1、Mat e at cal Problems and Proofs Combinatorics, Number Theory, and Geometry Mathematical Problems and Proofs Combinatorics, Number Theory, and Geometry This page intentionally left blank. Mathematical Problems and Proofs Combinatorics, Number Theory, and Geometry Branislav Kisacanin Delphi Delco Ele
2、ctronics Systems Kokomo, Indiana Kluwer Academic Publishers New York / Boston / Dordrecht / London / Moscow eBook ISBN: Print ISBN: 0-306-46963-4 0-306-45967-1 2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow All rights reserved No part of this eBook may be reproduced or t
3、ransmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwers eBookstore at: http:/ http:/www.ebooks To my love Saska and my hometown Novi Sad Pref ace F
4、or Whom Is this Book? This book is written for those who enjoy seeing mathematical formu las and ideas, int eresting probl ems, and elegant solutions . More specifically it is written for talented high-school students who are hungry for more mathematics and undergraduates who would like to see illus
5、 trations of abstract mathematical conc epts and to learn a bit about their historic ongm . It is written with that hope that many readers will learn how to read math ematicalliterature in genera l. How Do We Read Mathematics Books? Mathema tics books are read with pencil and paper at hand. The read
6、er some times wishes to check a deriva tion, co mpl ete some missing step s, or try a dif ferent solution. It is often very useful to compare one books explanation to another. It is also very useful to use the index and loca te some other ref erences to a theorem , formula, or a name. Many people do
7、 not know that mathematics books are read in more than one way: The first reading is just browsing - the reader makes the first contact with the book. At that time the reader forms a first impr ession about contents, reada bility, and illustratio ns. At the second reading the reader identif ies sect
8、ions or chapters to read. After such seco nd readings the reader may find the entire vii viii Preface book interesting and worth reading fr om co ver to cover. Every author aspires to be read in this way by more than just a few readers. The reader should not expect to understand every proof or idea
9、at once: It may be necessa ry to skip some details until other theorems or examples show the importance or further explain difficult parts. The reader will then disco ver that the previously unclea r concepts are much easie r to understand. Even when entire sections of a book are difficu lt to grasp
10、, it is useful to skim them, so that at the next reading this material will be easie r to understand. What Does this Book Contain? Besid es many basic and some adva nced theorems from co mbinato rics and numb er theory, this book contains more than 15 0 thoroughly solved exam ples and prob lems that
11、 ill ustrate theorems and ideas and develo p the readers pro blem-s olving ability and sense for elegant solutio ns. His toric notes and biographies of the four most important mathematicians of all time - Archimedes, Newton, Euler, and Gauss - will spark the reade rs imagination and interest for mat
12、hematics and its history. The main co ntents of the book are as follows: Chapter 1 defines and explains set theory terminology and concep ts. Several historically important examples are included. Chapter 2 introduces the reader to elem entary combinato rics. Bef ore defin ing combinations and permu
13、tatio ns, the reader is led through several exam ples to stimulate interest. Several illustrative exam ples in a separate sectio n explain how to use the method of generating functi ons. Chapter 3 introduces number theory, once the most theoretical of all math ema tical disciplines and today the hea
14、rt of cryptography. Amon g other topics the reader can find the Eucli dean algorithm, La mes theorem, the Chinese remainder theorem, and a few words about the Fermats last theorem. Four app endixes at the end of the book provide additional inf ormation. Appendix A explains mathematical induction, an
15、 important mathematical tool. Appendix B provides many fa scin ating details and historic fa cts about four impo rtant mathematical co nstants: 1t, e, y, and. Appendix C presents brief biographies of Archimedes, Newton, Euler, and Ga uss, follo wed by a chrono logical list of many other important na
16、mes fr om the history of mathematic s. Appendix D gives the Greek alphabet. Extensive references and index are provided for the benefit of the reader. Preface ix Acknowledgments I would like to thank many people who supported me and helped me while I was writing this book. Above all my parents, Lj i
17、ljana and Mi odrag; my brother, Miroslav; the love of my life, Saska and my former and present professors, in particular Prof . Rade Doroslovacki from the University of Novi Sad, and Prof . Gyan C. Agarwal from the University of Illinois at Chicago. Mrs. Apolonia Dugich and Dr. Miodrag Radulovacka w
18、ere great friends, and I wish to acknowledge their support, too . Finally I wish to thank Plenum and its mathematics ed itor, Mr. Thomas Cohn, for their interest in pu blishing my work, and their referees, whose names I will never know, for their com ments and suggestions. Ms. Marilyn Buckingham did
19、 a wonderfuljob copyediting the manuscript. My wife, Saska, helped me compile the index. Kok omo, IN Branislav Kisacanin Contents Key to Symbols xiii 1. Se t Theo ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Sets and Ele mentary Set Operations . . . . . .
20、 . . . . . . . . . . . . . . 3 1.2. Ca rtesian Product and Relations . . . . . . . . . . . . . . . . . . . . . . 7 1.3. Functions and Operations . . 9 104. Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5. Prob lems . . 14 2. Com bin ato ri cs . . . . . .
21、. . . . 19 2. l. Four Enumeration Prin ciples . 21 2.2. Introductory Prob lems . . 22 2.3. Basic Definitions 30 204 . Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5. Prob lems . 52 3. Num ber Theo ry 73 3.1. Divisibility of Numb ers 75 3.2. Important Functions in
22、 Numb er Theory . 89 3.3. Congme nces 92 3.4. Di ophantine Equati ons . 101 3.5. Problems . 1 1 0 4. Ge om e tr y . 117 4.1. Properti es of Triangles . 11 9 xi xii Contents 4.2. Anal ogies in Geometry 140 4.3. Two Geometric Tricks 142 4.4. Pro blems . . . . . . . . . . . . . . . . . . . . . . . . .
23、. . . . . . . . . . . 145 Ap pen dixes A. Ma th e mat ica l In du ct io n 165 A.I. Overview 167 A.2. Examples . . . 167 A.3. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 A.4. Hints and Notes 18 0 B. Im po rtant Ma themati ca l Co nstan ts . . . . . . . . . . . . .
24、 . . . . 189 C. Grea t Ma them aticia ns . 199 D. Gr eek Alph ab et 207 Referen ce s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 In de x . 215 o aEA b AcB AcB AUB A nB A- B Ai1B IAI AxB 1,2,3 ( 1,273) 1,2,3) II 3 3! pl q Pvq bla bIa (a, b ) a, b 1t e Key to Symb
25、ols End of proof or example a is inA b is not inA A is a subset of B A is a proper subset of B Set union Set intersection Set difference Symmetric set dif f erence Set cardi nality Cartesian product of A and B Set Ordered trip le Multiset For all There exists There exists only one Logical and Logica
26、l or b divides a b does not divide a OCD of a andb LCM of a and b Ratio of the circu mfer ence to the diameter of a circle Base of natural logarithms xiii xiv P Y In Fn Mp Hij cp(n) J1( n) n(x) (j(n ) ten) L IT lim “ - =, = = LxJ n! G) AB AB -7 AB MBC LABe Key to Symbols Golden sect ion Eulers const
27、ant Fibonacci number Fermat number Merscnn e munber Harmonic number Eulers function Mobi us function Number of primes x Swn of divisors of n Number of div isors of n Sum Product Limit Equality Cong ruence in geometry Congruence in mnnbertheory Asym ptotic beha vior Largest integer:; x n metorial Bin
28、 omial coefficient Segment Line Vector Triangle Angle Mathematical Problems and Proofs Combinatorics, Number Theory, and Geometry 1 Set Theory Set Theory 3 Set theoretic termi nology is used in all parts of mathematics, even in everyday language and life. In Chapter 1 we introduce the notation and t
29、erminology fr om set theory that are used in later chapters. We also show a fe w historically important examples that had a large impact on the development of mathematic s. 1.1. Sets and Elementary Set Operations Set and set element s are basic mathematical notions. If a, b, and c are elem ents of t
30、he set A, then we write A = a, b,c In this case the element a is in A, while the element d is not in A . We write that as follows: aEA dEA Instead of naming all elements of A by their na mes, it is often more conve ni ent to define a set in the following analytic way: A = xIP(x) which means that A i
31、s the set of all elements having the property P. EXAMPLE 1.1. The set of natural numb ers less than 9 can be written in several equivalent ways, for example: A = I ,2 ,3 ,4 ,5 ,6 ,7 ,8 A = 1 ,2, . ,8 A= nln ENA n9 Some sets are used so often that there is a standard notation for them: N Set of natur
32、al numb ers No Set of natural numb ers along with zero Z Set of integers Q Set of rational numb ers R Set of real numb ers C Set of complex numb ers NOTE: Au thors often consid er zero a natural numb er. 4 Chapter I DE FINITIO N 1.1 ( SUB SET). If for every element of A it is true that it is in B, t
33、oo, we say that A is a subset of B, and write A cB. DE FINITION 1.2 (E QUALITY OF SETS). Sets A and B are equal if A ell and B cA. Then we write A = B. Many pro ofs of equality of two sets proceed just as in this definition: First we prove thatAcB, then thatB cA . DEFINITION 1.3 (PROPER SUB SE T). I
34、f A c B and A 7:- B, we say that A is a proper subset of B, and write ACB. EXAMPLE 1.2. If A = l ,2 ,3,4 ,5 ,6, 7,8,9 and B = l, 3,5, 7,9, then BCA. Since obviously A 7:- B, we can also write A C B . Through the following five defmitions we introduce the most important set operations: union, interse
35、ction, diff erence, symmetric diff erence, and comple ment of a set. Each definition is illustrated by the corresponding Euler-Venn diagram in Figs. 1.1 and 1.2. DEFINITION 1.4 (UNION ). The union of sets A and B is the set of elements contained in at least one of these two sets: A U B= xl x E Av xE
36、B E XAMP LE 1. 3. If A = 1,2, 3,4 and B = 1 ,3,5,7 ,9, the n: A UB = 1, 2,3,4, 5,7,9 EXAMP LE 1.4. The union of the sets of odd and even integers is the set of all integers, i.e., Zodd U Z even = Z EXAMPLE 1.5 . The set of ratio nal numbers Q consist of real numbers which can be represented as fract
37、ions of integers. Alternatively it is a set of rea Is with either a finite or periodic decimal representation. All other reals are called irr ation al ; the set of irrational numbers is often denoted by 1. Therefore we canwrite QU I=R Set Theory a) b) c) d) FIGURE 1.1. Euler-Venn diagrams of (a) uni
38、on, (b) intersection, (c) difference, and (d) symmetric difference of sets. DEFINITION I.S (INTERSE CTION). The intersection of sets A and B is the set of elements contained in both of these sets: AnB = xix EA Ax E B If the intersection of two sets is an empty set, i.e., if sets A and B do not have
39、common elements, we say they are disjoint and write An B=8 or AnB= EXAMPLE 1.6. If A = 1, 2,3, 4 andB = 1 ,3,S,7,9, then: An B = 1, 3 EXAMPLE 1.7. The intersection of the sets of odd and even integers is the empty set, i.e., Zodd n Zeven = 8. Sets Q and I are also disjoint. DEFINITION 1.6 (DIFFERENC
40、E ) . The diffe rence of sets A and B is the set of elements from A not contained in B: A - B = xlxE A A x E B EXAMPLE 1.8. If A = 1, 2,3,4 and B = 1 ,3,S,7,9, then: A- B = 2 ,4 DEFINITION 1.7 (SYMMETRIC DIFFERENCE ). The symmetric differ ence of sets A and B is the set of elements not contained in
41、both A and B: A!1.B = (A UB) - (A UB) EXAMPLE 1.9 . If A = 1, 2,3,4 andB = 1 ,3,S,7,9, then A!1.B = 2, 4,S, 7,9 5 6 Chapter 1 A. I FIGURE 1.2. The complement of A with respect to I. DEFINITI ON 1. 8 (COMPLEMENT ). If A is a subset of some set I, the complement of A with respect to I is the set of el
42、ements from I not contained in A : A=xlx E 11 x A ExAMPLE 1. 10. The complement of the set of even integers with respect to Z, the set of all integers, is the set of odd integers, i.e., EXAMPLE 1.11. Prove that A - B = A n B. SOLUTION: To prove this identity, the equality of these two sets, we must
43、show that x E A - B if and only if x E An jj .Indeed: X E A-B x E A 1 X B x E A / X E 11 X E A n lJ DEFINI TION 1.9 (POWER SET). The set of all subsets of A is called the power set of A, and it is denoted by P(A ): P(A) = XlXc A Since by defmition 0 c A and A c A, then: G E peA ) A EP (A ) N01E: Lat
44、er in the chapter on combinatorics, we prove that if A has n elements, thenp(A) has 2“ elements. For example, if A = l,2 ,3 , then: P( A ) = 0, 1, 2 , 3 , 1 ,2 , 1,3 , 2 ,3 , 1 ,2 ,3 DEFIN ITION 1.10 (SET PARTITION ). Pa rtition ofa setA is a set of its nonempty, mutually disjoint subsets, whose uni
45、on is A . Set Theory 7 EXAMPLE 1.12. If A = 1, 2,3 , then all partitions of A are 1,2, 3, 1 , 2,3 , 2, 1 ,3 , 3, 1,2 , and 1 ,2 ,3) . When derming a set, the order of its elements is irrelevant. It also does not matter if we list some element more than once, for example: 1 ,2,3 =1 ,3 ,2 = 1, 1, 1 ,2
46、,3 If we do care about the order and repetition of elements, we use ordered pairs, triples, etc., for example: (1,2) 7:-(2,1) (1,1,2) 7:-(1,2 ) Here we define only the ordered pair because ordered triples, etc., are defined similarly. DEFINITION 1.11 (ORDERED PAIR). The ordered pair (a, b ) is defin
47、ed as (a,b ) = a, a,b The element a is its first component, while b is its second component. EXAMPLE 1.13. The ordered pair ( a, b) is equal to another ordered pair (c, d) if and only if a = c and b = d. NOTE: In mathematics we often work with objects for which the order of their elements is irrelev
48、ant, but the repetition is not. To model such objects we use the so-called mu ltisets . There will be more about them in Chapter 2. 1.2. Cartesian Product and Rel ations DEFINITION 1.12 (CARTESIAN PRODUCT). The Cartesian product of sets A andB is the set of all ordered pairs in which the first compo
49、nent is fromA and the second component is from B: A xB = ( a, b ) I a EO A 1 b EO B NOTE: The name Cartesian is derived from the Latin name of the French mathematician and philosopher Rene Descartes - Renatus Cartesius. 8 Chapter 1 EXAMPLE 1.14. If A = 1 ,2 ,3 and B = 7 ,9, then: AxB= (1, 7),( 1,9),( 2, 7),( 2,9 ),( 3, 7),( 3,9) DEFINITION 1.13 (RELATION ). The re
