Simplification of Boolean Functions


 Blaise Hutchinson
 3 years ago
 Views:
Transcription
1 Simplification of Boolean Functions Contents: Why simplification? The Map Method Two, Three, Four and Five variable Maps. Simplification of two, three, four and five variable Boolean function by Map method. Product of sums and Sum of products simplification. NAND and NOR implementation. Course Instructor Mohammed Abdul kader Assistant Professor, EEE, IIUC
2 The complexity of the digital logic gates that implement a Boolean function is directly related to the complexity of the algebraic expression from which the function is implemented. Simplification of Boolean Functions Although the truth table representation of a function is unique but the algebraic expression appears in many different form. Before implementation of logic circuit, the choice of simplest boolean expression from different representation results the minimum number of gates and less complexity in the digital circuit. There are different ways of simplification of Boolean function. In this section we will discuss the Map Method of simplifying Boolean function. 2
3 The map method provides a simple straight forward procedure for minimizing Boolean functions. The map method, first proposed by Veitch and slightly modified by Karnaugh, is also known as the Veitch diagram or the Karnaugh map. Karnaugh maps provide an alternative way of simplifying logic circuits. Instead of using Boolean algebra simplification techniques, you can transfer logic values froma Boolean statement or a truth table into a Karnaugh map. The arrangement of 0's and 1's with in the map helps you to visualize the logic relationships between the variables and leads directly to a simplified Boolean statement. The Map Method
4 Two Variable Map There are four minterms for two variables; hence the map consists of fours quares, one of each minterm. Representation of functions in two variable map. 4 F=xy F=x+y = x y+xy +xy = m1+ m2 + m3
5 Three Variable Map There are eight minterms for three variables, i.e. the map consist of eight squares. Minterms are not arranged in binary sequence but in a sequence similar to gray code/reflected code. There are four squares where each variable is equal to 1 and four where each is equal to 0. 5
6 Three Variable Map Understanding the usefulness of map for simplification of Boolean function Isolated 1 in map F = m1 + m7 = x y z + xyz Adjacent Pair/ 1 s in two adjacent square F = m5 + m7 = xy z + xyz = xz (y +y) = xz 6
7 Three Variable Map Understanding the usefulness of map for simplification of Boolean function Adjacent Quad/ 1 s in four adjacent square F = m1 + m3 + m5 + m7 = x y z + x yz+ xy z + xyz = x z (y +y) +xz (y +y) = x z +xz = z (x +x) = z Adjacent Octet/ 1 s in eight adjacent square F = 1 7
8 Three Variable Map: Examples Example 32: Simplify the Boolean function F= x yz + xy z + xyz + xyz Solution : yz xz So, after simplification, F = yz+ xz 8
9 Three Variable Map: Examples Example 33: Simplify the Boolean function F= A C+ A B+ AB C+BC Solution : F = A C+ A B+ AB C+BC =A BC+ A B C +A BC+ A BC + AB C+ABC+A BC A B C So, after simplification, F = C+ A B 9
10 Three Variable Map: Examples Example 34: Simplify the Boolean function F (x,y,z)= (0, 2, 4, 5, 6) Solution : z xy So, after simplification, F = z + xy 10
11 Four Variable Map
12 Four Variable Map The map for Boolean function of four binary variables has 16 minterms and the squares assigned to each. The rows and columns are numbered in a reflected code sequence, with only one digit changing value between two adjacent rows and columns. The minterm corresponding to each square can be obtained from the concatenation of the row number with the column number. The combination of adjacent squares that is useful during the simplification process is easily determined from inspection of the four variable map One square represents one minterm, giving a term of four literals. Two adjacent squares represent a term of three literal. Four adjacent squares represent a term of two literals. Eight adjacent squares represent a term of one literals. Sixteen adjacent squares represent the function equal to 1. wxyz: 1010 (10 in binary), so this square represent m10 12
13 Four Variable Map: Examples Example 35: Simplify the Boolean function F (x,y,z)= (0, 1,2, 4, 5, 6, 8, 9, 12, 13, 14) Solution : So, after simplification, F = y + w z +x z 13
14 Four Variable Map: Examples Example 36: Simplify the Boolean function F = A B C +B CD + A BCD +AB C Solution : So, after simplification, F = B D + B C +A C D 14
15 Five Variable Map
16 Five Variable Map: Examples Example 37: Simplify the Boolean function F (A, B, C, D, E) = (0, 2, 4, 6, 9, 11, 13, 15, 17, 21, 25, 27, 29, 31) Solution : A B E So, after simplification, F = BE+AD E + A B E AD E BE 16
17 Product of Sums Simplification All previous examples are in sumofproducts form [F = BE+AD E + A B E ] How to obtain the productofsum form * Simplify F in the form of sum of products. [If we mark the empty squares by 0 s and combine them into valid adjacent squares, we obtained a simplified expression of the complement of the function, i.e. of F ] * Apply DeMorgan's theorem F = (F ') * F': sum of products => F : product of sums 17
18 Example 36: Simplify the Boolean function in (a) sum of products and (b) product of sums F (A, B, C, D, E) = (0, 1, 2, 5, 8, 9, 10) Solution : Product of Sums Simplification (Cont.) (a) Sum of products simplification (b) product of sums simplification 18 F = B D + B C + A C D F = AB + CD + BD So, F = (A +B ) (C +D ) (B + D ) by DeMorgan theorem
19 Product of Sums Simplification (Cont.) Gate implementation of the function Example 38 Sum of products product of sums (a) F = B D + B C + A C D (b) F = (A +B ) (C +D ) (B + D ) Simplify the Boolean function F(A,B,C,D)=Π(0,1,2,3,4,10,11) 19
20 Product of Sums Simplification (Cont.) Example: Simplify the Boolean function F(A,B,C,D)=Π (0,1,2,3,4,10,11) AB CD From Map we get, F = A B + B C + A C D Using DeMorgan theorem F = (A B + B C + A C D ) = (A + B ) ( B +C ) ( A + C + D ) 20
21 NAND and NOR Implementation Why NAND and NOR implementation? Digital circuits are frequently constructed with NAND/NOR rather than with AND/OR gates. NAND and NOR gates are easier to fabricate with electronic components than AND/OR. Cheaper(lower cost) and faster(less delay). Any Boolean function can be constructed using only NAND or only NOR gates. That s why NAND and NOR are known as universal gates. 21
22 NAND and NOR Implementation Implementation of basic gates by NAND gate NOT gate by NAND gate x x AND gate by NAND gate x y (xy) ((xy) ) = xy x x OR gate by NAND gate y y (x y ) = x+y 22
23 NAND and NOR Implementation Implementation of basic gates by NOR gate NOT gate by NOR gate x x OR gate by NOR gate x y (x+y) ((x+y) ) = x+y x x AND gate by NOR gate y y (x +y ) = xy 23
24 NAND and NOR Implementation Two graphic symbols for NAND gate Two graphic symbols for NOR gate 24
25 NAND Implementation Implementation F=AB+CD+E by NAND gate only. 25
26 NAND Implementation Rules for obtaining the NAND logic diagram from a Boolean function 1. Simplify the function and express it in sum of products. 2. Draw a NAND gate for each product term of the function that has at least two literals. The inputs to each NAND gate are the literals of the term. This constitutes a group of firstlevel gates. 3. Draw a single NAND gate (using the ANDinvert or invertor graphic symbol) in the second level, with inputs coming from outputs of the 1 st level. 4. A term with a single literal requires an inverter in the first level or may be complemented and applied as an input to the secondlevel NAND gate. Note: If we simplify the function combining 0 s in a map, we obtain the simplified expression of the complement of the function in sum of product. The complement of the function can then be implemented with two levels of NAND gates using the rules stated above. If the normal output is desired, it would be necessary to insert a oneinput NAND gate. 26
27 NAND Implementation Example 39: Implement the following function with NAND gates F(x,y,z) = (0, 6) 27
28 NOR Implementation Implement the function F= (A+B) (C+D) E with NOR gates 28
29 NOR Implementation Example 39: Implement the following function with NAND gates F(x,y,z) = (0, 6) Sum of Product Product of sums Sum of Product Product of sums 29
30 30 Don tcare Conditions You don t always need all 2 n input combinations in an nvariable function. If you can guarantee that certain input combinations never occur. If some outputs aren t used in the rest of the circuit. A four bit decimal code, for example, has six combinations which are not used. Any digital circuit using this code operates under the assumption that these unused combinations will never occur as long as the system is working properly. The unused combinations is known as don t care conditions and can be used on the map to provide further simplification of the boolean expression. It should be realized that a don t care minterm is a combination of variables whose logical value is not specified. It cannot be marked with a 1 or, 0 in the map as it is not specified as 0 or 1. To distinguish the don t care condition from 1 s and 0 s, an X is used. Thus, an X inside a square in the map indicates that we don t care whether the value of 0 or 1 is assigned to F for the particular minterms.
31 NAND Implementation Example 312: Simplify F(w,x,y,z) = Σ(1,3,7,11,15) which has the don t care conditions d(w,x,y,z) = Σ (0,2,5) 31
32 NAND Implementation Exercise : Implement the following function with either NAND or NOR gates. Use only four gates. F = w xz+w yz+x yz +wxy z and d= wyz NAND Implementation yz wx Sum of products: Combine 1 s and some of X s F= x y+xz x x (x y) y F z (x z) 32
33 NAND Implementation NOR Implementation wx yz Products of sums: Combine 0 s and some of X s F = x y +xz So, F= (x+y) (x +z) x x (x +z) z F y (x +y ) 33
Chapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard
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 informationA B AB CD Objectives:
Objectives:. Four variables maps. 2. Simplification using prime implicants. 3. "on t care" conditions. 4. Summary.. Four variables Karnaugh maps Minterms A A m m m3 m2 A B C m4 C A B C m2 m8 C C m5 C m3
More informationChapter 3. GateLevel Minimization. Outlines
Chapter 3 GateLevel Minimization Introduction The Map Method FourVariable Map FiveVariable Map Outlines Product of Sums Simplification Don tcare Conditions NAND and NOR Implementation Other TwoLevel
More informationGateLevel Minimization
MEC520 디지털공학 GateLevel Minimization JeeHwan Ryu School of Mechanical Engineering GateLevel MinimizationThe Map Method Truth table is unique Many different algebraic expression Boolean expressions may
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 informationGateLevel Minimization
GateLevel Minimization Mano & Ciletti Chapter 3 By Suleyman TOSUN Ankara University Outline Intro to GateLevel Minimization The Map Method 2345 variable map methods ProductofSums Method Don t care
More informationS1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017
S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 Karnaugh Map Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and
More informationGateLevel Minimization
GateLevel Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2011 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method
More informationReview: Standard forms of expressions
Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and NOT. These operations can be combined to form complex expressions, which can
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 informationGate Level Minimization
Gate Level Minimization By Dr. M. Hebaishy Digital Logic Design Ch Simplifying Boolean Equations Example : Y = AB + AB Example 2: = B (A + A) T8 = B () T5 = B T Y = A(AB + ABC) = A (AB ( + C ) ) T8 =
More informationChapter 2 Combinational
Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean Equations Binary Logic
More informationGateLevel Minimization
GateLevel Minimization ( 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 The Map Method
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 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 information2008 The McGrawHill Companies, Inc. All rights reserved.
28 The McGrawHill Companies, Inc. All rights reserved. 28 The McGrawHill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y 28
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 informationLOGIC CIRCUITS. Kirti P_Didital Design 1
LOGIC CIRCUITS Kirti P_Didital Design 1 Introduction The digital system consists of two types of circuits, namely (i) Combinational circuits and (ii) Sequential circuit A combinational circuit consists
More informationGateLevel Minimization. BME208 Logic Circuits Yalçın İŞLER
GateLevel Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to
More informationCode No: 07A3EC03 Set No. 1
Code No: 07A3EC03 Set No. 1 II B.Tech I Semester Regular Examinations, November 2008 SWITCHING THEORY AND LOGIC DESIGN ( Common to Electrical & Electronic Engineering, Electronics & Instrumentation Engineering,
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 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 informationCombinational Logic with MSI and LSI
1010101010101010101010101010101010101010101010101010101010101010101010101010101010 1010101010101010101010101010101010101010101010101010101010101010101010101010101010 1010101010101010101010101010101010101010101010101010101010101010101010101010101010
More informationCSCI 220: Computer Architecture I Instructor: Pranava K. Jha. Simplification of Boolean Functions using a Karnaugh Map
CSCI 22: Computer Architecture I Instructor: Pranava K. Jha Simplification of Boolean Functions using a Karnaugh Map Q.. Plot the following Boolean function on a Karnaugh map: f(a, b, c, d) = m(, 2, 4,
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 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 informationPoints Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map
Points Addressed in this Lecture Lecture 4: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London Standard form of Boolean Expressions SumofProducts (SOP),
More informationLiteral Cost F = BD + A B C + A C D F = BD + A B C + A BD + AB C F = (A + B)(A + D)(B + C + D )( B + C + D) L = 10
Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm
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 informationDigital Logic Design. Outline
Digital Logic Design GateLevel Minimization CSE32 Fall 2 Outline The Map Method 2,3,4 variable maps 5 and 6 variable maps (very briefly) Product of sums simplification Don t Care conditions NAND and NOR
More informationB.Tech II Year I Semester (R13) Regular Examinations December 2014 DIGITAL LOGIC DESIGN
B.Tech II Year I Semester () Regular Examinations December 2014 (Common to IT and CSE) (a) If 1010 2 + 10 2 = X 10, then X is  Write the first 9 decimal digits in base 3. (c) What is meant by don
More informationDKT 122/3 DIGITAL SYSTEM 1
Company LOGO DKT 122/3 DIGITAL SYSTEM 1 BOOLEAN ALGEBRA (PART 2) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan s Theorems Boolean analysis of logic circuits
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 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 informationCHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey
CHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input
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 Design. Chapter 4. Principles Of. Simplification of Boolean Functions
Principles Of Digital Design Chapter 4 Simplification of Boolean Functions Karnaugh Maps Don t Care Conditions Technology Mapping Optimization, Conversions, Decomposing, Retiming Boolean Cubes for n =,
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 informationDigital Logic Lecture 7 Gate Level Minimization
Digital Logic Lecture 7 Gate Level Minimization By Ghada AlMashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. Kmap principles. Simplification using Kmaps. Don tcare
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 informationChapter 3 Simplification of Boolean functions
3.1 Introduction Chapter 3 Simplification of Boolean functions In this chapter, we are going to discuss several methods for simplifying the Boolean function. What is the need for simplifying the Boolean
More informationIncompletely Specified Functions with Don t Cares 2Level Transformation Review Boolean Cube KarnaughMap Representation and Methods Examples
Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2Level Transformation Review Boolean Cube KarnaughMap Representation and Methods Examples Incompletely specified functions
More informationDate Performed: Marks Obtained: /10. Group Members (ID):. Experiment # 04. Boolean Expression Simplification and Implementation
Name: Instructor: Engr. Date Performed: Marks Obtained: /10 Group Members (ID):. Checked By: Date: Experiment # 04 Boolean Expression Simplification and Implementation OBJECTIVES: To understand the utilization
More informationCombinational Logic Circuits
Chapter 3 Combinational Logic Circuits 12 Hours 24 Marks 3.1 Standard representation for logical functions Boolean expressions / logic expressions / logical functions are expressed in terms of logical
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 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 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 informationContents. Chapter 3 Combinational Circuits Page 1 of 34
Chapter 3 Combinational Circuits Page of 34 Contents Contents... 3 Combinational Circuits... 2 3. Analysis of Combinational Circuits... 2 3.. Using a Truth Table... 2 3..2 Using a Boolean unction... 4
More informationChapter 2: Combinational Systems
Uchechukwu Ofoegbu Chapter 2: Combinational Systems Temple University Adapted from Alan Marcovitz s Introduction to Logic and Computer Design Riddle Four switches can be turned on or off. One is the switch
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 informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2006 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : STLD(16EC402) Year & Sem: IIB.Tech & ISem Course & Branch: B.Tech
More informationSpring 2010 CPE231 Digital Logic Section 1 Quiz 1A. Convert the following numbers from the given base to the other three bases listed in the table:
Section 1 Quiz 1A Convert the following numbers from the given base to the other three bases listed in the table: Decimal Binary Hexadecimal 1377.140625 10101100001.001001 561.24 454.3125 111000110.0101
More informationUNIT II. Circuit minimization
UNIT II Circuit minimization The complexity of the digital logic gates that implement a Boolean function is directly related to the complexity of the algebraic expression from which the function is implemented.
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 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 informationGateLevel Minimization. section instructor: Ufuk Çelikcan
GateLevel Minimization section instructor: Ufuk Çelikcan Compleity of Digital Circuits Directly related to the compleity of the algebraic epression we use to build the circuit. Truth table may lead to
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science
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 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 information數位系統 Digital Systems 朝陽科技大學資工系. Speaker: FuwYi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷
數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: FuwYi Yang 楊伏夷 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象,
More informationA graphical method of simplifying logic
45 Karnaugh Map Method A graphical method of simplifying logic equations or truth tables. Also called a K map. Theoretically can be used for any number of input variables, but practically limited to 5
More informationSUBJECT CODE: IT T35 DIGITAL SYSTEM DESIGN YEAR / SEM : 2 / 3
UNIT  I PART A (2 Marks) 1. Using Demorgan s theorem convert the following Boolean expression to an equivalent expression that has only OR and complement operations. Show the function can be implemented
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in View Show
More informationSimplification of Boolean Functions
COM111 Introduction to Computer Engineering (Fall 20062007) NOTES 5  page 1 of 5 Introduction Simplification of Boolean Functions You already know one method for simplifying Boolean expressions: Boolean
More informationPhiladelphia University Faculty of Information Technology Department of Computer Science. Computer Logic Design. By Dareen Hamoudeh.
Philadelphia University Faculty of Information Technology Department of Computer Science Computer Logic Design By Dareen Hamoudeh Dareen Hamoudeh 1 Canonical Forms (Standard Forms of Expression) Minterms
More informationDigital logic fundamentals. Question Bank. Unit I
Digital logic fundamentals Question Bank Subject Name : Digital Logic Fundamentals Subject code: CA102T Staff Name: R.Roseline Unit I 1. What is Number system? 2. Define binary logic. 3. Show how negative
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 information10EC33: DIGITAL ELECTRONICS QUESTION BANK
10EC33: DIGITAL ELECTRONICS Faculty: Dr.Bajarangbali E Examination QuestionS QUESTION BANK 1. Discuss canonical & standard forms of Boolean functions with an example. 2. Convert the following Boolean function
More informationR10. II B. Tech I Semester, Supplementary Examinations, May
SET  1 1. a) Convert the following decimal numbers into an equivalent binary numbers. i) 53.625 ii) 4097.188 iii) 167 iv) 0.4475 b) Add the following numbers using 2 s complement method. i) 48 and +31
More informationLecture 4: Implementation AND, OR, NOT Gates and Complement
EE210: Switching Systems Lecture 4: Implementation AND, OR, NOT Gates and Complement Prof. YingLi Tian Feb. 13, 2018 Department of Electrical Engineering The City College of New York The City University
More informationSpecifying logic functions
CSE4: Components and Design Techniques for Digital Systems Specifying logic functions Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic and Dr.Pietro Mercati We have seen various concepts: Last
More informationENGINEERS ACADEMY. 7. Given Boolean theorem. (a) A B A C B C A B A C. (b) AB AC BC AB BC. (c) AB AC BC A B A C B C.
Digital Electronics Boolean Function QUESTION BANK. The Boolean equation Y = C + C + C can be simplified to (a) (c) A (B + C) (b) AC (d) C. The Boolean equation Y = (A + B) (A + B) can be simplified to
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 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 informationNODIA AND COMPANY. GATE SOLVED PAPER Computer Science Engineering Digital Logic. Copyright By NODIA & COMPANY
No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Computer
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 informationSlide Set 5. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary
Slide Set 5 for ENEL 353 Fall 207 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 207 SN s ENEL 353 Fall 207 Slide Set 5 slide
More informationCS6201DIGITAL PRINCIPLE AND SYSTEM DESIGN I YEAR/II SEM PARTB UNITI BOOLEAN ALGEBRA AND LOGIC GATES.
CS6201DIGITAL PRINCIPLE AND SYSTEM DESIGN I YEAR/II SEM PARTB UNITI BOOLEAN ALGEBRA AND LOGIC GATES. 1) Simplify the boolean function using tabulation method. F = (0, 1, 2, 8, 10, 11, 14, 15) List all
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 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 LOGIC CIRCUITS
COMBINATIONAL LOGIC CIRCUITS 4.1 INTRODUCTION The digital system consists of two types of circuits, namely: (i) Combinational circuits and (ii) Sequential circuits A combinational circuit consists of logic
More informationCombinational Logic Circuits Part III Theoretical Foundations
Combinational Logic Circuits Part III Theoretical Foundations Overview Simplifying Boolean Functions Algebraic Manipulation Karnaugh Map Manipulation (simplifying functions of 2, 3, 4 variables) Systematic
More informationMODULE 5  COMBINATIONAL LOGIC
Introduction to Digital Electronics Module 5: Combinational Logic 1 MODULE 5  COMBINATIONAL LOGIC OVERVIEW: For any given combination of input binary bits or variables, the logic will have a specific
More informationCprE 281: Digital Logic
CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Minimization CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev Administrative
More informationKarnaugh Map (KMap) Karnaugh Map. Karnaugh Map Examples. Ch. 2.4 Ch. 2.5 Simplification using Kmap
Karnaugh Map (KMap) Ch. 2.4 Ch. 2.5 Simplification using Kmap A graphical map method to simplify Boolean function up to 6 variables A diagram made up of squares Each square represents one minterm (or
More informationCS8803: Advanced Digital Design for Embedded Hardware
CS883: Advanced Digital Design for Embedded Hardware Lecture 2: Boolean Algebra, Gate Network, and Combinational Blocks Instructor: Sung Kyu Lim (limsk@ece.gatech.edu) Website: http://users.ece.gatech.edu/limsk/course/cs883
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 informationBOOLEAN ALGEBRA. 1. State & Verify Laws by using :
BOOLEAN ALGEBRA. State & Verify Laws by using :. State and algebraically verify Absorption Laws. (2) Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X (i) X + XY = X LHS = X + XY = X( + Y)
More informationCMPE223/CMSE222 Digital Logic
CMPE223/CMSE222 Digital Logic Optimized Implementation of Logic Functions: Strategy for Minimization, Minimum ProductofSums Forms, Incompletely Specified Functions Terminology For a given term, each
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems
More informationBOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.
COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless
More informationProgrammable Logic Devices. Programmable Read Only Memory (PROM) Example
Programmable Logic Devices Programmable Logic Devices (PLDs) are the integrated circuits. They contain an array of AND gates & another array of OR gates. There are three kinds of PLDs based on the type
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 information2. (a) Compare the characteristics of a floppy disk and a hard disk. (b) Discuss in detail memory interleaving. [8+7]
Code No: A109211202 R09 Set No. 2 1. (a) Explain the purpose of the following registers: i. IR ii. PC iii. MDR iv. MAR. (b) Explain with an example the steps in subtraction of two ndigit unsigned numbers.
More informationECE380 Digital Logic
ECE38 Digital Logic Optimized Implementation of Logic Functions: Strategy for Minimization, Minimum ProductofSums Forms, Incompletely Specified Functions Dr. D. J. Jackson Lecture 8 Terminology For
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 informationENGIN 112 Intro to Electrical and Computer Engineering
ENGIN 2 Intro to Electrical and Computer Engineering Lecture 8 Minimization with Karnaugh Maps Overview Kmaps: an alternate approach to representing oolean functions Kmap representation can be used to
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 information