CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

Size: px
Start display at page:

Download "CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation"

Transcription

1 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 to oserve logic function, DC chrcteristics, nd AC chrcteristics The MOS Trnsistor Almost ll of modern digitl electronics re uilt from one simple device, the metloide-semiconductor (MOS) field-effect trnsistor. MOS trnsistors come in two flvors, n-chnnel trnsistors sometimes clled NFETs, nd p-chnnel trnsistors (PFETs). A FET hs three terminls the gte, the, nd the drin. The schemtic symol for n NFET is shown elow with the terminls leled. The schemtic symol does not distinguish etween the nd drin terminls. This is ecuse they re interchngele, for n NFET, whichever terminl is more negtive is the nd the other is the drin. gte drin gte drin closed if gte= nd = open if gte= otherwise undefined For the purposes of understnding logic circuits, we cn model the NFET s switch etween the nd the drin tht is controlled y the gte. When the gte voltge is high (logic ) nd the voltge is low (logic ), the switch is closed effectively connecting the nd drin. When the gte is low (logic ), the switch is open disconnecting the nd drin. When the gte is high (logic ) nd the more negtive of the other two terminls is not t zero, the stte of the trnsistor is undefined. For this reson, we cn use n NFET to pss zeros, ut not ones we use PFET to pss s. gte drin gte closed if gte= nd = open if gte= otherwise undefined drin The PFET works ectly the sme wy n NFET does if you reverse nd. The PFET is open (or off) when the gte is high nd closed (or on) when the gte is low nd the is high hence it cn pss logic or not under control of the gte. The schemtic symol for PFET is shown ove long with our switch model for the PFET. For the PFET the is the more positive of the two interchngele terminls, so we tend to drw it on the top (y convention we drw schemtics with voltge decresing from top to ottom nd signls propgting from left to right). The symol for the PFET is the sme s the NFET ecept tht it includes n inversion ule on the gte terminl.

2 This ule indictes logicl inversion from to or to indicting tht the device is on when the gte is s opposed to the NFET which is on when the gte is. Series nd prllel switches We cn perform logicl function y connecting two switches or two FETs together in series or prllel. For emple, terminl c t the left elow will e connected to if AND re oth. Similrly, terminl f t the right elow will e connected to if either d OR e (or oth of them) is. c d e f We cn uild similr series nd prllel circuits out of PFETs. Also, while these re 2- input series nd prllel networks, we cn clerly etend these circuits to hndle n ritrry numer of inputs. Prep question : Drw n NFET network tht connects the output to when AND ( OR c) is true ( ( c) in shorthnd). Mking n Inverter To e composle circuit must generte n output tht is suitle for use s n input to similr circuit. A simple series or prllel comintion of NFETs s shown ove won t do this since the input needs to e either or, ut the output is either or open circuited. We need to dd PFETs (or resistor) to generte on the output in this stte. The simplest composle circuit is the inverter, shown ove long with its schemtic symol. When input is, the NFET is on nd the PFET is off, so output is. Similrly, when input is, output is connected to vi the PFET nd the NFET is off. To sve time in drwing, we use the schemtic symol on the left to indicte n inverter nd omit the power supply connections (to nd ). In short, the inverter genertes the logicl inverse of its input. We sy tht = NOT(), or = ~ (for shorthnd).

3 Logic Functions nd Boolen Alger While inverters re useful (signls never seem to e the polrity you wnt) we need to comine multiple logic signls to compute most functions. We cn specify logicl function of severl vriles in severl wys including n eqution or truth tle. For emple, the eqution for the NAND (not nd) function is = NOT( AND ), or = ~( ) for shorthnd. A truth tle shows the vlue of the function for every possile comintion of input vlues. For emple, the truth tle for the NAND function is: Sometimes it is convenient to write our truth-tle out in two dimensions with the vlues of long one is nd the vlues of long the other. This form of truth tle for the NAND is shown elow nd is clled Krnugh (pronounced Cr-gnw) mp. Just like we re used to using lgeric identities to simplify equtions using + nd * nd rel numer vriles, we cn use the identities of Boolen Alger to simplify equtions using NOT (~) AND ( ), nd OR ( ) nd inry-vlued vriles. The sic lws of Boolen Alger re Zero = = Identity = = Negtion ~ = ~ = ~(~) = ~ = ~ = Commuttivity = = Associtivity ( c) = ( ) c ( c) = ( ) c Distriutivity ( c) = ( ) ( c) ( c) = ( ) ( c) Idempotence = = De Morgn s ~( ) = ~ ~ ~( ) = ~ ~ Note tht these rules re similr to, ut not identicl to those of lger over + nd *. For emple, we cn distriute OR over AND nd AND over OR, ut we cn only distriute * over + nd not the other wy round. Also, + nd * re certinly not idempotent. For this reson I discourge people from using + nd * to represent OR nd AND respectively lthough it is common prctice. We cn esily convert ck nd forth etween truth tles nd equtions. To write the truth tle for n eqution, just sustitute ll possile vlues of the input vriles into the eqution nd evlute the resulting epression. Ech set of vlues gives one row of the

4 truth tle. To convert from truth tle to n eqution, we cn write the logic function in norml form s sum of products (n OR of ANDs). For ech line of the truth tle for which the output is, write down the AND tht corresponds to tht line (e.g., ~ ~ corresponds to the line =, =) nd OR the resulting ANDs together. For emple, the norml form for NAND gte is (~ ~) (~ ) ( ~). We cn then pply the lws of Boolen lger to simplify this epression to ~ ~. Prep question 2: Write down the norml form nd simplify the eqution for the following truth tle. Logic Gtes To implement logicl function of multiple signls, we uild logic gte, s shown elow, y replcing the PFET of the inverter with network (e.g., series or prllel) of PFETs tht pulls up (connects the output to ) when some logicl function, f, of the inputs is true, so the output,, is when f(,, ) is true. To hndle the cse when f is flse, we replce the NFET of the inverter with network of NFETs tht pulls down the output (connects it to ) when f is flse, so the output is when ~f(,, ) is true. PFET Network NFET Network We mke NAND gte (or not-nd) gte, s shown elow, y using series network for the pull-down side of the gte so the output is when AND re oth (in shorthnd, when = ). We use prllel network of PFETs to pull up the output when OR is zero (in shorthnd when ~ ~ = ). Thus, for this connection, f = ~( ) = ~ ~. nd ~f =. The circuit digrm for the NAND nd its schemtic symols re shown elow. The schemtic symol on the left reflects the = ~( ) interprettion the squred gte symol mens AND nd the ule (s ove) mens NOT, so = NOT( AND ) or ~( ). The symol on the right is the = ~ ~ interprettion, the rounded gte symol mens OR nd the inversion ules gin men not, so = (NOT ) OR (NOT

5 ) or ~ ~. This is the sme function s ~( ) write out the truth tles to convince yourself if you re not sure. Another useful gte is the NOR gte which hs the eqution = ~( ). The two symols for the NOR gte re shown elow long with the truth tle for NOR Prep question 3: Sketch trnsistor digrm for 2-input NOR gte. At this point, you my hve noticed tht ll of the gtes we re mking re inverting. This is ecuse the NFETs tke input to generte output nd the PFETs tke input to mke output. Any sttic CMOS gte you mke ccording to the rules ove will e inverting. To mke non-inverting gte, like n AND we need to use two levels of gtes s shown elow. The figure t the left shows how we relize the AND function with NAND gte nd n inverter. Note tht we cn drw n inverter with the ule on either side nd here we oey the convention tht we connect ules to ules to preserve the polrity of the logic. The figure t the right is the schemtic symol for n AND gte. Finlly we lso show the truth tle for n AND. ~ As noted in the reding, n inverting gte like NAND is complete in tht we cn uild ny function out of just NAND gtes. This is not true of AND gtes since there is no wy to mke n inverter out of n AND.

6 Prep question 4: Show how to crete n OR gte y composing inverting gtes. Another useful logic function is eclusive-or, sometimes clled XOR. The eqution for XOR is = ( ~) (~ ) nd we sometimes revite XOR y writing =. The truth tle for XOR is shown elow in oth forms. Prep question 5 (optionl chllenge question): Sketch n implementtion of n XOR gte using NFETs nd PFETs. A prize goes to the solution with the minimum numer of trnsistors. The CD47 To eperiment with uilding sttic CMOS logic gtes from MOS trnsistors, we will e using the Firchild CD47 integrted circuit. As shown in the pinout elow, this device consists of three NFETs nd three PFETs with some of their terminls tied together. Note tht you must tie pin 4 to (Vdd) nd pin 7 to (GND) for this prt to work properly. Using this device, we will wire up some simple logic gtes nd chrcterize their AC nd DC chrcteristics. The gtes we will wire up will e. An inverter 2. A 2-input NAND gte 3. A ~( ( c)) gte (see prep question ). 4. (optionl) Your XOR gte from prep question 5

7 In preprtion for the l, sketch how you will wire up these gtes using the CD47. Specificlly, drw schemtic showing the pins used y the, gte, nd drin of ech trnsistor. Once you wire up these three gtes, you will evlute ech of them using the following three steps.. Verify their logicl opertion using switch inputs nd n LED s output. Note tht due to the low drive strength of the CD47, you my need to uffer the LED using your 74AC4. 2. Trce the DC trnsfer curve input voltge vs. output voltge for t lest one of your gtes. 3. Oserve the AC trnsfer curve input nd output wveforms vs time.

Boolean Algebra. Boolean Algebra

Boolean 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 information

Combinational Logic. Precedence. Quick Quiz 25/9/12. Schematics à Boolean Expression. 3 Representations of Logic Functions. Dr. Hayden So.

Combinational 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 information

Fast Boolean Algebra

Fast 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 information

Overview of Today s Lecture:

Overview 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 information

expression simply by forming an OR of the ANDs of all input variables for which the output is

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

More information

Boolean algebra.

Boolean 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 information

Introduction to Electrical & Electronic Engineering ENGG1203

Introduction 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

ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Fall Notes - Unit 1

ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Fall Notes - Unit 1 INTRODUTION TO LOGI IRUITS Notes - Unit 1 OOLEN LGER This is the oundtion or designing nd nlyzing digitl systems. It dels with the cse where vriles ssume only one o two vlues: TRUE (usully represented

More information

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors: Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

More information

Control with binary code. William Sandqvist

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 information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

EECS 141 Due 04/19/02, 5pm, in 558 Cory

EECS 141 Due 04/19/02, 5pm, in 558 Cory UIVERSITY OF CALIFORIA College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Lst modified on April 8, 2002 y Tufn Krlr (tufn@eecs.erkeley.edu) Jn M. Rey, Andrei Vldemirescu Homework

More information

ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-378: Computer Hardware Design Winter Notes - Unit 1

ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-378: Computer Hardware Design Winter Notes - Unit 1 ELETRIL ND OMPUTER ENGINEERING DEPRTMENT, OKLND UNIVERSIT EE-78: omputer Hrdwre Design Winter 016 INTRODUTION TO LOGI IRUITS Notes - Unit 1 OOLEN LGER This is the oundtion or designing nd nlyzing digitl

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

September 13 Homework Solutions

September 13 Homework Solutions College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Note 12. Introduction to Digital Control Systems

Note 12. Introduction to Digital Control Systems Note Introduction to Digitl Control Systems Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd . Introduction A digitl control system is one in which the

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

M A T H F A L L CORRECTION. Algebra I 2 1 / 0 9 / U N I V E R S I T Y O F T O R O N T O

M A T H F A L L CORRECTION. Algebra I 2 1 / 0 9 / U N I V E R S I T Y O F T O R O N T O M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #1 CORRECTION Alger I 2 1 / 0 9 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O 1. Suppose nd re nonzero elements of field F. Using only the field xioms,

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set 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 information

Lecture Solution of a System of Linear Equation

Lecture Solution of a System of Linear Equation ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Boolean Algebra. Boolean Algebras

Boolean 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 information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 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 information

AP Calculus AB Summer Packet

AP Calculus AB Summer Packet AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

More information

Reasoning and programming. Lecture 5: Invariants and Logic. Boolean expressions. Reasoning. Examples

Reasoning and programming. Lecture 5: Invariants and Logic. Boolean expressions. Reasoning. Examples Chir of Softwre Engineering Resoning nd progrmming Einführung in die Progrmmierung Introduction to Progrmming Prof. Dr. Bertrnd Meyer Octoer 2006 Ferury 2007 Lecture 5: Invrints nd Logic Logic is the sis

More information

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and. Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

How 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?

How 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 out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Quadratic Forms. Quadratic Forms

Quadratic Forms. Quadratic Forms Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

IST 4 Information and Logic

IST 4 Information and Logic IST 4 Informtion nd Logic T = tody x= hw#x out x= hw#x due mon tue wed thr 28 M1 oh 1 4 oh M1 11 oh oh 1 2 M2 18 oh oh 2 fri oh oh = office hours oh 25 oh M2 2 3 oh midterms oh Mx= MQx out 9 oh 3 T 4 oh

More information

Digital Control of Electric Drives

Digital Control of Electric Drives igitl Control o Electric rives Logic Circuits - Comintionl Boolen Alger, escription Form Czech Technicl University in Prgue Fculty o Electricl Engineering Ver.. J. Zdenek Logic Comintionl Circuit Logic

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Convert the NFA into DFA

Convert the NFA into DFA Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:

More information

Designing Information Devices and Systems I Spring 2018 Homework 7

Designing Information Devices and Systems I Spring 2018 Homework 7 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

More information

Chapter E - Problems

Chapter E - Problems Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current

More information

CS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions

CS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 9 1. (4pts) ((p q) (q r)) (p r), prove tutology using truth tles. p

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

AP Calculus AB Summer Packet

AP Calculus AB Summer Packet AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

3 Regular expressions

3 Regular expressions 3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

More information

Network Analysis and Synthesis. Chapter 5 Two port networks

Network Analysis and Synthesis. Chapter 5 Two port networks Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

More information

Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chapter 3 Single Random Variables and Probability Distributions (Part 2) Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

Resistive Network Analysis

Resistive Network Analysis C H A P T E R 3 Resistive Network Anlysis his chpter will illustrte the fundmentl techniques for the nlysis of resistive circuits. The methods introduced re sed on the circuit lws presented in Chpter 2:

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Lecture 08: Feb. 08, 2019

Lecture 08: Feb. 08, 2019 4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Chapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis

Chapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis Chpter 4: Techniques of Circuit Anlysis Terminology Node-Voltge Method Introduction Dependent Sources Specil Cses Mesh-Current Method Introduction Dependent Sources Specil Cses Comprison of Methods Source

More information

5.2 Exponent Properties Involving Quotients

5.2 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 information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

Lecture 3. Introduction digital logic. Notes. Notes. Notes. Representations. February Bern University of Applied Sciences.

Lecture 3. Introduction digital logic. Notes. Notes. Notes. Representations. February Bern University of Applied Sciences. Lecture 3 Ferury 6 ern University of pplied ciences ev. f57fc 3. We hve seen tht circuit cn hve multiple (n) inputs, e.g.,, C, We hve lso seen tht circuit cn hve multiple (m) outputs, e.g. X, Y,, ; or

More information

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 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 information

Fachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap)

Fachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap) . Boolen Alger Fchgeiet Rechnersysteme. Boolen Alger Veriiction Technology Content. Boolen lger sics (recp).2 Resoning out Boolen expressions . Boolen Alger 2 The prolem o logic veriiction: Show tht two

More information

Lecture 3: Equivalence Relations

Lecture 3: Equivalence Relations Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

More information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010 CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

More information

Designing Information Devices and Systems I Spring 2018 Homework 8

Designing Information Devices and Systems I Spring 2018 Homework 8 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission

More information

DIRECT CURRENT CIRCUITS

DIRECT 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 information

EE273 Lecture 15 Asynchronous Design November 16, Today s Assignment

EE273 Lecture 15 Asynchronous Design November 16, Today s Assignment EE273 Lecture 15 Asynchronous Design Novemer 16, 199 Willim J. Dlly Computer Systems Lortory Stnford University illd@csl.stnford.edu 1 Tody s Assignment Term Project see project updte hndout on we checkpoint

More information

CS 330 Formal Methods and Models

CS 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 information

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

Special Relativity solved examples using an Electrical Analog Circuit

Special Relativity solved examples using an Electrical Analog Circuit 1-1-15 Specil Reltivity solved exmples using n Electricl Anlog Circuit Mourici Shchter mourici@gmil.com mourici@wll.co.il ISRAE, HOON 54-54855 Introduction In this pper, I develop simple nlog electricl

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

1 2 : 4 5. Why Digital Systems? Lesson 1: Introduction to Digital Logic Design. Numbering systems. Sample Problems 1 5 min. Lesson 1-b: Logic Gates

1 2 : 4 5. Why Digital Systems? Lesson 1: Introduction to Digital Logic Design. Numbering systems. Sample Problems 1 5 min. Lesson 1-b: Logic Gates Leon : Introduction to Digitl Logic Deign Computer ided Digitl Deign EE 39 meet Chvn Fll 29 Why Digitl Sytem? ccurte depending on numer of digit ued CD Muic i digitl Vinyl Record were nlog DVD Video nd

More information

UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences

UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Eld Alon Homework #3 Solutions EECS4 PROBLEM : CMOS Logic ) Implement the logic function

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1 MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee

CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Eercise

More information

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims

More information

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

More information

7. Indefinite Integrals

7. Indefinite Integrals 7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find

More information

Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

Exercises with (Some) Solutions

Exercises with (Some) Solutions Exercises with (Some) Solutions Techer: Luc Tesei Mster of Science in Computer Science - University of Cmerino Contents 1 Strong Bisimultion nd HML 2 2 Wek Bisimultion 31 3 Complete Lttices nd Fix Points

More information

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018 CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

y = f(x) This means that there must be a point, c, where the Figure 1

y = f(x) This means that there must be a point, c, where the Figure 1 Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

More information

Section 20.1 Electric Charge and Static Electricity (pages )

Section 20.1 Electric Charge and Static Electricity (pages ) Nme Clss Dte Section 20.1 Electric Chrge nd Sttic (pges 600 603) This section explins how electric chrge is creted nd how positive nd negtive chrges ffect ech other. It lso discusses the different wys

More information

HW3, Math 307. CSUF. Spring 2007.

HW3, 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 information