MC9211 Computer Organization


 Morgan Atkins
 2 years ago
 Views:
Transcription
1 MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (292/ODD) (29  / A&B)
2 Coverage Lesson2 Introduces the basic postulates of Boolean Algebra and shows the correlation between Boolean expressions and their corresponding logic diagrams. All possible logic operations for two variables are investigated and from that, the most useful logic gates used in the design of digital systems are determined. KSB67 2
3 Lesson2 Digital Logic Circuits. Basic definitions 2. Basic theorems and properties of Boolean algebra 3. Boolean Functions 4. Canonical and standard forms 5. Other logic operations 6. Digital Logic Gates KSB67 3
4 .Basic Definitions Boolean Algebra deals with binary variables and logic operations Binary variables are represented by A, B, x, y etc Logic Operations are AND, OR, NOT etc A Boolean function can be expressed algebraically with binary variables, the logic operation symbols, parentheses and equal sign ( ex: F = x + y z ) For a given value of variables the function can be either or The relationship between a function and its binary variables can be represented in a truth table KSB67 4
5 Boolean Algebra (contd..) From truth table a Boolean function can be obtained (and vice versa) A Boolean function can be transformed from an algebraic expression in to a logic diagram composed of AND, OR, NOT etc gates KSB67 5
6 Boolean Algebra (contd..) Truth Table x y z F Boolean Function F = x + y z Logic Diagram x y z F KSB67 6
7 Boolean Algebra (contd..) The purpose of Boolean Algebra is to facilitate the analysis and design of digital circuits It provides a convenient tool to.express in algebraic form a truth table relationship between binary variables 2.Express in algebraic form the inputoutput relationship of logic diagrams 3.Find simpler circuits for the same function KSB67 7
8 2.Basic Theorems of Boolean Algebra.Different letters in Boolean expressions are called variables Example: A.B + A.C +A.(D+E) has 5 variables 2.Each occurrence of a variable or its complement is called a literal Example: in the above expression there are 7 literals 3.Two expressions are equivalent if one expression equals only when the other equals, and one equals only when the other equals KSB67 8
9 Basic Definitions (contd..) 4.Two expressions are complements of each other if one expression equals only when the other equals, and vice versa The complement of a Boolean expression is obtained by changing all. s to + s changing all + s to. s changing all s to s changing all s to s and complementing each literal Example:.A + B C +? KSB67 9
10 Basic Definitions (contd..) 5.The dual of a Boolean expression is obtained by changing all. s to + s changing all + s to. s changing all s to s changing all s to s but not complementing any literal Example:.A + B C +? KSB67
11 Postulates. X = or else X = 2. AND 3. OR represented by. represented by + X Y X.Y. =. =. =. = X Y X+Y + = + = + = + = KSB67
12 Basic Theorems a.. X = b. + X = 2a.. X = X 2b. + X = X 3a. X X = X 3b.X + X = X 4a. X X = 4b. X + X = 5a. X Y = Y X 5b. X + Y = Y + X 6a.XYZ=X(YZ)=(XY)Z 6b.X+Y+Z= X+(Y+Z)= (X+Y)+Z DeMorgan s Theorem 7a. (X Y... Z) = X + Y + +Z 7b. (X + Y+ +Z) = X Y..Z KSB67 2
13 Basic Theorems (contd..) 8a.XY + XZ = X(Y+Z) 8b. (X+Y)(X+Z) = X + YZ 9a. XY + XY = X 9b. (X+Y)(X+Y ) = X a. X + XY = X b. X(X+Y) = X 2a. X + X Y = X + Y 2b. X (X +Y) = XY 3a. XY + X Z + YZ = XY + X Z 3b. (X+Y)(X +Z)(Y+Z) = (X+Y)(X +Z) 4a. XY + X Z = (X+Z)(X +Y) 4b. (X+Y)(X +Z) = XZ + X Y 5.a. X(Y+Z) = XY + XZ Distributive Law 5b. X + YZ = (X+Y)(X+Z) KSB67 3
14 Exercise.Determine by means of a truth table the validity of DeMorgan s theorem for three variables: (ABC) = A + B + C 2.Simplify the following expressions using Boolean algebra a) A + AB b) AB + AB c) A BC + AC d) A B + ABC + ABC 3.Simplify the following expressions using Boolean algebra a) AB + A(CD + CD ) b) (BC + A D)(AB + CD ) KSB67 4
15 Exercise (contd..) 2c) Solution: A BC + AC = A BC + AC(B + B ) = A BC + ABC + AB C = A BC + ABC + ABC + AB C = BC(A + A) + AC(B + B ) = BC + AC = C(A + B) KSB67 5
16 Exercise (contd..) 4.Using DeMorgan s theorem, show that: a) (A + B ) (A + B ) = b) A + A B + A B = 5.Given the Boolean function F = x y + xyz : a) Derive an algebraic expression for the complement F b) Show that F.F = c) Show that F + F = KSB67 6
17 6.Given the Boolean function F = xy z + x y z + xyz Exercise (contd..) a) List the truth table of the function b) Draw the logic diagram using the original Boolean expression c) Simplify the algebraic expression using Boolean algebra d) List the truth table of the function from the simplified expression and show that it is the same as the truth table in part a) e) Draw the logic diagram from the simplified expression and compare the total number of gates with the diagram of part b) KSB67 7
18 3. Boolean Functions A Binary variable can take value of or A Boolean Function is an expression formed with  Binary variables  Two binary operators OR and AND  A unary operator NOT  Parentheses and  An equal (=) sign For a given value of the variables the function can be either or Example: F = xyz KSB67 8
19 Boolean Functions (contd..) Any Boolean function can be represented by a TRUTH TABLE The number of rows in the table is 2 n where n is the number of binary variables in the function The and combinations for each row is obtained from binary numbers by counting from to 2 n  For each row of the table there is a value for the function equal to either or KSB67 9
20 Boolean Functions (contd..) Truth tables for Boolean functions F = XYZ F 2 = X+Y Z F 3 = X Y Z F 4 = XY +X Z Is given on the next slide KSB67 2
21 Boolean Functions (contd..) X Y Z F F 2 F 3 F 4 KSB67 2
22 Boolean Functions (contd..) A Boolean Function may be transformed from algebraic expression in to a logic diagram composed of AND, OR, and NOT gates Exercise: Convert the four functions given previously in to logic gates Algebraic manipulation: The minimization of number of literals and the number of terms results in a circuit with minimum equipment KSB
23 4. Canonical and Standard Forms There are two canonical or standard forms by which we can express any combinational logic network:  The SUM of PRODUCTs form (SOP)  The PRODUCT of SUMs form (POS) Before studying SOP and POS we have to study MINTERM and MAXTERM KSB
24 MINTERM A MINTERM of n variables is a product of the variables in which each variable appears exactly once in true or complemented form Each MINTERM is obtained from an AND term of the n variables, with each variable being  primed if the corresponding bit of the binary number is and  unprimed if it is a The symbol for each MINTERM is of the form m j, where j denotes the decimal equivalent of the binary number of the minterm designated KSB
25 MINTERM (contd..) A MINTERM of n variables is a product of the variables in which each variable appears exactly once in true or complemented form Each MINTERM is obtained from an AND term of the n variables, with each variable being  primed if the corresponding bit of the binary number is and  unprimed if it is a The symbol for each MINTERM is of the form m j, where j denotes the decimal equivalent of the binary number of the minterm designated KSB
26 MINTERM (contd..) X Y Z TERM Designation X Y Z m X Y Z m X YZ m 2 X YZ m 3 XY Z m 4 XY Z m 5 XYZ m 6 XYZ m 7 KSB
27 MINTERM (CONTD..) A Boolean function may be expressed algebraically from a given truth table by  forming a minterm for each combination of the variables that produces a in the function  and then taking OR of all those terms KSB
28 MINTERM Example X Y Z f f 2 f 2 f = X Y Z + XY Z + XYZ = m + m4 + m7 f(x,y,z) = (, 4, 7) f2 = X YZ + XY Z + XYZ + XYZ = m3 + m5 + m6 + m7 f2(x,y,z) = ( 3, 5, 6, 7) Note: Any Boolean function can be expressed as a sum of MITERM s KSB
29 Examples: MINTERM (contd..).express Boolean Function F = A+B C in a sum of MINTERMs KSB
30 MAXTERM A MAXTERM of n variables is a sum of the variables in which each variable appears exactly once in true or complemented form Each MAXTERM is obtained from an OR term of the n variables, with each variable being  unprimed if the corresponding bit of the binary number is and  primed if it is a The symbol for each MAXTERM is of the form M j, where j denotes the decimal equivalent of the binary number of the maxterm designated KSB67 3
31 MAXTERM (contd..) X Y Z TERM Designation X+Y+Z M X+Y+Z M X+Y +Z M 2 X+Y +Z M 3 X +Y+Z M 4 X +Y+Z M 5 X +Y +Z M 6 X +Y +Z M 7 KSB67 3
32 MAXTERM ( contd..) A Boolean function may be expressed algebraically from a given truth table by  forming a maxterm for each combination of the variables that produces a in the function  and then taking AND of all those terms KSB
33 MAXTERM Example X Y Z f f 2 f = (X+Y+Z)( X+Y +Z)(X+Y +Z ) (X +Y+Z )(X +Y +Z) = M + M2 +M3+M5+M6 f(x,y,z) = π (,2,3,5,6) f2 =? Note: Any Boolean function can be expressed as a product of MAXTERM s KSB
34 Examples: MAXTERM (contd..).express Boolean Function F = XY+X Z in a product of of MAXTERMs KSB
35 Conversion between canonical forms To convert from canonical form to another  interchange the symbols and π and  list those numbers missing from the original form Example: F(X,Y,Z) = (,3,6,7) = π (,2,4,5) Standard Form : The terms that form the function may contain one, two, or any number of literals in sum of product form or product of sum form Ex: F=Y +XY+X YZ (SoP) or KSB F =X(X +Y)(X Y Z) (PoS)
36 5 Other Logic Operations X Y F F F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F F F 2 F 3 F 4 F 5 KSB
37 KSB
38 6. Digital Logic Gates A gate is a circuit with one or more input signals but only one output signal Gates are twostate digital circuits because the input and output signals are either low ( Volts) or high (+5 Volts) Gates are usually called logic circuits because they can be analyzed with Boolean algebra KSB
39 Binary Digital Input Signal BASIC LOGIC GATE... Gate Binary Digital Output Signal Types of Basic Logic Gates Combinational Logic Gates Logic Gates whose output logic value depends only on the input logic values Sequential Logic Gates Logic Gates whose output logic value depends on the input values and the state (stored information) of the Gates Functions of Gates can be described by Truth Table Boolean Function Karnaugh Map There are two types of Combinational Logic Gates Basic gates NOT, OR and AND Derived gates NOR, NAND, XOR, XNOR KSB
40 Basic Gates  AND Gate Definition: An AND gate has two or more input signals but only one output signal: all inputs must be high to get a high output Truth table Three Input Two Input A B x Algebraic Function : x = A. B Symbol A x B A B C x KSB67 4
41 Basic Gates  OR Gate Definition: An OR gate has two or more input signals but only one output signal: if any input signal is high, the output signal is high Truth table Three Input Two Input A B X Algebraic Function : x = A + B A B C X Symbol A B x KSB67 4
42 Basic gates NOT or Inverter Definition: An Inverter (NOT) is a gate with only one input signal and one output signal the output state is always the opposite of the input state Symbol A Truth Table x A x Algebraic Function : x = A KSB
43 Basic gates Buffer Definition: A Buffer is gate with only one input signal and one output signal the output state is always the same as the input state the circuit is used only for power amplification Algebraic Function : x = A Symbol A Truth Table A x x KSB
44 Exercise. The following register has an output of. Show how to complement each bit. 6Bit Register 2.Show the truth table of a 4input OR gate KSB
45 3.What are the values of Y3 Y2 Y Y when each of the switches is pressed? KSB
46 4. What does the following circuit do? Y5 Y4 Y3 Y2 Y Y KSB
47 5.Write down the truth table for the following circuit KSB
48 6. BCD Decoder: For each of the combinations of ABCD which output line is high KSB
49 Derived Gates  NOR Gate Definition: A NOR gate has two or more input signals but only one output signal: all input signals must be low to get a high output Truth table Two Input A B X A Three Input B C X Algebraic Function : x = (A + B) =A.B Symbol A x B KSB
50 NOR Gate Application Control = A x Control = A x Control A x Observations:.When Control is, the output is the complement of input 2.When Control is, the output is always KSB67 5
51 Derived Gates  NAND Gate Definition: A NAND gate has two or more input signals but only one output signal: all input signals must be high to get a low output Truth table Two Input Three Input A B X A B C X Algebraic Function : x = (A. B) =A +B Symbol A B x KSB67 5
52 NAND Gate Application A x Control = A x Control Control = A x Observations:.When Control is, the output is always 2.When Control is, the output is the complement of input KSB
53 Derived Gates XOR (Exclusive OR) Gate Definition: An XOR gate has two or more input signals but only one output signal: it recognizes words that have odd number of s Truth table Two Input A Three Input B C X A B X Algebraic Function : x = A Symbol A x B + B = AB +A B KSB
54 XOR Gate Application A x INVERT INVERT = A x INVERT = A x Observations:.When INVERT is, the output is same as input 2.When Control is, the output is the complement of input KSB
55 Odd Parity Generator XOR Application (contd..) KSB
56 XOR Application (contd..) INVERTER KSB
57 Derived Gates  XNOR (Exclusive NOR) Gate Definition: An XNOR gate has two or more input signals but only one output signal: it recognizes words that have even number of s (also number with all s) Truth table Two Input A B X Algebraic Function X = (A B) Symbol A B x A Three Input KSB B C X
58 Digital Logic Gates  Summary Name Symbol Function Truth Table AND OR NOT A X = A B X or B X = AB A B A B X X = A + B A X X = A Buffer A X X = A NAND NOR XOR Exclusive OR XNOR Exclusive NOR or Equivalence A B X X = (AB) X X = (A + B) A X = A B X or B X = A B + AB A X = (A B) X or B X = A B + AB A B X A B X A X A X A B X A B X A B X A B X KSB
CS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman
CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions
More 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 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 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 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 TwoLevel Optimization
More informationChapter2 BOOLEAN ALGEBRA
Chapter2 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 informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III2 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 By Electrical Engineering Department College of Engineering King Saud University 14311432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean
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 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 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 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 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 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 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 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 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 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 (kmap). 1. Deriving of logical expression from
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 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 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 informationEC121 Digital Logic Design
EC121 Digital Logic Design Lecture 2 [Updated on 020418] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What
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 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 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 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 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 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 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 informationGateLevel Minimization
GateLevel 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. Fourvariable
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 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 41 Boolean Operations and Expressions 42 Laws and Rules of Boolean
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 informationWeekI. Combinational Logic & Circuits
WeekI 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 informationSpiral 1 / Unit 3
3. Spiral / Unit 3 Minterm and Maxterms Canonical Sums and Products 2 and 3Variable Boolean Algebra Theorems DeMorgan's Theorem Function Synthesis use Canonical Sums/Products 3.2 Outcomes I know 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 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. Combinationalcircuit analysis 4. Combinationalcircuit 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 informationChapter 2 Boolean Algebra and Logic Gates
Ch1: Digital Systems and Binary Numbers Ch2: Ch3: GateLevel Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan
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 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 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 informationTotal Time = 90 Minutes, Total Marks = 50. Total /50 /10 /18
University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Midterm Examination Instructor: M. Sachdev October 23rd, 2007 Total Time = 90 Minutes, Total
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 21 Basic Definitions 22 Axiomatic Definition of Boolean Algebra 23
More informationBoolean Algebra, Gates and Circuits
Boolean Algebra, Gates and Circuits Kasper Brink November 21, 2017 (Images taken from Tanenbaum, Structured Computer Organization, Fifth Edition, (c) 2006 Pearson Education, Inc.) Outline Last week: Von
More informationSystems I: Computer Organization and Architecture
Systems I: Computer Organization and Architecture Lecture 6  Combinational Logic Introduction A combinational circuit consists of input variables, logic gates, and output variables. The logic gates accept
More informationEx: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.
Boolean Expression Forms: Sumofproducts (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 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 informationUNIT 3 BOOLEAN ALGEBRA (CONT D)
UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions ExclusiveOR and ExclusiveNOR operations The consensus theorem Summary of algebraic
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 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 informationChapter 3. Boolean Algebra. (continued)
Chapter 3. Boolean Algebra (continued) Algebraic structure consisting of: set of elements B binary operations {+, } unary operation {'} Boolean Algebra such that the following axioms hold:. B contains
More 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 informationBoolean Algebra and Digital Logic 2009, University of Colombo School of Computing
IT 204 Section 3.0 Boolean Algebra and Digital Logic Boolean Algebra 2 Logic Equations to Truth Tables X = A. B + A. B + AB A B X 0 0 0 0 3 Sum of Products The OR operation performed on the products of
More informationChapter 2. Digital Logic Basics
Chapter 2 Digital Logic Basics 1 2 Chapter 2 2 1 Implementation using NND gates: We can write the XOR logical expression B + B using double negation as B+ B = B+B = B B From this logical expression, we
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 informationChapter 7 Logic Circuits
Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary
More 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 3 BOOLEAN ALGEBRA
CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 ExclusiveOR and Equivalence Operations 3.3 The Consensus
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter
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 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  Lec19cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationLearning Objectives. Boolean Algebra. In this chapter you will learn about:
Ref. Page Slide /78 Learning Objectives In this chapter you will learn about: oolean algebra Fundamental concepts and basic laws of oolean algebra oolean function and minimization Logic gates Logic circuits
More informationMark Redekopp, All rights reserved. Lecture 5 Slides. Canonical Sums and Products (Minterms and Maxterms) 23 Variable Theorems DeMorgan s Theorem
Lecture 5 Slides Canonical Sums and Products (Minterms and Materms) 23 Variable Theorems DeMorgan s Theorem Using products of materms to implement a function MAXTERMS Question Is there a set of functions
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 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 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 informationEvery time has a value associated with it, not just some times. A variable can take on any value within a range
Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»
More 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 ProductofSums
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 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 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 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 informationEE40 Lec 15. Logic Synthesis and Sequential Logic Circuits
EE40 Lec 15 Logic Synthesis and Sequential Logic Circuits Prof. Nathan Cheung 10/20/2009 Reading: Hambley Chapters 7.47.6 Karnaugh Maps: Read following before reading textbook http://www.facstaff.bucknell.edu/mastascu/elessonshtml/logic/logic3.html
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 nonempty set, + and. are binary operations in B, is a unary operation in B, 0 and 1 are special elements of
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 informationCircuits & Boolean algebra.
Circuits & Boolean algebra http://xkcd.com/730/ CSCI 255: Introduction to Embedded Systems Keith Vertanen Copyright 2011 Digital circuits Overview How a switch works Building basic gates from switches
More 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 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 informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 4 Following the slides of Dr. Ahmed H. Madian محرم 439 ه Winter 28
More informationCHAPTER 7. Exercises 17/ / /2 2 0
CHAPTER 7 Exercises E7. (a) For the whole part, we have: Quotient Remainders 23/2 /2 5 5/2 2 2/2 0 /2 0 Reading the remainders in reverse order, we obtain: 23 0 = 0 2 For the fractional part we have 2
More informationNumber System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary
Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two and threevariable Fourvariable Determination of minimum expressions using essential prime implicants Fivevariable
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 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 information1. Name the person who developed Boolean algebra
MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA
More information7.1. Unit 7. Minterm and Canonical Sums 2 and 3Variable Boolean Algebra Theorems DeMorgan's Theorem Simplification using Boolean Algebra
7.1 Unit 7 Minterm and Canonical Sums 2 and 3Variable 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 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 informationCombinatorial Logic Design Principles
Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 41 Boolean algebra a.k.a. switching algebra deals with boolean values  0, 1 Positivelogic convention analog voltages
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Overview Part 1 Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra
More informationReview for Test 1 : Ch1 5
Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8
More informationELC224C. Karnaugh Maps
KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to Kmaps How to use Kmaps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on
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 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 informationWhy digital? Overview. Number Systems. Binary to Decimal conversion
Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not
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 informationUNIT 4 MINTERM AND MAXTERM EXPANSIONS
UNIT 4 MINTERM AND MAXTERM EXPANSIONS Spring 2 Minterm and Maxterm Expansions 2 Contents Conversion of English sentences to Boolean equations Combinational logic design using a truth table Minterm and
More informationEECS150  Digital Design Lecture 4  Boolean Algebra I (Representations of Combinational Logic Circuits)
EECS150  Digital Design Lecture 4  Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4bool1 Page 1, 9/5 9am Outline Review of
More information