Math 155 (Lecture 3)
|
|
- Malcolm Claud Day
- 6 years ago
- Views:
Transcription
1 Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,, }? The aswer to this questio is deoted by (, which is typically read as choose To obtai a formula for (, we first cosider a differet coutig problem: how may ways are there to choose a sequece of distict elemets of the set {,, }? I other words, how may ijective fuctios f are there from the set {,, } to the set {,,, }? This is easy to determie: there are possible values for f(), ( ) possible values for f(), ad so forth, so the umber of such fuctios is give by ( )( ) ( + ) = ( )( ) (3)()() ( ( ) (3)()() =! (! Of course, this is differet from the aswer to Questio, because we are coutig ordered sequeces of legth, rather tha elemet subsets Every elemet subset of {,,, } ca be ordered i precisely! differet ways We therefore obtai the idetity! ( ) =! (!, or ( )! =!(! Remar The above formula maes sese for ; we obviously have ( = for > Let us ow cosider aother method of determiig the itegers (, which proceeds ot by coutig directly but istead by establishig a recurrece relatio Let X, deote the collectio of all subsets of {,, } havig size, so that X, = ( We ca partitio the set X, ito two subsets Let X +, deote the collectio of all -elemet subsets of {,, } which cotai the iteger Such a subset is give by the uio of {} with a ( )-elemet subset of {,, } It follows that ( ) X +, = X, = Let X, deote the collectio of -elemet subsets of {,, } which do ot cotai the iteger We the have X, = X,, so that ( ) X, = We therefore have at least for > ( ) ( ) = X, = X +, + X, = + ( ),
2 Exercise 3 Prove the idetity ( ) ( = ) ( + ) ( directly from the formula ) =! It may be istructive to orgaize the itegers ( ito a table ( )!(! ( ) ( ) ( ) ( ) ( ) ( 3 3) ( 4 3 4), which is called Pascal s triagle The recurrece relatio ( ) ( = ( ) + dictates how to fill this table i: each etry is the sum of the two etries diagoally above it More specifically, we obtai , Let s ow see what we ca lear usig the method of geeratig fuctios itroduced i the last lecture Fix a iteger, ad defie a power series F (x) by the formula F (x) = ( ) x Sice ( = for >, this is actually a fiite sum: that is, F (x) is a polyomial fuctio of x For
3 example, whe =, we obtai F = For >, we ca use our recurrece relatio to obtai F (x) = ( ) x = ( ) x + ( ) x = x( ( ) x l + ( ) x l = (x + )F (x) It follows that F (x) = (x + ) Now tae variables y ad z, ad write (y + z) = y ( + z y ) = y F ( z y ) = y ( ) ( z y ) = ( ) y z This is a idetity of polyomials, ad therefore remais valid after substitutig ay umbers we lie for y ad z We have prove: Theorem 4 (Biomial Theorem) For ay quatities y ad z (belogig to the itegers, or to the real umbers, or more geerally ay commutative rig) ad ay, we have (y + z) = ( ) y z Because of Theorem 4, the itegers ( are ofte called biomial coefficiets I the last lecture, we metioed that geeratig fuctios should geerally be viewed as formal power series: that is, we geerally do ot care whether or ot they coverge However, we ca sometimes get iformatio by evaluatig a geeratig fuctio at a poit Let us close this lecture by givig a illustratio of this priciple Questio 5 Fix a iteger Compute the sum ( We ow describe a few differet approaches to this questio First, let s use the method of geeratig fuctios Let F (x) be defied as above, so that ( is the coefficiet of x i F (x) Let F (x) deote the derivative of F (x), so that the coefficiet of x i F (x) is ( ) It follows that ( is the sum of the coefficiets of the polyomial F (x): that is, it is the iteger F () We saw above that F (x) = (x+) It follows that F (x) = (x + ), so that F () = Let us ow try to arrive at the same aswer by combiatorial meas The idea is to iterpret ( ) as the solutio to a coutig problem Questio 6 Suppose we are give a group of people How may ways are there to select a committee (that is, a subset of the collectio of people) together with a leader of that committee (who we require belogs to the committee) Let s try aswer Questio 6 i two differet ways Suppose first that we are iterested i committees of size I this case, there are ( choices for the members of the committee, ad choices for its leader: a total of ( choices i all The aswer to Questio 6 is the give by summig over : that is, by ( Here is aother way to solve the coutig problem posed i Questio 6 First, there are choices for the leader of the committee Oce the leader is fixed, the remaiig people ca each be assiged to the committee or ot, for a total of choices We ca therefore give as a aswer to Questio 6 We coclude that ( ) = 3
4 Let us describe oe more way of arrivig at the aswer The idea is to iterpret the sum appearig i Questio 5 ot the solutio to a coutig problem, but as the solutio to a expected value problem Questio 7 Suppose that a subset S of {,, } is chose at radom What is the expected umber of elemets of S? The aswer is evidetly O the other had, we ca compute the aswer as S {,,} S Here there ( terms i the umerator where the associated summad is, so we ca rewrite the umerator as ( ) We therefore obtai ( ) = = Recall that a deragemet of the set {,, } is a permutatio of {,, } with o fixed poits: that is, a permutatio π such that π(i) i for all i Let s begi with the followig questio: Questio 8 How may deragemets are there of the set {,, }? permutatio chose at radom is a deragemet? What is the probability that a To fix ideas, let D deote the umber of deragemets of the set {,, } We have D = (the idetity permutatio of the empty set has o fixed poits), D = (the idetity permutatio of {} does have a fixed poit), D = (there is a uique deragemet of {, }, give by the o-idetity permutatio), D 3 = (the deragemets of the set {,, 3} are precisely the cyclic permutatios) Our goal ow is to obtai a formula for the itegers D As a startig poit, we ow that the total umber of permutatios of the set {,, } is! We therefore have! = D + X, where X is the set of permutatios of {,, } which have at least oe fixed poit I fact, we ca say somethig more refied For >, let X deote the collectio of all permutatios of {, } havig exactly fixed poits The X ca be writte as the disjoit uio of the subsets X, X,, X We therefore have! = D + X + X + X X Remar 9 We ca write this equatio more aturally as! = X + X + + X, where X deotes the set of permutatios with o fixed poits: that is, the set of deragemets To determie D from the above formula, we eed to ow how big the sets X are How ca we describe a permutatio π with exactly fixed poits? First, we eed to specify which elemets of {,, } are fixed poits of π: there are ( possibilities for this i all Secod, we eed to specify what our permutatio does o the remaiig elemets This restricted permutatio must be fixed poit free (otherwise, π would have more tha fixed poits), so the umber of choices is give by D We therefore obtai the formula ( ) X = D so that our earlier equatio reads! = ( ) D 4
5 Usig our formula for the biomial coefficiets ( ) =!!(!, we see that both sides of this expressio are divisible by! Dividig out, we get = D!(! The aalysis above gives a series of equatios (oe for each ) which ca be used to successively solve for the itegers D Rather tha treatig all of these equatios separately, it will be coveiet to thi about them all at oce, usig the method of geeratig fuctios For this, let us itroduce a formal variable x Multiplyig our previous equatio by x, we get x = D x!(! Summig over all, we obtai a idetity of formal power series x = D x!(! It is ow coveiet to rearrage the sum o the right had side: ote that givig a iteger ad aother iteger betwee ad is equivalet to givig a pair of oegative itegers ad l, with = +l We therefore obtai x =,l D l x +l! The right had side of this expressio factors as a product ( x! )( l =,l ) x! The first factor should be familiar: it the power series expasio for the expoetial fuctio e x Let us itroduce a otatio for the secod factor: F (x) = l = + x + x3 3 + The power series F (x) is called the expoetial geeratig fuctio for the sequece of itegers {D l } l It differs from the geeratig fuctios we have met so far because of the additio of a auxiliary factor of o the coefficiet of x l We ca ow rewrite our equatio as x = ex F (x), which is easy to solve: we get F (x) = e x x 5
Lecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 9 Set partitios ad permutatios It could be said that the mai objects of iterest i combiatorics are subsets, partitios ad permutatios of a fiite set. We have spet some time coutig
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More information(ii) Two-permutations of {a, b, c}. Answer. (B) P (3, 3) = 3! (C) 3! = 6, and there are 6 items in (A). ... Answer.
SOLUTIONS Homewor 5 Due /6/19 Exercise. (a Cosider the set {a, b, c}. For each of the followig, (A list the objects described, (B give a formula that tells you how may you should have listed, ad (C verify
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More information1. n! = n. tion. For example, (n+1)! working with factorials. = (n+1) n (n 1) 2 1
Biomial Coefficiets ad Permutatios Mii-lecture The followig pages discuss a few special iteger coutig fuctios You may have see some of these before i a basic probability class or elsewhere, but perhaps
More informationHomework 3. = k 1. Let S be a set of n elements, and let a, b, c be distinct elements of S. The number of k-subsets of S is
Homewor 3 Chapter 5 pp53: 3 40 45 Chapter 6 p85: 4 6 4 30 Use combiatorial reasoig to prove the idetity 3 3 Proof Let S be a set of elemets ad let a b c be distict elemets of S The umber of -subsets of
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationREVIEW FOR CHAPTER 1
REVIEW FOR CHAPTER 1 A short summary: I this chapter you helped develop some basic coutig priciples. I particular, the uses of ordered pairs (The Product Priciple), fuctios, ad set partitios (The Sum Priciple)
More informationarxiv: v1 [math.co] 23 Mar 2016
The umber of direct-sum decompositios of a fiite vector space arxiv:603.0769v [math.co] 23 Mar 206 David Ellerma Uiversity of Califoria at Riverside August 3, 208 Abstract The theory of q-aalogs develops
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationChapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:
Chapter 7 COMBINATIONS AND PERMUTATIONS We have see i the previous chapter that (a + b) ca be writte as 0 a % a & b%þ% a & b %þ% b where we have the specific formula for the biomial coefficiets: '!!(&)!
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 1 Coutig subsets I this sectio, we cout the subsets of a -elemet set. The coutig umbers are the biomial coefficiets, familiar objects but there are some ew thigs to say about
More informationLet us consider the following problem to warm up towards a more general statement.
Lecture 4: Sequeces with repetitios, distributig idetical objects amog distict parties, the biomial theorem, ad some properties of biomial coefficiets Refereces: Relevat parts of chapter 15 of the Math
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationCombinatorics and Newton s theorem
INTRODUCTION TO MATHEMATICAL REASONING Key Ideas Worksheet 5 Combiatorics ad Newto s theorem This week we are goig to explore Newto s biomial expasio theorem. This is a very useful tool i aalysis, but
More informationCourse : Algebraic Combinatorics
Course 18.312: Algebraic Combiatorics Lecture Notes # 18-19 Addedum by Gregg Musier March 18th - 20th, 2009 The followig material ca be foud i a umber of sources, icludig Sectios 7.3 7.5, 7.7, 7.10 7.11,
More informationCourse : Algebraic Combinatorics
Course 8.32: Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier February 4th - 6th, 2009 Recurrece Relatios ad Geeratig Fuctios Give a ifiite sequece of umbers, a geeratig fuctio is a compact
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationCIS Spring 2018 (instructor Val Tannen)
CIS 160 - Sprig 2018 (istructor Val Tae) Lecture 5 Thursday, Jauary 25 COUNTING We cotiue studyig how to use combiatios ad what are their properties. Example 5.1 How may 8-letter strigs ca be costructed
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More informationIntroduction To Discrete Mathematics
Itroductio To Discrete Mathematics Review If you put + pigeos i pigeoholes the at least oe hole would have more tha oe pigeo. If (r + objects are put ito boxes, the at least oe of the boxes cotais r or
More informationChapter 1 : Combinatorial Analysis
STAT/MATH 394 A - PROBABILITY I UW Autum Quarter 205 Néhémy Lim Chapter : Combiatorial Aalysis A major brach of combiatorial aalysis called eumerative combiatorics cosists of studyig methods for coutig
More informationExercises 1 Sets and functions
Exercises 1 Sets ad fuctios HU Wei September 6, 018 1 Basics Set theory ca be made much more rigorous ad built upo a set of Axioms. But we will cover oly some heuristic ideas. For those iterested studets,
More informationLINEAR ALGEBRAIC GROUPS: LECTURE 6
LINEAR ALGEBRAIC GROUPS: LECTURE 6 JOHN SIMANYI Grassmaias over Fiite Fields As see i the Fao plae, fiite fields create geometries that are uite differet from our more commo R or C based geometries These
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationThe Binomial Theorem
The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationHomework 9. (n + 1)! = 1 1
. Chapter : Questio 8 If N, the Homewor 9 Proof. We will prove this by usig iductio o. 2! + 2 3! + 3 4! + + +! +!. Base step: Whe the left had side is. Whe the right had side is 2! 2 +! 2 which proves
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationInjections, Surjections, and the Pigeonhole Principle
Ijectios, Surjectios, ad the Pigeohole Priciple 1 (10 poits Here we will come up with a sloppy boud o the umber of parethesisestigs (a (5 poits Describe a ijectio from the set of possible ways to est pairs
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More informationChapter 6. Advanced Counting Techniques
Chapter 6 Advaced Coutig Techiques 6.: Recurrece Relatios Defiitio: A recurrece relatio for the sequece {a } is a equatio expressig a i terms of oe or more of the previous terms of the sequece: a,a2,a3,,a
More information1 Counting and Stirling Numbers
1 Coutig ad Stirlig Numbers Natural Numbers: We let N {0, 1, 2,...} deote the set of atural umbers. []: For N we let [] {1, 2,..., }. Sym: For a set X we let Sym(X) deote the set of bijectios from X to
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationSeries: Infinite Sums
Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More information3 Gauss map and continued fractions
ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ]
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationDavid Vella, Skidmore College.
David Vella, Skidmore College dvella@skidmore.edu Geeratig Fuctios ad Expoetial Geeratig Fuctios Give a sequece {a } we ca associate to it two fuctios determied by power series: Its (ordiary) geeratig
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Stirlig ad Lagrage Sprig 2003 This sectio of the otes cotais proofs of Stirlig s formula ad the Lagrage Iversio Formula. Stirlig s formula Theorem 1 (Stirlig s
More informationCOMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n.
COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where
More information2.4 - Sequences and Series
2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationLecture Notes for CS 313H, Fall 2011
Lecture Notes for CS 313H, Fall 011 August 5. We start by examiig triagular umbers: T () = 1 + + + ( = 0, 1,,...). Triagular umbers ca be also defied recursively: T (0) = 0, T ( + 1) = T () + + 1, or usig
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationTopic 5: Basics of Probability
Topic 5: Jue 1, 2011 1 Itroductio Mathematical structures lie Euclidea geometry or algebraic fields are defied by a set of axioms. Mathematical reality is the developed through the itroductio of cocepts
More informationDiscrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions
CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationSOLVED EXAMPLES
Prelimiaries Chapter PELIMINAIES Cocept of Divisibility: A o-zero iteger t is said to be a divisor of a iteger s if there is a iteger u such that s tu I this case we write t s (i) 6 as ca be writte as
More information14 Classic Counting Problems in Combinatorics
14 Classic Coutig Problems i Combiatorics (Herbert E. Müller, May 2017, herbert-mueller.ifo) Combiatorics is about coutig objects, or the umber of ways of doig somethig. I this article I preset some classic
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationUNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST. First Round For all Colorado Students Grades 7-12 November 3, 2007
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST First Roud For all Colorado Studets Grades 7- November, 7 The positive itegers are,,, 4, 5, 6, 7, 8, 9,,,,. The Pythagorea Theorem says that a + b =
More informationGenerating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
More informationBasic Counting. Periklis A. Papakonstantinou. York University
Basic Coutig Periklis A. Papakostatiou York Uiversity We survey elemetary coutig priciples ad related combiatorial argumets. This documet serves oly as a remider ad by o ways does it go i depth or is it
More informationP1 Chapter 8 :: Binomial Expansion
P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationLegendre-Stirling Permutations
Legedre-Stirlig Permutatios Eric S. Egge Departmet of Mathematics Carleto College Northfield, MN 07 USA eegge@carleto.edu Abstract We first give a combiatorial iterpretatio of Everitt, Littlejoh, ad Wellma
More informationBINOMIAL COEFFICIENT AND THE GAUSSIAN
BINOMIAL COEFFICIENT AND THE GAUSSIAN The biomial coefficiet is defied as-! k!(! ad ca be writte out i the form of a Pascal Triagle startig at the zeroth row with elemet 0,0) ad followed by the two umbers,
More informationLinear chord diagrams with long chords
Liear chord diagrams with log chords Everett Sulliva Departmet of Mathematics Dartmouth College Haover New Hampshire, U.S.A. everett..sulliva@dartmouth.edu Submitted: Feb 7, 2017; Accepted: Oct 7, 2017;
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationIs mathematics discovered or
996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies
More informationGenerating Functions
Geeratig Fuctios Geeratig Fuctios are a powerful tool i combiatorics Wilf describes them, quite appropriately, as a clotheslie o which we hag up a sequece of umber for display What does he mea by that?
More information( ) GENERATING FUNCTIONS
GENERATING FUNCTIONS Solve a ifiite umber of related problems i oe swoop. *Code the problems, maipulate the code, the decode the aswer! Really a algebraic cocept but ca be eteded to aalytic basis for iterestig
More informationMath 4707 Spring 2018 (Darij Grinberg): homework set 4 page 1
Math 4707 Sprig 2018 Darij Griberg): homewor set 4 page 1 Math 4707 Sprig 2018 Darij Griberg): homewor set 4 due date: Wedesday 11 April 2018 at the begiig of class, or before that by email or moodle Please
More information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More informationMath 172 Spring 2010 Haiman Notes on ordinary generating functions
Math 72 Sprig 200 Haima Notes o ordiary geeratig fuctios How do we cout with geeratig fuctios? May eumeratio problems which are ot so easy to hadle by elemetary meas ca be solved usig geeratig fuctios
More informationMath 5705: Enumerative Combinatorics, Fall 2018: Homework 3
Uiversity of Miesota, School of Mathematics Math 5705: Eumerative Combiatorics, all 2018: Homewor 3 Darij Griberg October 15, 2018 due date: Wedesday, 10 October 2018 at the begiig of class, or before
More informationDiscrete Math Class 5 ( )
Discrete Math 37110 - Class 5 (2016-10-11 Istructor: László Babai Notes tae by Jacob Burroughs Revised by istructor 5.1 Fermat s little Theorem Theorem 5.1 (Fermat s little Theorem. If p is prime ad gcd(a,
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More information2.4 Sequences, Sequences of Sets
72 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.4 Sequeces, Sequeces of Sets 2.4.1 Sequeces Defiitio 2.4.1 (sequece Let S R. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For each
More informationCSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber
More informationLecture 10: Mathematical Preliminaries
Lecture : Mathematical Prelimiaries Obective: Reviewig mathematical cocepts ad tools that are frequetly used i the aalysis of algorithms. Lecture # Slide # I this
More informationSection 6.4: Series. Section 6.4 Series 413
ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationLecture notes for Enumerative Combinatorics
Lecture otes for Eumerative Combiatorics Aa de Mier Uiversity of Oxford Michaelmas Term 2004 Cotets Subsets, multisets, ad balls-i-bis 3. Words ad permutatios............................. 3.2 Subsets ad
More informationIntermediate Math Circles November 4, 2009 Counting II
Uiversity of Waterloo Faculty of Mathematics Cetre for Educatio i Mathematics ad Computig Itermediate Math Circles November 4, 009 Coutig II Last time, after lookig at the product rule ad sum rule, we
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationPermutations, Combinations, and the Binomial Theorem
Permutatios, ombiatios, ad the Biomial Theorem Sectio Permutatios outig methods are used to determie the umber of members of a specific set as well as outcomes of a evet. There are may differet ways to
More information