Boolean Algebra. Boolean Algebra

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Boolean Algebra. Boolean Algebra"

Transcription

1 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 one, such tht for every element of B: nd In ddition, certin ioms must e stisfied: - closure properties for oth inry opertions nd the unry opertion - ssocitivity of ech inry opertion over the other, - commuttivity of ech ech inry opertion, - distriutivity of ech inry opertion over the other, - sorption rules, - eistence of complements with respect to ech inry opertion We will ssume tht hs higher precedence thn +; however, this is not generl rule for ll Boolen lgers. Computer Orgniztion McQuin

2 Boolen Alger Computer Orgniztion McQuin Aioms of Boolen Alger Associtive Lws: for ll, nd c in B, ( ( c c ( ( c c ( ( ( c c ( ( ( c c ( ( Commuttive Lws: for ll nd in B, Distriutive Lws: for ll, nd c in B, Asorption Lws: for ll, nd c in B, Eistence of Complements: for ll in B, there eists n element ā in B such tht

3 Emples of Boolen Algers Boolen Alger 3 The clssic emple is B = {true, flse} with the opertions AND, OR nd NOT. An isomorphic emple is B = {, } with the opertions +, nd ~ defined y: + * ~ Given set S, the power set of S, P(S is Boolen lger under the opertions union, intersection nd reltive complement. Other, interesting emples eist Computer Orgniztion McQuin

4 More Properties It's lso possile to derive some dditionl fcts, including: - the elements nd re unique - the complement of n element is unique - nd re complements of ech other Boolen Alger 4 Computer Orgniztion McQuin

5 DeMorgn's Lws & More Boolen Alger 5 DeMorgn's Lws re useful theorems tht cn e derived from the fundmentl properties of Boolen lger. For ll nd in B, Of course, there s lso doule-negtion lw: And there re idempotency lws: Boundedness properties: Computer Orgniztion McQuin

6 Logic Epressions nd Equtions Boolen Alger 6 A logic epression is defined in terms of the three sic Boolen opertors nd vriles which my tke on the vlues nd. For emple: z : y y y : ( 2 ( 2 3 ( 2 3 A logic eqution is n ssertion tht two logic equtions re equl, where equl mens tht the vlues of the two epressions re the sme for ll possile ssignments of vlues to their vriles. For emple: y y y y Of course, equtions my e true or flse. Wht out the one ove? Computer Orgniztion McQuin

7 Boolen Alger Computer Orgniztion McQuin Why do they cll it "lger"? A Boolen epression cn often e usefully trnsformed y using the theorems nd properties stted erlier: y y y y y y y y Tht is reltively simple emple of reduction. Try showing the following epressions re equl: z y z y

8 Why do they cll it "lger"? Here's nother tht hppens to e relted to inry ddition: Boolen Alger 8 Z A B C A B C A B C A B C Given A B C A B C A B C A B C A B C A B C Idempotence A B C A B C A B C A B C A B C A B C Commuttivity, Associtivity A AB C AC B B AB C C Commuttivity, Distriutivity B C AC A B Complements AC B C AB Boundedness Computer Orgniztion McQuin

9 Tutologies, Contrdictions & Stisfiles Boolen Alger 9 A tutology is Boolen epression tht evlutes to true ( for ll possile vlues of its vriles. A contrdiction is Boolen epression tht evlutes to flse ( for ll possile vlues of its vriles. A Boolen epression is stisfile if there is t lest one ssignment of vlues to its vriles for which the epression evlutes to true (. Computer Orgniztion McQuin

10 Truth Tles Boolen Alger A Boolen epression my e nlyzed y creting tle tht shows the vlue of the epression for ll possile ssignments of vlues to its vriles: Computer Orgniztion McQuin

11 Proving Equtions with Truth Tles Boolen Alger Boolen equtions my e proved using truth tles (dull nd mechnicl: c c + c ~(**c ~*~*~c Computer Orgniztion McQuin

12 Proving Equtions Algericlly Boolen Alger 2 Boolen equtions my e proved using truth tles, which is dull nd oring, or using the lgeric properties: B, sorption, with lw of complements B, sorption, with lw of complements Note the dulity Computer Orgniztion McQuin

13 Proving Equtions Algericlly Boolen Alger 3 B, sorption, with ( lw of complements sorption, with = Computer Orgniztion McQuin

14 Sum-of-Products Form Boolen Alger 4 A Boolen epression is sid to e in sum-of-products form if it is epressed s sum of terms, ech of which is product of vriles nd/or their complements: It's reltively esy to see tht every Boolen epression cn e written in this form. Why? The summnds in the sum-of-products form re clled minterms. - ech minterm contins ech of the vriles, or its complement, ectly once - ech minterm is unique, nd therefore so is the representtion (side from order Computer Orgniztion McQuin

15 Emple Boolen Alger 5 Given truth tle for Boolen function, construction of the sum-of-products representtion is trivil: - for ech row in which the function vlue is, form product term involving ll the vriles, tking the vrile if its vlue is nd the complement if the vrile's vlue is - tke the sum of ll such product terms y z F y z y z y z yz F y z y z y z y z Computer Orgniztion McQuin

16 Product-of-Sums Form Boolen Alger 6 A Boolen epression is sid to e in product-of-sums form if it is epressed s product of terms, ech of which is sum of vriles: Every Boolen epression cn lso e written in this form, s product of mterms. Fcts similr to the sum-of-products form cn lso e sserted here. The product-of-sums form cn e derived y epressing the complement of the epression in sum-of-products form, nd then complementing. Computer Orgniztion McQuin

17 Emple Boolen Alger 7 Given truth tle for Boolen function, construction of the product-of-sums representtion is trivil: - for ech row in which the function vlue is, form product term involving ll the vriles, tking the vrile if its vlue is nd the complement if the vrile's vlue is - tke the sum of ll such product terms; then complement the result y z F y z y z y z y z F y z y z y z y z F y z y z y z y z ( y z ( y z ( y z ( y z Computer Orgniztion McQuin

18 Boolen Functions Boolen Alger 8 A Boolen function tkes n inputs from the elements of Boolen lger nd produces single vlue lso n element of tht Boolen lger. For emple, here re ll possile 2-input Boolen functions on the set {, }: A B zero nd A B or or A B nor eq B' A' nnd one Computer Orgniztion McQuin

19 Universlity Any Boolen function cn e epressed using: - only AND, OR nd NOT - only AND nd NOT - only OR nd NOT - only AND nd XOR - only NAND - only NOR Boolen Alger 9 The first ssertion should e entirely ovious. The remining ones re ovious if you consider how to represent ech of the functions in the first set using only the relevnt functions in the relevnt set. Computer Orgniztion McQuin

INF1383 -Bancos de Dados

INF1383 -Bancos de Dados 3//0 INF383 -ncos de Ddos Prof. Sérgio Lifschitz DI PUC-Rio Eng. Computção, Sistems de Informção e Ciênci d Computção LGER RELCIONL lguns slides sedos ou modificdos dos originis em Elmsri nd Nvthe, Fundmentls

More information

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.

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

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

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

ECON 331 Lecture Notes: Ch 4 and Ch 5

ECON 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 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

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

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

Lecture 9: LTL and Büchi Automata

Lecture 9: LTL and Büchi Automata Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

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

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

More information

The Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2

The Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2 The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published

More information

a 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.

a 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 information

4.1. Probability Density Functions

4.1. Probability Density Functions STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

Fractions arise to express PART of a UNIT 1 What part of an HOUR is thirty minutes? Fifteen minutes? tw elve minutes? (The UNIT here is HOUR.

Fractions arise to express PART of a UNIT 1 What part of an HOUR is thirty minutes? Fifteen minutes? tw elve minutes? (The UNIT here is HOUR. 6- FRACTIONS sics MATH 0 F Frctions rise to express PART of UNIT Wht prt of n HOUR is thirty minutes? Fifteen minutes? tw elve minutes? (The UNIT here is HOUR.) Wht frction of the children re hppy? (The

More information

8 factors of x. For our second example, let s raise a power to a power:

8 factors of x. For our second example, let s raise a power to a power: CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,

More information

Parallel Projection Theorem (Midpoint Connector Theorem):

Parallel Projection Theorem (Midpoint Connector Theorem): rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages 5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

More information

Fact: All polynomial functions are continuous and differentiable everywhere.

Fact: All polynomial functions are continuous and differentiable everywhere. Dierentibility AP Clculus Denis Shublek ilernmth.net Dierentibility t Point Deinition: ( ) is dierentible t point We write: = i nd only i lim eists. '( ) lim = or '( ) lim h = ( ) ( ) h 0 h Emple: The

More information

Complexity in Modal Team Logic

Complexity in Modal Team Logic ThI Theoretische Informtik Complexity in Modl Tem Logic Julin-Steffen Müller Theoretische Informtik 18. Jnur 2012 Theorietg 2012 Theoretische Informtik Inhlt 1 Preliminries 2 Closure properties 3 Model

More information

Partial Differential Equations

Partial Differential Equations Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry

More information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix Eigenvalues and Eigenvectors September 13, 2017 Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

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

Continuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim

Continuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim Recll the following properties of limits. Theorem. Suppose tht lim f() =L nd lim g() =M. Then lim [f() ± g()] = L + M, lim [f()g()] = LM, if M = 0, lim f() g() = L M. Furthermore, if f() g() for ll, then

More information

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1 CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

ANALYTIC SOLUTION OF QUARTIC AND CUBIC POLYNOMIALS. A J Helou, BCE, M.Sc., Ph.D. August 1995

ANALYTIC SOLUTION OF QUARTIC AND CUBIC POLYNOMIALS. A J Helou, BCE, M.Sc., Ph.D. August 1995 ANALYTIC SOLUTION OF QUARTIC AND CUBIC POLYNOMIALS By A J Helou, BCE, M.Sc., Ph.D. August 995 CONTENTS REAL AND IMAGINARY ROOTS OF CUBIC AND QUARTIC POLYNOMIALS. INTRODUCTION. COMPUTER PROGRAMS. REAL AND

More information

Fundamentals of Electrical Circuits - Chapter 3

Fundamentals of Electrical Circuits - Chapter 3 Fundmentls of Electricl Circuits Chpter 3 1S. For the circuits shown elow, ) identify the resistors connected in prllel ) Simplify the circuit y replcing prllel connect resistors with equivlent resistor.

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

Waveguide Guide: A and V. Ross L. Spencer

Waveguide Guide: A and V. Ross L. Spencer Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Solutions to Problem Set #1

Solutions to Problem Set #1 CSE 233 Spring, 2016 Solutions to Prolem Set #1 1. The movie tse onsists of the following two reltions movie: title, iretor, tor sheule: theter, title The first reltion provies titles, iretors, n tors

More information

Convex Sets and Functions

Convex Sets and Functions B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5. Suggested Solution MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

More information

Mapping the delta function and other Radon measures

Mapping the delta function and other Radon measures Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

More information

6.5 Plate Problems in Rectangular Coordinates

6.5 Plate Problems in Rectangular Coordinates 6.5 lte rolems in Rectngulr Coordintes In this section numer of importnt plte prolems ill e emined ug Crte coordintes. 6.5. Uniform ressure producing Bending in One irection Consider first the cse of plte

More information

Automata and Languages

Automata and Languages Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive

More information

E1: CALCULUS - lecture notes

E1: CALCULUS - lecture notes E1: CALCULUS - lecture notes Ştefn Blint Ev Kslik, Simon Epure, Simin Mriş, Aureli Tomoiogă Contents I Introduction 9 1 The notions set, element of set, membership of n element in set re bsic notions of

More information

Lesson 4 Linear Algebra

Lesson 4 Linear Algebra Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,

More information

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

More information

Exam 2 Solutions ECE 221 Electric Circuits

Exam 2 Solutions ECE 221 Electric Circuits Nme: PSU Student ID Numer: Exm 2 Solutions ECE 221 Electric Circuits Novemer 12, 2008 Dr. Jmes McNmes Keep your exm flt during the entire exm If you hve to leve the exm temporrily, close the exm nd leve

More information

EQUIVARIANT CHERN-SCHWARTZ-MACPHERSON CLASSES IN PARTIAL FLAG VARIETIES: INTERPOLATION AND FORMULAE

EQUIVARIANT CHERN-SCHWARTZ-MACPHERSON CLASSES IN PARTIAL FLAG VARIETIES: INTERPOLATION AND FORMULAE EQUIVARIANT CHERN-SCHWARTZ-MACPHERSON CLASSES IN PARTIAL FLAG VARIETIES: INTERPOLATION AND FORMULAE R. RIMÁNYI AND A. VARCHENKO Dedicted to Piotr Prgcz on the occsion of his 60th irthdy Astrct. Consider

More information

METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS

METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist

More information

On the degree of regularity of generalized van der Waerden triples

On the degree of regularity of generalized van der Waerden triples On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of

More information

7 - Continuous random variables

7 - Continuous random variables 7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

More information

arxiv: v3 [math.rt] 28 Mar 2017

arxiv: v3 [math.rt] 28 Mar 2017 Reltive cluster tilting ojects in tringulted ctegories Wuzhong Yng nd Bin Zhu Astrct rxiv:504.00093v3 [mth.rt] 8 Mr 07 Assume tht is Krull-Schmidt, Hom-finite tringulted ctegory with Serre functor nd cluster-tilting

More information

Discrete Time Process Algebra with Relative Timing

Discrete Time Process Algebra with Relative Timing Discrete Time Process Alger with Reltive Timing J.C.M. Beten nd M.A. Reniers Deprtment of Mthemtics nd Computing Science, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlnds

More information

Using integration tables

Using integration tables Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl

More information

Nondeterministic Finite Automata

Nondeterministic Finite Automata Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)

More information

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints) C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

More information

From the Numerical. to the Theoretical in. Calculus

From the Numerical. to the Theoretical in. Calculus From the Numericl to the Theoreticl in Clculus Teching Contemporry Mthemtics NCSSM Ferury 6-7, 003 Doug Kuhlmnn Phillips Acdemy Andover, MA 01810 dkuhlmnn@ndover.edu How nd Why Numericl Integrtion Should

More information

Exponents and Polynomials

Exponents and Polynomials C H A P T E R 5 Eponents nd Polynomils ne sttistic tht cn be used to mesure the generl helth of ntion or group within ntion is life epectncy. This dt is considered more ccurte thn mny other sttistics becuse

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

Plate Theory. Section 11: PLATE BENDING ELEMENTS

Plate Theory. Section 11: PLATE BENDING ELEMENTS Section : PLATE BENDING ELEMENTS Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A

More information

1.3 The log laws. Inverse functions. We have defined logx as the index needed to write x as a power of 10. For example:

1.3 The log laws. Inverse functions. We have defined logx as the index needed to write x as a power of 10. For example: 1.3 The log lws Inverse functions. We hve defined log s the inde needed to write s power of. For emple: since 1.8 = 66.07 we hve log66.07 = 1.8 Note tht wht this sys is tht the functions f() = nd g() =

More information

Diophantine Steiner Triples and Pythagorean-Type Triangles

Diophantine Steiner Triples and Pythagorean-Type Triangles Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 1534-1178 Diophntine Steiner Triples nd Pythgoren-Type Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer

More information

π,π is the angle FROM a! TO b

π,π is the angle FROM a! TO b Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two

More information

Ordinal diagrams. By Gaisi TAKEUTI. (Received April 5, 1957) $(a)\frac{s_{1}s_{2}}{(b)\frac{s_{3}s_{4}}{(c)\frac{s}{s}6\underline{6}}}$ Fig.

Ordinal diagrams. By Gaisi TAKEUTI. (Received April 5, 1957) $(a)\frac{s_{1}s_{2}}{(b)\frac{s_{3}s_{4}}{(c)\frac{s}{s}6\underline{6}}}$ Fig. cn Journl the Mthemticl Society Jpn Vol 9, No 4, Octor, 1957 Ordinl digrms By Gi TAKEUTI (Received April 5, 1957) In h pper [2] on the constency-pro the theory nturl numrs, G Gentzen ssigned to every pro-figure

More information

PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule

PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule PHYS 75: Clssicl echnics Sll Oscilltions: Exple A Liner Tritoic olecule A Liner Tritoic olecule x b b x x3 x Experientlly, one ight be interested in the rdition resulted fro the intrinsic oscilltion odes

More information

C1M14. Integrals as Area Accumulators

C1M14. Integrals as Area Accumulators CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms

More information

Orthogonal Polynomials and Least-Squares Approximations to Functions

Orthogonal Polynomials and Least-Squares Approximations to Functions Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

More information

Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading

Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method

More information

CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS

CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics

More information

ON A CERTAIN PRODUCT OF BANACH ALGEBRAS AND SOME OF ITS PROPERTIES

ON A CERTAIN PRODUCT OF BANACH ALGEBRAS AND SOME OF ITS PROPERTIES HE PULISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY Series A OF HE ROMANIAN ACADEMY Volume 5 Number /04 pp. 9 7 ON A CERAIN PRODUC OF ANACH ALGERAS AND SOME OF IS PROPERIES Hossossein JAVANSHIRI Mehdi

More information

PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes. Paul Carnig. January ODE s vs PDE s 1 PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

More information

Entropy and Ergodic Theory Notes 10: Large Deviations I

Entropy and Ergodic Theory Notes 10: Large Deviations I Entropy nd Ergodic Theory Notes 10: Lrge Devitions I 1 A chnge of convention This is our first lecture on pplictions of entropy in probbility theory. In probbility theory, the convention is tht ll logrithms

More information

Non-Linear & Logistic Regression

Non-Linear & Logistic Regression Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find

More information

11.1 Exponential Functions

11.1 Exponential Functions . Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

More information

Simpson s 1/3 rd Rule of Integration

Simpson s 1/3 rd Rule of Integration Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?

More information

A little harder example. A block sits at rest on a flat surface. The block is held down by its weight. What is the interaction pair for the weight?

A little harder example. A block sits at rest on a flat surface. The block is held down by its weight. What is the interaction pair for the weight? Neton s Ls of Motion (ges 9-99) 1. An object s velocit vector v remins constnt if nd onl if the net force cting on the object is zero.. hen nonzero net force cts on n object, the object s velocit chnges.

More information

This enables us to also express rational numbers other than natural numbers, for example:

This enables us to also express rational numbers other than natural numbers, for example: Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers

More information

Riemann Stieltjes Integration - Definition and Existence of Integral

Riemann Stieltjes Integration - Definition and Existence of Integral - Definition nd Existence of Integrl Dr. Adity Kushik Directorte of Distnce Eduction Kurukshetr University, Kurukshetr Hryn 136119 Indi. Prtition Riemnn Stieltjes Sums Refinement Definition Given closed

More information

Functions of bounded variation

Functions of bounded variation Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics C-level thesis Dte: 2006-01-30 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054-700 10

More information

Complementing Büchi Automata with a Subset-tuple Construction

Complementing Büchi Automata with a Subset-tuple Construction DEPARTEMENT D INFORMATIQUE DEPARTEMENT FÜR INFORMATIK Bd de Pérolles 90 CH-1700 Friourg www.unifr.ch/informtics WORKING PAPER Complementing Büchi Automt with Suset-tuple Construction J. Allred & U. Ultes-Nitsche

More information

Section 7.1 Area of a Region Between Two Curves

Section 7.1 Area of a Region Between Two Curves Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

More information

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VII

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VII Tutoril VII: Liner Regression Lst updted 5/8/06 b G.G. Botte Deprtment of Chemicl Engineering ChE-0: Approches to Chemicl Engineering Problem Solving MATLAB Tutoril VII Liner Regression Using Lest Squre

More information

Non-blocking Supervisory Control of Nondeterministic Systems via Prioritized Synchronization 1

Non-blocking Supervisory Control of Nondeterministic Systems via Prioritized Synchronization 1 Non-blocking Supervisory Control of Nondeterministic Systems vi Prioritized Synchroniztion 1 Rtnesh Kumr Deprtment of Electricl Engineering University of Kentucky Lexington, KY 40506-0046 Emil: kumr@engr.uky.edu

More information

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch.

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch. Overview H9 Vertlerouw H 9: Prsing: op-down & LL(1) do 3 mei 2001 56 heo Ruys h. 8 - Prsing 8.1 ontext-free Grmmrs 8.2 op-down Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough

More information

Chapter 2 Fundamental Concepts

Chapter 2 Fundamental Concepts Chpter 2 Fundmentl Concepts This chpter describes the fundmentl concepts in the theory of time series models In prticulr we introduce the concepts of stochstic process, men nd covrince function, sttionry

More information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We 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 information

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA CHPTER 3 LOGIC GTES & OOLEN LGER C H P T E R O U T C O M E S Upon completion of this chapter, student should be able to: 1. Describe the basic logic gates operation 2. Construct the truth table for basic

More information

Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.

Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100. Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte

More information

Point Groups and Space Groups in Geometric Algebra

Point Groups and Space Groups in Geometric Algebra Point Groups nd Spce Groups in Geometric Alger Dvid Hestenes Deprtment of Physics nd Astronomy Arizon Stte University, Tempe, Arizon, USA Astrct. Geometric lger provides the essentil foundtion for new

More information

Acta Universitatis Carolinae. Mathematica et Physica

Acta Universitatis Carolinae. Mathematica et Physica Act Universittis Croline. Mthemtic et Physic Thoms N. Vougiouklis Cyclicity in specil clss of hypergroups Act Universittis Croline. Mthemtic et Physic, Vol. 22 (1981), No. 1, 3--6 Persistent URL: http://dml.cz/dmlcz/142458

More information

Analogical Dissimilarity: definition, algorithms and first experiments in machine learning

Analogical Dissimilarity: definition, algorithms and first experiments in machine learning Anlogicl Dissimilrity: definition, lgorithms nd first experiments in mchine lerning Lurent Miclet, Arnud Delhy To cite this version: Lurent Miclet, Arnud Delhy. Anlogicl Dissimilrity: definition, lgorithms

More information

The Thermodynamics of Aqueous Electrolyte Solutions

The Thermodynamics of Aqueous Electrolyte Solutions 18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong

More information

Multiplicative Versions of Infinitesimal Calculus

Multiplicative Versions of Infinitesimal Calculus Multiplictive Versios o Iiitesiml Clculus Wht hppes whe you replce the summtio o stdrd itegrl clculus with multiplictio? Compre the revited deiitio o stdrd itegrl D å ( ) lim ( ) D i With ( ) lim ( ) D

More information

Wave Equation on a Two Dimensional Rectangle

Wave Equation on a Two Dimensional Rectangle Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c

More information

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals. MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

More information