 0.M 2.M 4.M 6 = M (0,2,4,6) In general, just replace the minterms with maxterms, using maxterm numbers that don t appear in the sum of minterms: F(A,B,C)= m(1,3,5,7) = M (0,2,4,6) The same thing works for converting from a product of maxterms to a sum of minterms. Chapter 2 page: 15
16 SOP and POS Forms Sum of Minterms Sometimes, its convenient to express a Boolean function in its sum-of-minterms form. If the function is not in this form, it can be made by first expanding the expression into a sum of AND terms. Each term is then inspected to see if it contains all the variables. If it misses one or more, it is ANDed with an expression such as x+x, where x is the missing variable. Example: Express F=A+B C as a sum of minterms or a SOP. Then first term A is missing 2 variables B and C A= A(B+B ) = AB + AB AB = AB(C+C ) = ABC + ABC AB =AB (C+C ) =AB C+AB C Then second term B C is missing one variables A B C = B C(A+A ) =B CA+B CA F(A,B,C)=ABC + ABC +AB C+AB C +A B C= m(1,4,5,6,7) Chapter 2 page: 16
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 informationBoolean 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 informationCS 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 informationCHAPTER 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 informationChapter 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 informationCHAPTER 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 informationThis 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 informationMC9211 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 informationBoolean 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 informationDigital 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 informationChapter 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 informationChapter 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 informationStandard 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 informationChapter 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 informationChapter-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 informationFunctions. 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 informationUnit 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 informationComputer 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 informationEC-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 informationMidterm1 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 informationChapter 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 informationCHAPTER 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 informationLogic 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 informationECEN 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 informationLecture 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 informationBOOLEAN 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 informationBinary 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 informationDIGITAL 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 informationBoolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE
Boolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE Today s Topics Boolean algebra applications in logic
More informationChapter 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 informationCombinational 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 informationECE 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 informationLecture 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 informationII. 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 informationChapter 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 informationChapter 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 informationE&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 informationWEEK 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 informationDigital 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 informationDigital 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 informationE&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 informationCombinational Logic. Fan-in/ Fan-out Timing. Copyright (c) 2012 Sean Key
Combinational Logic Fan-in/ Fan-out Timing Copyright (c) 2012 Sean Key Fan-in & Fan-out Fan-in The number of inputs to a logic gate Higher fan-in can lead to longer gate delays because of the higher input
More informationIf 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 informationBoolean 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 informationUnit 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 informationLogic 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 informationChapter 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 informationCHAPTER1: 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 informationTheorem/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 informationBoolean Algebra. Examples: (B=set of all propositions, or, and, not, T, F) (B=2 A, U,, c, Φ,A)
Boolean Algebra Definition: A Boolean Algebra is a math construct (B,+,.,, 0,1) where B is a non-empty set, + and. are binary operations in B, is a unary operation in B, 0 and 1 are special elements of
More informationSignals 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 informationChapter 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 informationCombinational Logic Circuits Part II -Theoretical Foundations
Combinational Logic Circuits Part II -Theoretical Foundations Overview Boolean Algebra Basic Logic Operations Basic Identities Basic Principles, Properties, and Theorems Boolean Function and Representations
More informationUnit 2 Session - 6 Combinational Logic Circuits
Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums
More informationLecture 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 informationBoolean 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 informationIn 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 informationEEA051 - 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 informationMark Redekopp, All rights reserved. Lecture 5 Slides. Canonical Sums and Products (Minterms and Maxterms) 2-3 Variable Theorems DeMorgan s Theorem
Lecture 5 Slides Canonical Sums and Products (Minterms and Materms) 2-3 Variable Theorems DeMorgan s Theorem Using products of materms to implement a function MAXTERMS Question Is there a set of functions
More informationENG2410 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 informationWeek-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 informationChapter 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 informationLecture 4: More Boolean Algebra
Lecture 4: More Boolean Algebra Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University ENGIN2
More informationNumber 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 informationECE 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 informationContents. 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 informationLecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University
Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Aby K George, ECE Department, Wayne State University Contents The Map method Two variable
More informationPart 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 informationCSE 140, Lecture 2 Combinational Logic CK Cheng CSE Dept. UC San Diego
CSE 140, Lecture 2 Combinational Logic CK Cheng CSE Dept. UC San Diego 1 Combinational Logic Outlines 1. Introduction 1. Scope 2. Review of Boolean lgebra 3. Review: Laws/Theorems and Digital Logic 2.
More informationCombinational 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 informationSpiral 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 informationDIGITAL 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/ M Morris Mano Digital Design Ahmad_911@hotmailcom / / / / wwwuqucscom Binary Systems Introduction - Digital Systems - The Conversion Between Numbering Systems - From Binary To Decimal - Octet To Decimal
More informationChapter 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 informationIntroduction to Digital Logic Missouri S&T University CPE 2210 Boolean Representations
Introduction to Digital Logic Missouri S&T University CPE 2210 Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology cetinkayae@mst.edu
More informationStandard & 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 informationBinary 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 informationCHAPTER 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 informationMinimization techniques
Pune Vidyarthi Griha s COLLEGE OF ENGINEERING, NSIK - 4 Minimization techniques By Prof. nand N. Gharu ssistant Professor Computer Department Combinational Logic Circuits Introduction Standard representation
More informationCPE100: 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 information211: 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 informationEEE130 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 informationBoolean Algebra. Sungho Kang. Yonsei University
Boolean Algebra Sungho Kang Yonsei University Outline Set, Relations, and Functions Partial Orders Boolean Functions Don t Care Conditions Incomplete Specifications 2 Set Notation $09,3/#0,9 438 v V Element
More informationUNIVERSITI 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 informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two- and three-variable Four-variable Determination of minimum expressions using essential prime implicants Five-variable
More informationBOOLEAN 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 informationSlide 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 informationIntroduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps
Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology
More information2. 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 information7.1. 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 informationAdministrative 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 informationMidterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes
Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name
More informationChap 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 informationCOE 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 informationDiscrete 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 informationBOOLEAN 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 informationCombinatorial 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 informationEECS150 - 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 informationGate-Level Minimization
Gate-Level Minimization Dr. Bassem A. Abdullah Computer and Systems Department Lectures Prepared by Dr.Mona Safar, Edited and Lectured by Dr.Bassem A. Abdullah Outline 1. The Map Method 2. Four-variable
More informationBinary 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