The keys to advanced mathematics: recurrent themes in abstract reasoning
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Mansfield, OH
BookMasters Distribution Center
1995
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XX, 476 S. graph. Darst. |
| ISBN: | 0964451905 |
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| adam_text | ILL ORDERING INFORMATION FOR THE KEYS TO ADVANCED MATHEMATICS: RECURRENT
THEMES IN ABSTRACT REASONING BY DANIEL SOLOW ISBN NUMBER: 0-9644519-0-5
ORDER FROM: BOOKMASTERS DISTRIBUTION CENTER 1444 U.S. ROUTE 42, RD # 11
MANSFIELD, OH 44903 FAX: (419) 281-6883 FOR CREDIT CARD, LIBRARY, AND
BOOKSTORE ORDERS CALL TOLL FREE (800) 247-6553. A COMPLIMENTARY DESK
COPY AND SOLUTIONS MANUAL FOR ALL THE EXERCISES ARE AVAILABLE TO
INSTRUCTORS WHO ADOPT THIS BOOK AS A REQUIRED TEXT BY SENDING THE COURSE
NAME AND NUMBER TOGETHER WITH THE INSTRUCTORS S NAME AND ADDRESS TO
PROFESSOR DANIEL SOLOW DEPARTMENT OF OPERATIONS RESEARCH WEATHERHEAD
SCHOOL OF MANAGEMENT CASE WESTERN RESERVE UNIVERSITY CLEVELAND, OH 44106
THE COMPANION BOOK, HOW TO READ AND DO PROOFS, SECOND ED., BY DANIEL
SOLOW IS AVAILABLE FROM JOHN WILEY & SONS, INC., NEW YORK, NY (ISBN
NUMBER: 0-471-51004-1). CONTENTS FOREWORD VII PREFACE TO THE INSTRUCTOR
IX ACKNOWLEDGMENTS XV 1 WHAT S THE USE OF MATHEMATICS? 1 1.1 USING
MATHEMATICS TO SOLVE SPECIFIC PROBLEMS 2 1.1.1 CLOSED-FORM SOLUTIONS 2
1.1.2 NUMERICAL METHODS 3 1.1.3 FINDING SOLUTIONS MORE EFFICIENTLY 4 1.2
UNIFICATION 5 1.2.1 UNIFICATION OF LINEAR EQUATIONS 6 1.2.2 UNIFICATION
OF THE TRAVELING-SALESPERSON PROBLEMS 8 SUMMARY OF UNIFICATION 11 1.3
IDENTIFYING SIMILARITIES AND DIFFERENCES 12 1.4 GENERALIZATION 16 1.4.1
A GENERALIZATION TO QUADRATIC EQUATIONS 17 1.4.2 A GENERALIZATION TO TWO
EQUATIONS IN TWO UNKNOWNS 20 1.4.3 ADDITIONAL EXAMPLES OF
GENERALIZATIONS 22 1.4.4 CORRECTING SYNTAX ERRORS IN GENERALIZATIONS 24
SUMMARY OF GENERALIZATION 24 1.5 ABSTRACTION 26 1.5.1 WORKING WITH
OBJECTS 26 1.5.2 ADDITIONAL EXAMPLES OF ABSTRACTION 29 1.5.3 CORRECTING
SYNTAX ERRORS WHEN DEVELOPING ABSTRACTIONS 32 SUMMARY OF ABSTRACTION 33
1.6 MATHEMATICAL PROOFS 34 1.6.1 MATHEMATICAL STATEMENTS AND
IMPLICATIONS 35 1.6.2 THE FORWARD-BACKWARD METHOD 37 1.6.3 THE
EXISTENTIAL QUANTIFIER AND THE CONSTRUCTION METHOD 41 1.6.4 THE
UNIVERSAL QUANTIFIER AND THE CHOOSE METHOD . 43 1.6.5 INDUCTION 45 XVN
XVIII CONTENTS 1.6.6 THE UNIVERSAL QUANTIFIER AND THE SPECIALIZATION
METHOD 47 1.6.7 NESTED QUANTIFIERS 50 1.6.8 THE CONTRAPOSITIVE METHOD 52
1.6.9 THE CONTRADICTION METHOD 54 1.6.10 NEGATIONS OF STATEMENTS 56
1.6.11 EITHER/OR METHODS 61 1.6.12 UNIQUENESS METHODS 64 CHAPTER SUMMARY
68 EXERCISES 69 2 WORKING WITH MATHEMATICAL CONCEPTS 81 2.1 CREATING AND
WORKING WITH IMAGES 82 2.1.1 CREATING IMAGES 82 2.1.2 TRANSLATING IMAGES
TO SYMBOLIC FORM 86 2.2 CREATING DEFINITIONS 90 2.2.1 WHAT IS A
MATHEMATICAL DEFINITION? . . . 91 2.2.2 CREATING YOUR OWN DEFINITIONS .
96 SUMMARY OF CREATING DEFINITIONS 106 2.3 DEVELOPING AXIOMATIC SYSTEMS
106 2.3.1 WHAT IS AN AXIOMATIC SYSTEM? 106 2.3.2 WORKING WITH AN
AXIOMATIC SYSTEM 110 2.3.3 OTHER EXAMPLES OF AXIOMATIC SYSTEMS 113
SUMMARY OF DEVELOPING AXIOMATIC SYSTEMS 123 CHAPTER SUMMARY 123
EXERCISES 124 3 SELECTED TOPICS IN DISCRETE MATHEMATICS 135 3.1 SETS 135
3.1.1 SETS AND THEIR REPRESENTATIONS 135 3.1.2 COMPARING SETS 139 3.1.3
UNARY OPERATIONS ON SETS . . 144 3.1.4 BINARY OPERATIONS ON SETS 147 3.2
FUNCTIONS 150 3.2.1 FUNCTIONS AND THEIR REPRESENTATIONS 151 3.2.2
COMPARISONS AND OPERATIONS INVOLVING FUNCTIONS 157 3.2.3 PROPERTIES OF
FUNCTIONS 161 3.3 GRAPHS 171 3.3.1 GRAPHS AND THEIR REPRESENTATIONS 171
3.3.2 OPERATIONS ON GRAPHS 175 3.3.3 THE PROPERTY OF A GRAPH BEING
CONNECTED . . . . . . 181 3.3.4 SOLVING PROBLEMS USING GRAPHS 182 3.4
PUTTING IT ALL TOGETHER 192 3.4.1 IDENTIFYING SIMILARITIES AND
DIFFERENCES 192 3.4.2 DEVELOPING AN ABSTRACT SYSTEM 193 3.4.3 STATING
THE PROBLEM IN TERMS OF THE ABSTRACT SYSTEM 198 CONTENTS XIX 3.4.4
DEVELOPING A SOLUTION PROCEDURE 199 3.4.5 DEVELOPING AN AXIOMATIC SYSTEM
200 CHAPTER SUMMARY 201 EXERCISES 202 4 SELECTED TOPICS IN LINEAR
ALGEBRA 213 4.1 VECTORS 213 4.1.1 SPECIAL VECTORS 215 4.1.2 COMPARING
TWO VECTORS 216 4.1.3 OPERATIONS ON VECTORS 218 4.1.4 PROPERTIES OF
VECTORS AND THEIR OPERATIONS 224 4.2 MATRICES 228 4.2.1 REPRESENTATIONS
OF MATRICES 228 4.2.2 SPECIAL MATRICES 230 4.2.3 COMPARING TWO MATRICES
232 4.2.4 OPERATIONS ON MATRICES 233 4.2.5 PROPERTIES OF MATRICES AND
THEIR OPERATIONS 236 4.3 THE ALGEBRA OF SOLVING LINEAR EQUATIONS 240
4.3.1 THE PROBLEM OF SOLVING A SYSTEM OF LINEAR EQUATIONS 240 4.3.2 THE
BASIC APPROACH TO SOLVING LINEAR EQUATIONS . . 242 4.3.3 THE INVERSE OF
A MATRIX 243 4.3.4 CLOSED-FORM INVERSES OF CERTAIN MATRICES 245 4.3.5 A
NUMERICAL METHOD FOR FINDING THE INVERSE OF AN (N X N) MATRIX: GAUSSIAN
ELIMINATION 249 4.4 THE THEORY OF SOLVING LINEAR EQUATIONS 254 4.4.1
PROPERTIES OF THE INVERSE 255 4.4.2 LINEARLY INDEPENDENT VECTORS 257 4.5
VECTOR SPACES 265 4.5.1 DEVELOPING AN ABSTRACT SYSTEM 266 4.5.2
DEVELOPING AN AXIOMATIC SYSTEM 270 4.5.3 DERIVING RESULTS FOR THE
AXIOMATIC SYSTEM 275 CHAPTER SUMMARY 277 EXERCISES 279 5 SELECTED TOPICS
IN ABSTRACT ALGEBRA 289 5.1 INTEGER ARITHMETIC 289 5.1.1 BINARY
OPERATIONS ON INTEGERS 289 5.1.2 DIVISIBILITY OF INTEGERS 293 5.1.3
INTEGERS MODULO N 299 5.1.4 PRIME NUMBERS 302 5.2 GROUP THEORY 304 5.2.1
GROUPS 305 5.2.2 ADDITIONAL EXAMPLES OF GROUPS 311 5.2.3 SUBGROUPS 317
5.3 RINGS AND FIELDS 319 5.3.1 RINGS 319 5.3.2 FIELDS 326 XX CONTENTS
CHAPTER SUMMARY 328 EXERCISES 329 6 SELECTED TOPICS IN REAL ANALYSIS 335
6.1 ALGEBRAIC AND ORDER PROPERTIES OF REAL NUMBERS 335 6.1.1 ALGEBRAIC
PROPERTIES OF REAL NUMBERS 336 6.1.2 ORDER PROPERTIES OF REAL NUMBERS
338 6.2 SOME DIFFERENCES BETWEEN Z, Q, AND 1R 346 6.2.1 THE INFIMUM
PROPERTY OF REAL NUMBERS 346 6.2.2 USING THE INFIMUM PROPERTY TO
DISTINGUISH THE REALS FROM THE RATIONALS 350 6.2.3 UPPER BOUNDS AND
SUPREMA 355 6.2.4 THE NUMBER OF ELEMENTS IN A SET 356 6.3 SEQUENCES OF
REAL NUMBERS 364 6.3.1 WHAT IS A SEQUENCE OF REAL NUMBERS? 365 6.3.2
OPERATIONS ON SEQUENCES 367 6.3.3 SEQUENCES WITH SPECIAL PROPERTIES:
BOUNDED AND MONOTONE SEQUENCES 369 6.3.4 COMPARISONS OF SEQUENCES AND
SETS 375 6.4 CONVERGENCE OF SEQUENCES OF REAL NUMBERS 377 6.4.1
CONVERGENCE OF A SEQUENCE TO A GIVEN REAL NUMBER 378 6.4.2 CONVERGENCE
RESULTS 386 6.4.3 CONVERGENCE OF A SEQUENCE TO SOME REAL NUMBER 391 6.5
GENERALIZATIONS OF SEQUENCES OF REAL NUMBERS 394 6.5.1 SEQUENCES OF
VECTORS 394 6.5.2 CONVERGENCE OF A SEQUENCE OF VECTORS 395 6.5.3 AN
ABSTRACT SYSTEM FOR SEQUENCES: METRIC SPACES 398 CHAPTER SUMMARY 401
EXERCISES 403 A SUMMARY OF PROOF TECHNIQUES 407 SOLUTIONS TO
ODD-NUMBERED EXERCISES 417 SOLUTIONS TO CHAPTER 1 EXERCISES 417
SOLUTIONS TO CHAPTER 2 EXERCISES 425 SOLUTIONS TO CHAPTER 3 EXERCISES
430 SOLUTIONS TO CHAPTER 4 EXERCISES 436 SOLUTIONS TO CHAPTER 5
EXERCISES 443 SOLUTIONS TO CHAPTER 6 EXERCISES 450 GLOSSARY . 455 INDEX
471
|
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| isbn | 0964451905 |
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| spelling | Solow, Daniel Verfasser (DE-588)141538406 aut The keys to advanced mathematics recurrent themes in abstract reasoning Daniel Solow Mansfield, OH BookMasters Distribution Center 1995 XX, 476 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiskunde gtt Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematik (DE-588)4037944-9 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007428331&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Solow, Daniel The keys to advanced mathematics recurrent themes in abstract reasoning Wiskunde gtt Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4037944-9 |
| title | The keys to advanced mathematics recurrent themes in abstract reasoning |
| title_auth | The keys to advanced mathematics recurrent themes in abstract reasoning |
| title_exact_search | The keys to advanced mathematics recurrent themes in abstract reasoning |
| title_full | The keys to advanced mathematics recurrent themes in abstract reasoning Daniel Solow |
| title_fullStr | The keys to advanced mathematics recurrent themes in abstract reasoning Daniel Solow |
| title_full_unstemmed | The keys to advanced mathematics recurrent themes in abstract reasoning Daniel Solow |
| title_short | The keys to advanced mathematics |
| title_sort | the keys to advanced mathematics recurrent themes in abstract reasoning |
| title_sub | recurrent themes in abstract reasoning |
| topic | Wiskunde gtt Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Wiskunde Mathematik Mathematics |
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