Mark Redekopp, All rights reserved. Lecture 4 Slides. Boolean Algebra Logic Functions Canonical Sums/Products
|
|
- Debra Nichols
- 5 years ago
- Views:
Transcription
1 Lecture 4 Slides Boolean Algebra Logic Functions Canonical Sums/Products
2 LOGIC FUNCTION REPRESENTATION
3 Logic Functions A logic function maps input combinations to an output value ( 1 or ) 3 possible representations of a function Equation Schematic Truth Table Can convert between representations Truth table is onl unique representation
4 Eample Convert the following description to a Boolean equation An automobile warning buer is activated if the headlights are on AND the door is open OR the ke is in the ignition and the door is open. B = HD + KD B = D(H+K) Mark Redekopp, All rights reserved Equation and circuit representations are not unique
5 Eample Convert the following description to a Boolean equation An automobile warning buer is activated if the headlights are on AND the door is open OR the ke is in the ignition and the door is open. B = HD + KD B = D(H+K) Mark Redekopp, All rights reserved Equation and circuit representations are not unique
6 Truth Tables to Equations/Circuits Given a circuit with n-inputs, 2 n possible combinations eist => 2 n rows in a T.T. A general approach to converting a T.T. to equation or circuit Build a checker/decoder circuit for each combination and include the combinations where the function should be 1
7 BOOLEAN ALGEBRA Mark Redekopp, All rights reserved
8 Boolean Algebra A set of theorems to help us manipulate logical epressions/equations Aioms = Basis / assumptions used Theorems = manipulations that we can use
9 Aioms Aioms are the basis for Boolean Algebra Notice that these aioms are simpl restating our definition of digital/binar logic A1/A1 = Binar variables (onl 2 values possible) A2/A2 = NOT operation A3,A4,A5 = AND operation A3,A4,A5 = OR operation (A1) X = if X 1 (A1 ) X = 1 if X (A2) If X =, then X = 1 (A2 ) If X = 1, then X = (A3) = (A3 ) = 1 (A4) 1 1 = 1 (A4 ) + = (A5) 1 = 1 = (A5 ) + 1 = 1 + = 1
10 Dualit Ever truth statement can ields another truth statement I eercise if I have time and energ (original statement) I don t eercise if I don t have time or don t have energ (dual statement) To epress the dual, swap 1 s s +
11 Dualit The dual of an epression is not equal to the original Original epression Taking the dual of both sides of an equation ields a new equation Dual X + 1 = 1 Original equation X = Dual
12 Single Variable Theorem (T1) X+ = X (T1) X 1 = X (T1 ) X Y Z OR Hold Y constant X Y Z AND Whenever a variable is OR ed with, the output will be the same as the variable OR Anthing equals that anthing Whenever a variable is AND ed with 1, the output will be the same as the variable 1 AND Anthing equals that anthing
13 Single Variable Theorem (T2) X+1 = 1 (T2) X = (T2 ) X Y Z OR Hold Y constant X Y Z AND Whenever a variable is OR ed with 1, the output will be 1 1 OR anthing equals 1 Whenever a variable is AND ed with, the output will be AND anthing equals
14 Single Variable Theorem (T3) X+X = X (T3) X X = X (T3 ) X Y Z OR Whenever a variable is OR ed with itself, the result is just the value of the variable X Y Z AND Whenever a variable is AND ed with itself, the result is just the value of the variable This theorem can be used to reduce two identical terms into one OR to replicate one term into two.
15 Single Variable Theorem (T4) (X ) = X (T4) (X) = X (T4) 1 Anthing inverted twice ields its original value
16 Single Variable Theorem (T5) X+X = 1 (T5) X X = (T5 ) X Y Z OR Whenever a variable is OR ed with its complement, one value has to be 1 and thus the result is 1 X Y Z AND Whenever a variable is AND ed with its complement, one value has to be and thus the result is This theorem can be used to simplif variables into a constant or to epand a constant into a variable.
17 ISEL ISEL1 ISEL2 ISEL3 OSEL OSEL1 OSEL2 OSEL3 Application: Channel Selector Given 4 input, digital music/sound channels and 4 output channels Given individual select inputs that select 1 input channel to be routed to 1 output channel Input channels ICH OCH ICH1 ICH2 Channel Selector OCH1 OCH2 4 Output channels ICH3 OCH3 Input Channel Select Output Channel Select
18 OSEL OSEL1 OSEL2 Application: Steering Logic 4-input music channels (ICH) Select one input channel (use ISEL inputs) Route to one output channel (use OSEL inputs) ICH ICH ICH ICH 3 ISEL ISEL1 ISEL2 ISEL3 OSEL3 OCH OCH 1 OCH 2 OCH 3
19 OSEL OSEL1 OSEL2 OSEL3 Application: Steering Logic 1 st Level of AND gates act as barriers onl passing 1 channel OR gates combines 3 streams of s with the 1 channel that got passed (i.e. ICH1) 2 nd Level of AND gates passes the channel to onl the selected output ICH ICH 1 ICH 2 ICH 3 1 ICH1 Connection Point ICH1 ICH1 ICH1 ICH1 ICH1 1 OCH OCH 1 OCH 2 ICH1 OCH 3 AND: 1 AND ICH = ICH AND ICH = 1 ISEL ISEL1 ISEL2 ISEL3 OR: + ICH1 + + = ICH1 1 AND: 1 AND ICH1 = ICH1 AND ICH1 =
20 Your Turn Build a circuit that takes 3 inputs: S, IN, IN1 and outputs a single bit Y. It s functions should be: If S =, Y = IN (IN passes to Y) If S = 1, Y = IN1 (IN1 passes to Y) IN S Y IN1
21 DEFINITIONS Mark Redekopp, All rights reserved
22 Definitions of the Da X Z X Y Z X Y Z NOT (Inverter) AND OR Z X ' or X or ~X Z X Y Z X Y Literal: A literal is an instance of a single variable or its complement. Correct Eamples:,,, ALARM, (LON) Incorrect Eamples: +, (these are epressions) Product Term: A logical product (AND ing) of two or more literals or a single literal b itself Correct Eamples:, w, w a b, c Incorrect Eamples: (+) w, ( ) Onl evaluates to 1 for a single input combination Sum Term: A single literal or a logical sum (OR ing) of two or more literals. Correct Eamples: +w+, +, Incorrect Eamples: ab+c, ( + ) Onl evaluates to for a single input combination
23 Check Yourself Epression Sum Term / Product Term / Both / Neither w ( ) w ( ) w++ w+ (w++) w
24 Check Yourself Epression w ( ) w ( ) w++ w+ (w++) w Sum Term / Product Term / Both / Neither Neither (Can t have complements of sub-epressions onl literal) Product Term (AND is associative) Neither (Mied epression) Sum Term Neither (Can t have complements of sub-epressions) Both
25 CHECKERS Mark Redekopp, All rights reserved
26 Gates Gates can have more than 2 inputs but the functions sta the same AND = output = 1 if ALL inputs are 1 Outputs 1 for onl 1 input combination OR = output = 1 if ANY input is 1 Outputs for onl 1 input combination X Y Z F X Y Z F F F input AND 3-input OR
27 Checkers / Decoders An AND gate onl outputs 1 for 1 combination That combination can be changed b adding inverters to the inputs We can think of the AND gate as checking or decoding a specific combination and outputting a 1 when it matches. X Y Z F X Y Z F 1 1 F F AND gate decoding (checking for) combination AND gate decoding (checking for) combination
28 Checkers / Decoders An AND gate onl outputs 1 for 1 combination That combination can be changed b adding inverters to the inputs We can think of the AND gate as checking or decoding a specific combination and outputting a 1 when it matches. X Y Z F X Y Z F F F AND gate decoding (checking for) combination AND gate decoding (checking for) combination
29 Checkers / Decoders Place inverters at the input of the AND gates such that F produces 1 onl for input combination {,,} = {1} G produces 1 onl for input combination {,,} = {11} X Y Z F X Y Z G 1 1 F G AND gate decoding (checking for) combination AND gate decoding (checking for) combination
30 Checkers / Decoders Place inverters at the input of the AND gates such that F produces 1 onl for input combination {,,} = {1} G produces 1 onl for input combination {,,} = {11} X Y Z F X Y Z G 1 1 F G AND gate decoding (checking for) combination AND gate decoding (checking for) combination
31 Checkers / Decoders An OR gate onl outputs for 1 combination That combination can be changed b adding inverters to the inputs Add inverters to create an OR gate decoding (checking for) combination 1 F X Y Z F Add inverters to create an OR gate decoding (checking for) combination 11 F X Y Z F
32 Checkers / Decoders An OR gate onl outputs for 1 combination That combination can be changed b adding inverters to the inputs We can think of the OR gate as checking or decoding a specific combination and outputting a when it matches. OR gate decoding (checking for) combination 1 F X Y Z F OR gate decoding (checking for) combination 11 F X Y Z F
33 Combining functions into others A LOGIC EXERCISE
34 Combining Functions Given intermediate functions F1 and F2, how could ou use AND, OR, NOT to make G X Y Z F1 F2 G X Y Z F1 F2 G
35 Combining Functions Given intermediate functions F1 and F2, how could ou use AND, OR, NOT to make G X Y Z F1 F2 G X Y Z F1 F2 G X Y Z F1 F2 F1 F2 F1*F G = F1*F2
36 Combining Functions Given intermediate functions F1 and F2, how could ou use AND, OR, NOT to make H X Y Z F1 F2 H X Y Z F1 F2 H
37 Combining Functions Given intermediate functions F1 and F2, how could ou use AND, OR, NOT to make H X Y Z F1 F2 H X Y Z F1 F2 H X Y Z F1 F2 F1 F2 F1+F H = F1 + F2
38 Combining Functions Given intermediate functions F1 and F2, how could ou use AND, OR, NOT to make J X Y Z F1 F2 J X Y Z F1 F2 J
39 Combining Functions Given intermediate functions F1 and F2, how could ou use AND, OR, NOT to make J X Y Z F1 F2 J X Y Z F1 F2 J X Y Z F1 F2 F1 + F J = F1 + F2
40 Combining Functions How I want to change J slightl to make K and I ll give ou an etra function F3 X Y Z F1 F2 K X Y Z F1 F2 J K F3 1 X Y Z F1 F2 F3 F1 + F2 + F K = F1 + F2 + F3
41 Question Is there a set of functions (F1, F2, etc.) that would allow ou to build ANY 3- variable function X Y Z F1 Fn?? Think simple, think man X Y Z F1 F2 Fn?? 1? 1? 1 1? 1? 1 1? 1 1? 1 1 1? X Y Z m m1 m2 m3 m4 m5 m6 m7? 1? 1 1? 1 1? 1 1 1? 1 1? 1 1 1? 1 1 1? ? OR together an combination of m I s
42 Checkers / Decoders The m i functions on the previous slide are just AND gate checkers That combination can be changed b adding inverters to the inputs We can think of the AND gate as checking or decoding a specific combination and outputting a 1 when it matches. X Y Z F X Y Z F m m AND gate decoding (checking for) combination AND gate decoding (checking for) combination
43 Minterm Definition Minterm: A product term where each input variable of a function appears as eactl one literal f(,,) => ++
44 Minterms A minterm can be generated for ever combination of inputs Each minterm is the AND ing of variables that will evaluate to 1 for onl that combination A minterm checks or decodes a specific input combination and outputs 1 when found Minterm 3 11 = = m 3 Minterm 5 11 = = m 5 To make the minterm, complement the variables that equal and leave the variables in their true form that equal 1. X Y Z F
45 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders/checkers X Y Z F
46 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F 1 A X Y Z A
47 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F 1 A X Y Z A B B
48 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F 1 A X Y Z A B C B C
49 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F 1 A X Y Z A B C F B F C
50 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F 1 1 A X Y Z A B C F B 1 F C F(1,,) = 1
51 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F 1 1 A 1 X Y Z A B C F B F C F(,1,) = 1
52 Using Decoders to Implement Functions Given an an logic function, it can be implemented with the superposition of decoders X Y Z F A X Y Z A B C F B F C F(,1,1) =
53 Minterms Onl one minterm can evaluate to 1 at an time. m m 1 m 2 m 3 X Y F X Y X Y X Y X Y
54 Venn Diagram of Minterms Onl one region ON and all others OFF m m2 X=,Y= X Y X=1,Y= X Y m1 X=,Y=1 X Y m3 X=1,Y=1 X Y
55 Venn Diagram of Minterms To compose a function we can OR the minterms from the function s ON-Set X Y F OR OR F= X=1,Y= X Y X=,Y=1 X Y = F = XY + X Y
56 Finding Equations/Circuits Given a function and checkers (called decoders) for each combination, we just need to OR together the checkers where F = 1 3-bit number {,,} Mark Redekopp, All rights reserved Checker for Checker for 1 Checker for 1 Checker for 11 Checker for 1 Checker for 11 Checker for 11 Checker for 111 Assume we use ANDgate decoders that output a 1 when the combination is found F X Y Z F
57 BACKUP Mark Redekopp, All rights reserved
58 Unique Representations Arithmetic => f( 1, 2,, n ) Representations Graph Equation Logic => f( 1, 2,, n ) Representations Schematic No Boolean Equation No, unless ou convert to canonical sum (minterms) or canonical product (materms) form Truth Table Yes Binar Decision Diagram (BDD) Yes (if variables are ordered)
59 Boolean Algebra Terminolog Literal: A literal is an instance of a single variable or its complement. Correct Eamples:,,, ALARM, (LON) Incorrect Eamples: +, (these are epressions) Product Term: A single literal or a logical product (AND ing) of two or more literals. Correct Eamples:, w, w a b, c Incorrect Eamples: (+) w, ( ) Onl evaluates to 1 for a single input combination Sum Term: A single literal or a logical sum (OR ing) of two or more literals. Correct Eamples: +w+, +, Incorrect Eamples: ab+c, ( + ) Onl evaluates to for a single input combination SOP (Sum of Products) Form: An SOP epression is a logical sum (OR) of product terms. Correct Eamples: [ + w + a b c], [w + + ] Incorrect Eamples: [ +w (a+b) ], [ + ( ) ] POS (Product of Sums) Form: A POS epression is a logical product (AND) of sum terms. Correct Eamples: [(+ +) (w +) (a)], [ (+) (w +)] Incorrect Eamples: [( +) (+w)], [(+) (+) ]
60 Using Gates as Decoders (Checkers, Filters) White light Prism Polariing Filters White light X Y Z F X Y Z F F F input AND 3-input OR
61 Making Colors An color can be made b re-combining certain wavelengths of visible light Polariing Filters White light Polariing Filters White light Polariing Filters White light
Mark 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 informationOutcomes. Spiral 1 / Unit 2. Boolean Algebra BOOLEAN ALGEBRA INTRO. Basic Boolean Algebra Logic Functions Decoders Multiplexers
-2. -2.2 piral / Unit 2 Basic Boolean Algebra Logic Functions Decoders Multipleers Mark Redekopp Outcomes I know the difference between combinational and sequential logic and can name eamples of each.
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 informationOutcomes. Spiral 1 / Unit 3. The Problem SYNTHESIZING LOGIC FUNCTIONS
-3. -3.2 Outcomes Spiral / Unit 3 Minterm and Materms Canonical Sums and Products 2 and 3 Variable oolean lgebra Theorems emorgan's Theorem unction Snthesis use Canonical Sums/Products Mark Redekopp I
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 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 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 informationLogic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 1 Gate Circuits and Boolean Equations
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hperlinks are active in
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 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 informationChapter 2 Combinational
Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations HOANG Trang Reference: 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean
More information8. BOOLEAN ALGEBRAS x x
8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical
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 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 informationSimplifying Logic Circuits with Karnaugh Maps
Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified
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 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 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 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 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 information2.1. Unit 2. Digital Circuits (Logic)
2.1 Unit 2 Digital Circuits (Logic) 2.2 Moving from voltages to 1's and 0's ANALOG VS. DIGITAL volts volts 2.3 Analog signal Signal Types Continuous time signal where each voltage level has a unique meaning
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 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 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 informationLearning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr.
/7/ CE 4 Digital ystem Design Dr. Arshad Aziz Fundamental of ogic Design earning Objectives Review the basic concepts of logic circuits Variables and functions Boolean algebra Minterms and materms ogic
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 informationSimplification of Boolean Functions. Dept. of CSE, IEM, Kolkata
Simplification of Boolean Functions Dept. of CSE, IEM, Kolkata 1 Simplification of Boolean Functions: An implementation of a Boolean Function requires the use of logic gates. A smaller number of gates,
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 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: 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 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 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 informationChapter 2 : Boolean Algebra and Logic Gates
Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 1431-1432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean
More informationEXPERIMENT #4: SIMPLIFICATION OF BOOLEAN FUNCTIONS
EXPERIMENT #4: SIMPLIFICATION OF BOOLEAN FUNCTIONS OBJECTIVES: Simplify Boolean functions using K-map method Obtain Boolean expressions from timing diagrams Design and implement logic circuits Equipment
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 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 informationDigital Design. Digital Design
Principles Of Digital Design Chapter 3 Boolean Algebra and Logic Design Boolean Algebra Logic Gates Digital Design Implementation Technology ASICs Gate Arrays Basic Algebraic Properties A set is a collection
More informationWEEK 3.1 MORE ON KARNAUGH MAPS
WEEK 3. MORE ON KARNAUGH MAPS Don t Cares Sometimes, we might have inputs and it doesn t matter what the output is; i.e., we don t care what the output is. These situations are called don t cares. Rather
More informationUnit 3 Session - 9 Data-Processing Circuits
Objectives Unit 3 Session - 9 Data-Processing Design of multiplexer circuits Discuss multiplexer applications Realization of higher order multiplexers using lower orders (multiplexer trees) Introduction
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 informationSlides for Lecture 10
Slides for Lecture 10 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 30 September, 2013 ENEL 353
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 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 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 informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More information14:332:231 DIGITAL LOGIC DESIGN. Combinational Circuit Synthesis
:: DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering all Lecture #: Combinational Circuit Synthesis I Combinational Circuit Synthesis Recall: Combinational circuit
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 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 informationOutcomes. Spiral 1 / Unit 5. Logic Function Synthesis KARNAUGH MAPS. Karnaugh Maps
-. -. Spiral / Unit Mark Redekopp Outcomes I know the difference between combinational and sequential logic and can name examples of each. I understand latency, throughput, and at least technique to improve
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 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 informationDIGITAL ELECTRONICS & it0203 Semester 3
DIGITAL ELECTRONICS & it0203 Semester 3 P.Rajasekar & C.M.T.Karthigeyan Asst.Professor SRM University, Kattankulathur School of Computing, Department of IT 8/22/2011 1 Disclaimer The contents of the slides
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Charles Kime & Thomas Kaminski 2004 Pearson Education, Inc. Terms of Use (Hyperlinks are active
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 informationUniversity of Technology
University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year معالجات دقيقة المرحلة الرابعة ھندسة الليزر والبصريات االلكترونية Lecture 3 & 4 Boolean Algebra and Logic Gates
More informationOutcomes. Spiral 1 / Unit 5. Logic Function Synthesis KARNAUGH MAPS. Karnaugh Maps
-. -. Spiral / Unit Mark Redekopp Outcomes I know the difference between combinational and sequential logic and can name examples of each. I understand latency, throughput, and at least technique to improve
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 informationCover Sheet for Lab Experiment #3
The University of Toledo EECS:1100 R2 Digital Logic Design Dr. Anthony D. Johnson Cover Sheet for Lab Experiment #3 Student Names Course Section Anthony Phillips 003 Alexander Beck 003 Report turned in
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 informationCombinational Logic. By : Ali Mustafa
Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output
More informationUnit 8A Computer Organization. Boolean Logic and Gates
Unit 8A Computer Organization Boolean Logic and Gates Announcements Bring ear buds or headphones to lab! 15110 Principles of Computing, Carnegie Mellon University - CORTINA 2 Representing and Manipulating
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 informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National Universit Prof. Wangrok Oh(CNU) / 5 Overview Aiomatic Definition of Boolean Algebra 2 Basic Theorems
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 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 information2 Application of Boolean Algebra Theorems (15 Points - graded for completion only)
CSE140 HW1 Solution (100 Points) 1 Introduction The purpose of this assignment is three-fold. First, it aims to help you practice the application of Boolean Algebra theorems to transform and reduce Boolean
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 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 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 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 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 informationEx: 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 informationXOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure.
XOR - XNOR Gates Lesson Objectives: In addition to AND, OR, NOT, NAND and NOR gates, exclusive-or (XOR) and exclusive-nor (XNOR) gates are also used in the design of digital circuits. These have special
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 informationIntroduction to Digital Logic
Introduction to Digital Logic Lecture 7: Design Goals (Metrics) 2-Level Logic Negative Logic Warmup Consider F(w,x,y,z). Show the algebraic form of m4 and M4? Use Boolean algebra to find the minimal SOP
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Winter Notes - Unit 2
ECE-7: Digital Logic Design Winter 8 Notes - Unit OPTIMIZED IMPLEMENTATION OF LOGIC FUNCTIONS BASIC TECHNIQUES: We can alas minimie logic unctions using the Boolean theorems. Hoever, more poerul methods
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 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 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 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 informationLogic Design Combinational Circuits. Digital Computer Design
Logic Design Combinational Circuits Digital Computer Design Topics Combinational Logic Karnaugh Maps Combinational uilding locks Timing 2 Logic Circuit logic circuit is composed of: Inputs Outputs Functional
More informationTextbook: Digital Design, 3 rd. Edition M. Morris Mano
: 25/5/ P-/70 Tetbook: Digital Design, 3 rd. Edition M. Morris Mano Prentice-Hall, Inc. : INSTRUCTOR : CHING-LUNG SU E-mail: kevinsu@yuntech.edu.tw Chapter 3 25/5/ P-2/70 Chapter 3 Gate-Level Minimization
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 informationPart 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life...
Part 1: Digital Logic and Gates Analog vs Digital waveforms An analog signal assumes a continuous range of values: v(t) ANALOG A digital signal assumes discrete (isolated, separate) values Usually there
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 informationThe Karnaugh Map COE 202. Digital Logic Design. Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals
The Karnaugh Map COE 202 Digital Logic Design Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals Presentation Outline Boolean Function Minimization The Karnaugh Map (K-Map) Two, Three,
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal
More information2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS
CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital
More informationCSC9R6 Computer Design. Practical Digital Logic
CSC9R6 Computer Design Practical Digital Logic 1 References (for this part of CSC9R6) Hamacher et al: Computer Organization App A. In library Floyd: Digital Fundamentals Ch 1, 3-6, 8-10 web page: www.prenhall.com/floyd/
More informationS C F F F T T F T T S C B F F F F F T F T F F T T T F F T F T T T F T T T
EECS 270, Winter 2017, Lecture 1 Page 1 of 6 Use pencil! Say we live in the rather black and white world where things (variables) are either true (T) or false (F). So if S is Mark is going to the Store
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 informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationDIGITAL LOGIC CIRCUITS
DIGITAL LOGIC CIRCUITS Introduction Logic Gates Boolean Algebra Map Specification Combinational Circuits Flip-Flops Sequential Circuits Memor Components Integrated Circuits BASIC LOGIC BLOCK - GATE - Logic
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 informationGoals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations
Introduction to Electrical Engineering, II LETURE NOTES #2 Instructor: Email: Telephone: Office: ndrew. Kahng (lecture) abk@ucsd.edu 858-822-4884 office 3802 P&M lass Website: http://vlsicad.ucsd.edu/courses/ece20b/wi04/
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 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 informationUniversity of Toronto Faculty of Applied Science and Engineering Department of Electrical and Computer Engineering Midterm Examination
University of Toronto Faculty of Applied Science and Engineering Department of Electrical and Computer Engineering Midterm Eamination ECE 241F - Digital Systems Wednesday October 11, 2006, 6:00 7:30 pm
More informationLOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D.
Basic Eperiment and Design of Electronics LOGIC GATES Ho Kyung Kim, Ph.D. hokyung@pusan.ac.kr School of Mechanical Engineering Pusan National University Outline Boolean algebra Logic gates Karnaugh maps
More 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 information