Good Review book ( ) ( ) ( )
|
|
- Arron Powers
- 6 years ago
- Views:
Transcription
1 7/31/ Boolen (Switching) Algebr Review Good Review book BeBop to the Boolen Boogie: An Unconventionl Guide to Electronics, 2 nd ed. by Clive Mxwell Hightext Publictions Inc. from Amzon.com for pprox. $27.14 ISBN Theorems nd Postultes + 0= 0 = 0 = ( ) ( ) + bc = + b + c + = 1 = 0 + = = + 1= 1 1 = ( ) + b = + b = + b = + b + b = ( ) b ( ) ( ) ( ) ( + ) = ( + ) ( + c) b + b = + b + b = b + bc = b + c + b + b c b + b= b b = + b b + c + bc = b + c + ( b) ( + c) ( b + c) = ( + b) ( + c) The bove tble is vilble for downlod in printble version from my homepge! While you re t it, print out the No. of 1's document which you will need for the Quine-McCluskey topics lter. In our review of Boolen Algebr, I ll be minly teching by exmple nd will be using the Theorems nd Postultes shown bove. A certin level of lgebr experience is ssumed, so the first six rows of rules re not going to be introduced.
2 7/31/ Boolen Exmple 1 (Truth tble): Problem Sttement: Use Truth tble to Prove : f (, ) b = + b = + b b b + b = + b = = = = 1 QE.. D.( Thus it is proven) Boolen Exmple 2 Problem Sttement Simplify the following switching function using: A + AB = A f (,b,c) b + = + b + c = + b + c = + c = = m - no gtes 1 level circuit - A WIR E! ( 4,5,6, 7 ) = M( 0,1,2, 3)
3 7/31/ This problem cn be solved nother wy. Boolen Exmple 3 Problem Sttement: Simplify the problem from the lst exmple by fctoring 1 st. f (,b,c ) = b+ + b+ c = x+ 1= 1 b 1 b c = ns Note tht the result for both exmples is the sme. Boolen Exmple 4 Problem Sttement: Simplify the following switching expression. (, b, c) Simplify f = c + bc fctor out the c = c ( + b ) ( b) = c + = c + bc ns
4 7/31/ MultiSIM Exmple 2-1 Problem Sttement: Use MultiSIMs Logic Converter to verify the results from the circuit shown. Once the circuit hs been built, connect the inputs nd output to the Logic Converter. Be creful to insure tht the inputs re connected with the MSB on the frthest input to the left. The input t the fr right end is where the circuit output is connected. This device is only used with circuits which only hve single output. Once the circuit is connected, select the Circuit to Tble button (see bove right). As cn be seen, the resulting tble mtches the tble on the previous pge.
5 7/31/ Boolen Exmple 5 Problem Sttement: Prove tht the following Theorem is true: ( ) ( ) ( ) ( ) ( ) f X,Y,Z = X + Y X + Z = X + YZ X + Y X + Z = X + YZ XX + XZ + XY + YZ = X + YZ X X + XZ + XY + YZ = X + YZ X X + XY X + YZ = X + YZ X + YZ = X + YZ Boolen Exmple 6 Problem Sttement: Simplify f( x, y,z) = m( 0, 3, 4,5, 7) ( x, y, z) f = x y z + xyz + xy z + xyz + xyz = xyz + x y z + xy z + xyz + xyz = xyz + (x + x) y z + x (y + y) z (+ = 1) (+ = 1) = xyz + y z + xz = yz + xz + xyz b b c (b+ bc = b+ c) = yz+ xz + yz reorder ns
6 7/31/ Note the second to lst line of the lst exmple. = yz+ x z + xyz b b c theorem (b + bc = b + c) = y z + xz + yz An ttempt hs been mde to show how tht prt of the expression mtches one of the theorems. Note how the (x) hs been ssigned to (b) in the theorem nd the ( X ) hs been ssigned to the ( b ) in the theorem. Let s tke look t how this ssignment might chnge if the problem chnged slightly: All tht hppened ws the (x) nd the ( X ) trded plces. Note tht the ssignment of (b) in the theorem is now mde to the ( X ) vice the (x) nd note the chnge in the nswer. The student might be tempted to ssign the ( b ) from the theorem to the ( X ), but this would be wrong. = y z + x z+ x yz b b c theorem ( b + bc = b + c ) = y z + x z + yz
7 7/31/ The following is n exmple of how n SOP (Sum of Products) expression cn be converted into n expression which would only require NAND gtes to implement by using De Morgn s Theorem. Boolen Exmple 7 (NAND ONLY LOGIC) Problem Sttement: Implement f( x, y,z) = y z + xz + yz in NAND only logic. st 1, double negte both sides of the eqution. We know tht = so if we double negte both sides, the expression will remin UNCHANGED! f(x, y,z) = y z + xz + yz Once this hs been done, perform DeMorgn's on the bottom negtion, only pplying the theorem to the signs of the expresssion ( y z) ( xz) ( yz) = the resulting expression cn be inplemented by using ONLY NAND gtes. ANSWER
8 7/31/ The designer cn use the sme technique to implement POS (Product of Sums) expression s NOR only circuit. Boolen Exmple 8 (NOR ONLY LOGIC) Problem Sttement: Implement the following expression in NOR only logic. f( x, y,z) (x y + z)( y ( x, y, z) = + + = f (x + y + z)(y + z) As before, only pply De'Morgn's to the bottom negtion nd then only to the signs in the expression. z) = (x + y + z) + (y + z) As noted erlier, the min-term list cn esily be converted into n lgebric SOP expression. If POS expression is desired, express the min-term list s mx-term list. Then the simplifiction cn be crried out. Boolen Exmple 9 Problem Sttement: Implement nd simplify the following expression in POS formt. f ( x, y, z) = m(0,3, 4,5,7) = M(1, 2, 6) = ( x + y + z)( x + y + z)( x + y + z) b b ( + b)( + b) = = ( x + y + z)( y + z) Answer This exmple lso shows simplifiction technique which cn be very helpful. Note tht the exmple equtes + y z to the in the theorem ( ) ( ) + b + b =.
9 7/31/ MultiSIM Exmple 2-2 Problem Sttement: Use MultiSIMs Logic Converter to verify the proof: Prove : f (, b) = + b = + b Open the Logic Converter nd select the two bits to the left s shown bove. Then type the eqution in the entry re t the bottom of the device. Use ( ) when negtion is required (see bove right). Once the eqution is complete, select the A B Tble button. The resulting tble is shown below:
10 7/31/ Boolen Exmple 1 0 Problem Sttement: Find the miniml POS Boolen Expression f ( xyz,, ) = ( x + z) ( x + y) ( x + y + z) ( x + y + z) reorder nd pply two different theorems x y x y z b = ( + b) = x + z x y z + + b b c ( + b) + b+ c = ( + b)( + c) = ( x + y) ( x + z) ( y + z )
11 7/31/ MultiSIM Exmple 2-3 Problem Sttement: Use MultiSIM s Logic Converter to verify the simplifiction from the previous exmple. In order to progrm this exmple into MultiSIM you first need to convert the (x, y, z) switching list into n (A, B, C) switching list.? ( xyz,, ) = ( + ) ( + ) ( + + ) ( + + ) = ( x + y) ( x + z) ( y + z) f x z x y x y z x y z? ( ABC,, ) ( ) ( ) + = ( ) f = A+ C A+ B A+ B + C A+ B C A+ B A+ C B + C Next, open the Logic Converter, select the A, B, nd C columns in the tble nd progrm the eqution into the entry box. Don t forget to use the ( ) s the negtion symbol. Once you progrm the eqution into the converter, select the A B Tble button. The result cn be seen in the figure bove. simple Finlly, select the " Tble A B " button. The simplified eqution will pper in the eqution entry re s shown below. Note tht the result is s predicted.
12 7/31/ Boolen Exmple 1 1 Problem Sttement: Simplify the following expression: (,,, ) = ( ) ( ) + ( ) ( ) f A B C D AC AD CD BD A C A D C D B D C A D B = ( A C) ( A D) ( C D) ( B D) = = AA + A D + AC + CD + BC + C D + BD + DD 0 0 = AD+ AC + CD + C D + BC + BD remember D ( C + C) 1 tht ny vrible AND'd with "1" is equl to tht vrible, so D = A D + AC + B C + B D + D b + b = + b B + D = A D + AC + BC + B + D = + + AC + + ( 1 ) reorder.... A D D B C B b b + b = = + b = = = D D B = D + B + AC nswer Implement s NAND only expression = D + B + AC = ( D) ( B) ( AC) ( D) ( B) ( AC) = nswer
13 7/31/ MultSIM Exmple 2-4 Problem Sttement: Verify the simplifiction in the previous exmple using Multisim.
Control with binary code. William Sandqvist
Control with binry code Dec Bin He Oct 218 10 11011010 2 DA 16 332 8 E 1.1c Deciml to Binäry binry weights: 1024 512 256 128 64 32 16 8 4 2 1 71 10? 2 E 1.1c Deciml to Binäry binry weights: 1024 512 256
More informationBoolean Algebra. Boolean Algebras
Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero
More informationOverview of Today s Lecture:
CPS 4 Computer Orgniztion nd Progrmming Lecture : Boolen Alger & gtes. Roert Wgner CPS4 BA. RW Fll 2 Overview of Tody s Lecture: Truth tles, Boolen functions, Gtes nd Circuits Krnugh mps for simplifying
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationLinearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationMathcad Lecture #1 In-class Worksheet Mathcad Basics
Mthcd Lecture #1 In-clss Worksheet Mthcd Bsics At the end of this lecture, you will be ble to: Evlute mthemticl epression numericlly Assign vrible nd use them in subsequent clcultions Distinguish between
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationCH 9 INTRO TO EQUATIONS
CH 9 INTRO TO EQUATIONS INTRODUCTION I m thinking of number. If I dd 10 to the number, the result is 5. Wht number ws I thinking of? R emember this question from Chpter 1? Now we re redy to formlize the
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 informationReview Factoring Polynomials:
Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
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 informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationMATH FIELD DAY Contestants Insructions Team Essay. 1. Your team has forty minutes to answer this set of questions.
MATH FIELD DAY 2012 Contestnts Insructions Tem Essy 1. Your tem hs forty minutes to nswer this set of questions. 2. All nswers must be justified with complete explntions. Your nswers should be cler, grmmticlly
More informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
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 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 informationBasic Derivative Properties
Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0
More informationBoolean algebra.
http://en.wikipedi.org/wiki/elementry_boolen_lger Boolen lger www.tudorgir.com Computer science is not out computers, it is out computtion nd informtion. computtion informtion computer informtion Turing
More informationHow do you know you have SLE?
Simultneous Liner Equtions Simultneous Liner Equtions nd Liner Algebr Simultneous liner equtions (SLE s) occur frequently in Sttics, Dynmics, Circuits nd other engineering clsses Need to be ble to, nd
More informationFast Boolean Algebra
Fst Boolen Alger ELEC 267 notes with the overurden removed A fst wy to lern enough to get the prel done honorly Printed; 3//5 Slide Modified; Jnury 3, 25 John Knight Digitl Circuits p. Fst Boolen Alger
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:45-10:45m SH 1607 Mth Lb Hours: TR 1-2pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
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 Exclusive-OR and Equivalence Operations 3.3 The Consensus
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point
More informationOperations with Matrices
Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationUNIT-4. File downloaded from For the things of this world cannot be made known without a knowledge of mathematics.
File downloded from http://jsuniltutoril.weely.com UNIT- For the things of this world cnnot e mde known without knowledge of mthemtics.. Solve y fctoriztion. - + ( ) = 0 Ans: + ( - ) = 0. [( + ) + ( -
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationLinear Algebra 1A - solutions of ex.4
Liner Algebr A - solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists - ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix
More informationExponentials - Grade 10 [CAPS] *
OpenStx-CNX module: m859 Exponentils - Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationCS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
More informationCOMPUTER SCIENCE TRIPOS
CST.2011.2.1 COMPUTER SCIENCE TRIPOS Prt IA Tuesdy 7 June 2011 1.30 to 4.30 COMPUTER SCIENCE Pper 2 Answer one question from ech of Sections A, B nd C, nd two questions from Section D. Submit the nswers
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two- and three-variable Four-variable Determination of minimum expressions using essential prime implicants Five-variable
More informationIntroduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationWhat else can you do?
Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright
More informationMath 130 Midterm Review
Mth 130 Midterm Review April 6, 2013 1 Topic Outline: The following outline contins ll of the mjor topics tht you will need to know for the exm. Any topic tht we ve discussed in clss so fr my pper on the
More informationCS 330 Formal Methods and Models
CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 2 1. Prove ((( p q) q) p) is tutology () (3pts) y truth tle. p q p q
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationIST 4 Information and Logic
IST 4 Informtion nd Logic T = tody x= hw#x out x= hw#x due mon tue wed thr fri 31 M1 1 7 oh M1 14 oh 1 oh 2M2 21 oh oh 2 oh Mx= MQx out Mx= MQx due 28 oh M2 oh oh = office hours 5 3 12 oh 3 T 4 oh oh 19
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationSection 3.1: Exponent Properties
Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted
More informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
More informationCombinational Logic. Precedence. Quick Quiz 25/9/12. Schematics à Boolean Expression. 3 Representations of Logic Functions. Dr. Hayden So.
5/9/ Comintionl Logic ENGG05 st Semester, 0 Dr. Hyden So Representtions of Logic Functions Recll tht ny complex logic function cn e expressed in wys: Truth Tle, Boolen Expression, Schemtics Only Truth
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationBoolean Algebra. Boolean Variables, Functions. NOT operation. AND operation. AND operation (cont). OR operation
oolean lgebra asic mathematics for the study of logic design is oolean lgebra asic laws of oolean lgebra will be implemented as switching devices called logic gates. Networks of Logic gates allow us to
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationDually quasi-de Morgan Stone semi-heyting algebras II. Regularity
Volume 2, Number, July 204, 65-82 ISSN Print: 2345-5853 Online: 2345-586 Dully qusi-de Morgn Stone semi-heyting lgebrs II. Regulrity Hnmntgoud P. Snkppnvr Abstrct. This pper is the second of two prt series.
More informationCS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation
CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationHQPD - ALGEBRA I TEST Record your answers on the answer sheet.
HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property
More informationMidterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes
Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name
More informationSection 3.2: Negative Exponents
Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.
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 informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More informationAn Introduction to Trigonometry
n Introduction to Trigonoetry First of ll, let s check out the right ngled tringle below. The LETTERS, B & C indicte the ngles nd the letters, b & c indicte the sides. c b It is iportnt to note tht side
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationIn this skill we review equations that involve percents. review the meaning of proportion.
6 MODULE 5. PERCENTS 5b Solving Equtions Mening of Proportion In this skill we review equtions tht involve percents. review the mening of proportion. Our first tsk is to Proportions. A proportion is sttement
More information7.5 Integrals Involving Inverse Trig Functions
. integrls involving inverse trig functions. Integrls Involving Inverse Trig Functions Aside from the Museum Problem nd its sporting vritions introduced in the previous section, the primry use of the inverse
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationdf dt f () b f () a dt
Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem
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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
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 informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationIntroduction to Electrical & Electronic Engineering ENGG1203
Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 27-8 Dr. Hden Kwok-H So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll
More information