0,1 is an accumulation

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Sectio Accumulatio Poits Sectio 5.4 Bolzao-Weierstrass ad Heie-Borel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real aalysis; the Bolzao- Weierstrass Theorem ad Heie-Borel Coverig Theorem. Both proofs are two of the most elegat i mathematics. Accumulatio Poits Every set of real umbers has associated with it a set of accumulatio limit poits, a cocept which allows for a precise aalysis of closeess; closeess of real umbers, closeess of poits i R, closeess of fuctios, closeess of operators. The accumulatio poits may be a subset of a give set, part of a give set or totally disjoit of the give set. Its defiig characteristic is that every accumulatio poit of a set is ear some poit of the set other tha itself. Defiitio: A umber a is a accumuatio poit (or limit poit) of a set A if ad oly if for ay δ > the there exists the δ -eighborhood of a cotais at least oe poit of A distict from a. I other words every eighborhood of a cotais poits of A differet from a. Keep i mid that a accumulatio poit of a set may or may ot belog to the set. A eighborhood of a poit (ay ope iterval cotaiig the poit) that does ot cotai the poit is called a deleted eighborhood of the poit. Thus, the, 2 1 is a deleted eighborhood of 1. set ( ) { } Margi Note: Ituitively, a accumulatio poit of a set (which may or may ot belog to the set) is a poit where o matter how little you wiggle away from the poit you itersect poits of the set. I other words, the set likes to suggle up to accumulatio poits. Example 1 (Accumulatio Poits) a) Every poit i the closed iterval [ ],1 is a accumulatio

2 Sectio Accumulatio Poits poit of the ope iterval(,1 ) sice every deleted eighborhood of a [,1] itersects some poit i (,1 ). b) Fiite sets have o accumulatio poits sice aroud every real umber (iside or outside the set) you ca fid a deleted eighborhood that does ot cotai elemets of the set. c) The set (,1) { 2} has accumulatio poits [,1 ]. The umber 2 is ot a accumulatio poit of the set sice there exists a deleted eighborhood aroud 2 that does ot itersect members of the set. d) The set { 1/ : = 1, 2,... } has oe accumulatio poit at. Aroud ay other poit i the set you ca fid a deleted eighborhood that does t itersect the set.. e) The itegers Z have o accumulatio poit eve though the set is ifiite. This is easy eough to see sice each iteger is cotaied i a deleted eighborhood of radius.25 that does ot itersect ay members of the set. Margi Note: The reader may recall limits of sequeces from calculus which are examples of accumulatio poits of the elemets i the sequece. What do the Real Numbers Really Look Like? What does the real lie look like if you look at it really close up? We thik of it as a cotiuum of poits extedig idefiitely i two directios, but what if you could look at it uder a microscope ad were able to tur up the magificatio higher ad higher. What would you begi to see? You might be disappoited sice you will ever get to a stage where you would see, oe ratioal umber, three irratioal umbers, oe ratioal umber, The real umbers are self-similar, they look alike o matter what the scale. So how do we visualize the real umbers i our mids? Well, we simply have to uderstad the may properties of the real umbers which ca be verified mathematically, the use your imagiatio to visualize them it i your mid s eye. We ow come to oe of the most importat theorems i aalysis, the Bolzao-Weierstrass theorem, but before we state ad prove the theorem we must itroduce ourselves to the cocept of ested closed itervals.

3 Sectio Accumulatio Poits Nested ed Itervals By a sequece of ested itervals I [ a, b ] = we mea a sequece of closed itervals with the left edpoit a movig towards the right, ad the right edpoit b movig towards the left. The questio we ask is, what ca be said about the itersectio of all the sets; i.e. the set of all poits commo to every iterval? The followig lemma, which will be used to prove both the Bolzao-Weierstrass ad Heie-Borel theorems, gives the aswer. Lemma 1 (Nested Iterval Lemma) If a, b a, b... a, b [ ] [ ] [ ] is a ested sequece of closed itervals whose legths coverge to, i.e. lim b a =, the their itersectio cosists of a sigle poit ( ) [ a, b ] = x. = 1 Bolzao-Weierstrass Theorem Here is a iterestig questio that will test your ituitio about the real umber system ad accumulatio poits. Some people will aswer this questio i the affirmative ad others i the egative, so the questio is ot trivial. Here is the questio. Suppose you begi markig off poits iside some bouded iterval, ope, closed, or either, lets say [,1 ] for coveiece, ad suppose you do this idefiitely. The questio is ca you do it i such a way that there will ever be a accumulatio poit? I other ca you mark off poits i such a way that they ever buch up aywhere? Of course it is possible to mark off poits so you do have a accumulatio poit, simply pick x = 1/, = 1, 2,... which has a accumulatio poit at. I fact if x = you are clever, you ca pick a sequece { } 1 that has 2 accumulatio poits, i fact 3, 4,, a fiite umber of accumulatio poits. It is also easy to see that if the iterval is ubouded, say [, ) the there eed ot be a accumulatio poit, as the example x =, = 1, 2,... illustrates. So ca you

4 Sectio Accumulatio Poits x = fid a sequece { } 1 poit? Thik hard. of umbers i [,1 ] that does ot have a accumulatio Historical Note: Berard Bolzao ( ) was a Czech philosopher/mathematicia ad theologia. He was a Catholic priest who is best remembered today for his views i the methodology ad mathematics ad logic. May of the ideas later developed by Cator were uderstood by Bolzao. The Bolzao-Weierstrass theorem was first prove by Bolzao but ufortuately the result was lost. It was re-prove by the great Germa mathematicia Karl Weierstrass ( ). Weierstrass is ofte called the father of moder aalysis, havig brought mathematical rigor to the level we see today. The aswer to the questio about the existece of a accumulatio poit of a bouded ifiite set of real umbers is the statemet of the Bolzao- Weierstrass theorem. The theorem is importat ad the proof igeious. Theorem 1: Bolzao-Weierstrass Theorem Every bouded ifiite set S of real umbers has a accumulatio poit ir. Proof:, S a, b, Sice S is bouded, there is a closed iterval [ a b ] such that [ ] where the midpoit x = ( a + b) / 2 of [ a, b ] divides [, ] a b ito two closed subitervals. The subitervals overlap at the midpoit but all that matters is a, b. Now oe of these subitervals that their legths are half the legth of [ ] (or possibly both) cotais a ifiite umber of poits, else the set S is the uio of two fiite sets, cotrary to the assumptio that S is ifiite. Lettig I be the subiterval that cotais a ifiite umber of poits (if both 1 subitervals cotai a ifiite umber of poits we pick oe at radom), we cotiue by dividig I 1 ito two closed subitervals of equal legth, were we call I 2 a subiterval that has cotais a ifiite umber of poits. See

5 Sectio Accumulatio Poits Figure 1. itervals Cotiuig i this maer, we arrive at a sequece of closed [ ] a b I I I I, each of whose legth is half that of the previous iterval. Hece by Lemma 1 the set Ik cosists of a sigle poit, say k = 1 x. Strategy for the Bolzao-Weierstrass Theorem; Divide ad Coquer Figure 1 We are ot yet doe; we must show that x is a accumulatio poit of S. To show this let ( α, β ) by ay eighborhood of x ( α, β ), ad let k mi ( x α, β x ) is less tha k, we observe I ( α, β ). But I cotais a ifiite umber of poits ad hece so does the arbitrary eighborhood ( α, β ). Thus x is a = ad ote k >. Selectig a iterval I whose legth accumulatio poit of S. Margi Note: I terms of sequeces, the Bolzao-Weierstrass theorem says that ay bouded sequece x 1, x 2,... has at least oe coverget subsequece. Margi Note: Cator claimed that the Bolzao-Weierstrass is the basis for most importat results i aalysis. Realize the theorem is false if oe restricts oeself to ifiite bouded subsets of the ratioal umbers. The Bolzao- Weierstrass theorem states somethig iheret about the real umber system. Example 2 (Accumulatio Poits) a) The set A = 1,,,, is a bouded ifiite set so the Bolzao Weierstrass theorem guaratees at least oe accumulatio poit, which i this

6 Sectio Accumulatio Poits case there is exactly oe accumulatio poit, amely. The accumulatio poit of this set does ot belog to the set. b) set of itegers Z is a ifiite set but is ot bouded ad so the coditios of the Bolzao-Weierstrass theorem are ot satisfied, hece there is o guaratee of ay accumulatio poits. I this case the set has o accumulatio poits. c) The set A = (,1) { 2,3,4,... } is a ifiite set but ot bouded so the coditios of the Bolzao-Weierstrass theorem are ot satisfied; hece there is o guaratee the set has ay accumulatio poits. However the set does have accumulatio poits, amely all poits i the closed iterval [,1 ]. d) The set A = { 1, 2,3,4,5} is bouded but ot ifiite ad thus the coditios of the Bolzao-Weierstrass theorem are ot satisfies, hece there is o guaratee the set has ay accumulatio poits. I this case the set does ot have ay accumulatio poits. Fiite sets ever have accumulatio poits. Fiite Ope Covers C for Sets There are cocepts i mathematics which make you ask what does this have to do with aythig, but after further study you say wow, who ever thought of this! Oe such cocept is the idea of coverigs of sets ad i particular fiite ope coverigs. We will see that sets that have fiite ope coverigs behave to a great degree like fiite sets, ad of course aythig fiite is a lot simpler tha somethig ifiite.

7 Sectio Accumulatio Poits Defiitio Let A be a set of real umbers. A collectio C = { J : J } C of subsets of R is called a cover (or coverig rig) of A if A is a subset of the uio of the members of C, i.e. A { J : J C }. If each elemet J C i the coverig is a ope set, the coverig is called a ope cover, ad if C cotais oly a fiite umber of sets, the coverig C is called a fiite ope cover. You might thik of a coverig ituitively as a collectio of umbrellas providig shade from a summer su as draw below. Example 3 1 a) The family C =,1, : N is a ope cover for (,1 ), but o fiite sub-collectio of these itervals will cover (,1). Do you agree? b) The closed bouded iterval [,1 ] has a ope cover 1 1,1 +, : N the member of C whe = 1. { } C=, a example beig the sigle set ( 1,2 ) which is {,, : N} c) The real umbers R is covered by the ope cover C= ( ) but there is o fiite subcover forr.

8 Sectio Accumulatio Poits So why is the cocept of fiite coverigs so importat i aalysis? It has to do with the fact that sets which ca be covered with a fiite collectio of ope sets behave to a extet like fiite sets, which of course are easier to study tha ifiite sets. Compactess ad the Heie-Borel Theorem The cocept of ope covers of sets, i particular ope covers that have fiite subcovers, leads directly to oe of the most importat cocepts i aalysis, compactess. Defitio: A set A of real umbers is called compact if wheever it is cotaied i a uio of a family C of ope sets, it is also cotaied i the uio of a fiite umber of the sets i C. So ow the questio remais, how do we kow if a set is compact; i.e. every ope cover has a fiite subcover? As we have see i Example 3 some sets have this desirable property, some do ot. Offhad it would seem to be a very hard property to determie. Fortuately two mathematicias, Eduard Heie ad Emile Borel, foud a simple characterizatio of these fiite-like sets, whereby oe ca tell at a glace if they have this fiite property. This characterizatio is called the Heie-Borel (Coverig) Theorem. Theorem 2 (Heie ( Heie-Borel Coverig Theorem) a, b is compact. Proof: Every closed ad bouded iterval [ ] We must prove that for ay closed ad bouded iterval [ a, b ] that is covered by a collectio { J : J } subcollectio of itervals i C, say 1, 2,..., is for every x [ a, b] C= C of ope itervals, the there is a fiite J J J which also covers [, ] a b ; that we have x J k for some 1 k. The proof is by cotradictio. Assume the theorem false; that is, there, C= J : J C for which exists a ope cover of [ a b ] cosistig of itervals { } there is o fiite subcover J1, J 2,..., J. This beig true, the the midpoit of [ a, b ] divides the iterval ito two closed itervals, where at least oe iterval, which we call I 1, is ot covered by a fiite sub-collectio of members of the coverigc. We the divide I 2 i a similar maer ad arrive at a ew closed subiterval I 2, whose legth is half that of I 2 ad also is ot C= J : J C. covered by a fiite umber of members of the coverig { } Cotiuig i this maer, we arrive at a oicreasig sequece of closed itervals

9 Sectio Accumulatio Poits [ ] 1 2 a b I I I, where each iterval I k is half as log as its predecessor ad is ot covered by a fiite umber of members of the coverig { J : J } 1 we kow that the itersectio sice x [ a, b] ope itervals, say ( α, β ) such that x ( α β ) C= C. But from Theorem Ik cosists of a sigle poit, say k= 1 x. Now we kow there exists (at least) oe member of the family C of. But from the way the, itervals I k are formed there exists a iterval I whose legth is so small that x I ( α β ) { J J }, C= : C. A cotradictio; we kow I caot be covered by a fiite umber of members of C, but it is covered by ( α, β ), a sigle elemet of C. Figure 1 But this is a cotradictio sice we have said that I caot be covered by a J : J I α, β C= J : J C. fiite umber of coverigs of C= { C }, but ( ) { } Hece, we coclude every ope cover of a closed ad bouded iterval [ a, b ] does have a fiite subcover. Margi Note: The observatio that subsets of real umbers have fiite covers is equivalet to beig closed ad bouded was first observed by Germa mathematicia Heirich Eduard Heie i the 187s ad later i 1894 formulated precisely by Frech mathematicia Emile Borel.

11 Sectio Accumulatio Poits Problems 1. Fid the accumulatio poits of the followig sets (if ay). State whether the coditios of the Bolzao-Weierstrass theorem hold. a) N b) Q c) R 2, 4 4,5 d) ( ) ( ) { 1 : N } e) ( ) f) Q g) (,1) m h) : m, N 2 1 i) m + : m, N 2. (Covers of Sets) What does it mea for a family of sets ot to be a cover C for a set A? What does it mea for a cover C of a set A ot to have a fiite sub-cover. Give examples of each. 3. (Compactess) Use the geeral Heie-Borel theorem, which states that a set of real umbers is compact if ad oly if it is closed ad bouded, to determie which of the followig sets are compact. a) { 1,2,3, 4,5 } b) [,1] [ 2,3] c) { x : x 2 = 2} d) [,1 ) e) [,1] { 2,3, 4,5} 4. (Closed Sets) A set is closed if it cotais its accumulatio poits. Fid the accumulatio poits of the followig sets ad verify that those sets that are closed do cotai their accumulatio poits. a) N b) Q c) R 2, 4 4,5 d) ( ) ( )

12 Sectio 5.4 { 1 : N } e) ( ) f) Q g) (,1) m h) : m, N 2 1 i) m + : m, N 12 Accumulatio Poits 5. (Ope Subcover) (Ope Subcover) Fid a fiite ope subcover of the set [ ],1 for the cover C =,1,. j 1 4 j= 1 6. (Itersectios of Closed Itervals) The itersectio of a fiite umber of closed itervals is oe of three types of sets. What are they? 7. (Itersectios of Ope Itervals) The itersectio of a fiite umber of ope itervals is oe of two types of sets. What are they? 8. (Examples) Give examples of the followig. a) A bouded set with o accumulatio poits. b) A ubouded set with oe accumulatio poit. c) A set with two accumulatio poits. d) A ubouded set with a ifiite umber of accumulatio poits. e) A ubouded with oe accumulatio poit. f) A ope set with o accumulatio poits.

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

1 Set Theory and Functions

Set Theory ad Fuctios. Basic De itios ad Notatio A set A is a collectio of objects of ay kid. We write a A to idicate that a is a elemet of A: We express this as a is cotaied i A. We write A B if every

Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

Sequences II. Chapter 3. 3.1 Convergent Sequences

Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

Convexity, Inequalities, and Norms

Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

Module 4: Mathematical Induction

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

Section 11.3: The Integral Test

Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

TAYLOR SERIES, POWER SERIES

TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

Metric, Normed, and Topological Spaces

Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolute-value metric

Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

3. Greatest Common Divisor - Least Common Multiple

3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

4.3. The Integral and Comparison Tests

4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

PART TWO. Measure, Integration, and Differentiation

PART TWO Measure, Itegratio, ad Differetiatio Émile Félix-Édouard-Justi Borel (1871 1956 Émile Borel was bor at Sait-Affrique, Frace, o Jauary 7, 1871, the third child of Hooré Borel, a Protestat miister,

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

1. C. The formula for the confidence interval for a population mean is: x t, which was

s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

Chapter 5: Inner Product Spaces

Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

Section 1.6: Proof by Mathematical Induction

Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

Using Excel to Construct Confidence Intervals

OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

3 Basic Definitions of Probability Theory

3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio

I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

Overview of some probability distributions.

Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

5 Boolean Decision Trees (February 11)

5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

Math C067 Sampling Distributions

Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

8.1 Arithmetic Sequences

MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

1.3 Binomial Coefficients

18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

The Field Q of Rational Numbers

Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

Approximating the Sum of a Convergent Series

Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

Recursion and Recurrences

Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

Permutations, the Parity Theorem, and Determinants

1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

Determining the sample size

Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

Confidence Intervals

Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

The second difference is the sequence of differences of the first difference sequence, 2

Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

1.3. VERTEX DEGREES & COUNTING

35 Chapter 1: Fudametal Cocepts Sectio 1.3: Vertex Degrees ad Coutig 36 its eighbor o P. Note that P has at least three vertices. If G x v is coected, let y = v. Otherwise, a compoet cut off from P x v

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

Chapter One BASIC MATHEMATICAL TOOLS

Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is

The Stable Marriage Problem

The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

Notes on exponential generating functions and structures.

Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

Math 113 HW #11 Solutions

Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

Analysis Notes (only a draft, and the first one!)

Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that

I. Why is there a time value to money (TVM)?

Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

NOTES ON INEQUALITIES FELIX LAZEBNIK

NOTES ON INEQUALITIES FELIX LAZEBNIK Order ad iequalities are fudametal otios of moder mathematics. Calculus ad Aalysis deped heavily o them, ad properties of iequalities provide the mai tool for developig

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

Betting on Football Pools

Bettig o Football Pools by Edward A. Beder I a pool, oe tries to guess the wiers i a set of games. For example, oe may have te matches this weeked ad oe bets o who the wiers will be. We ve put wiers i

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + \$ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

Estimating the Mean and Variance of a Normal Distribution

Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS

ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS JÓZSEF BALOGH, DHRUV MUBAYI, AND ANDRÁS PLUHÁR Abstract The edge-badwidth of a graph G is the badwidth of the lie graph of G We show asymptotically tight bouds o

Revised Edition (P

Revised Editio - 05 (P (ii) Preface It is gratifyig to ote that educatio as a whole ad school educatio i particular witess marked chages i the state of Tamil Nadu resultig i the implemetatio of uiform

Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

Unit 20 Hypotheses Testing

Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

Sum and Product Rules. Combinatorics. Some Subtler Examples

Combiatorics Sum ad Product Rules Problem: How to cout without coutig. How do you figure out how may thigs there are with a certai property without actually eumeratig all of them. Sometimes this requires

7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y