The elements of advanced mathematics:
"The gap between the rote, calculational learning mode of calculus and ordinary differential equations and the more theoretical learning mode of analysis and abstract algebra grows ever wider and more distinct, and students' need for a well-guided transition grows with it. For more than si...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla.
Chapman & Hall
2002
|
| Edition: | 2. ed. |
| Series: | Studies in advanced mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "The gap between the rote, calculational learning mode of calculus and ordinary differential equations and the more theoretical learning mode of analysis and abstract algebra grows ever wider and more distinct, and students' need for a well-guided transition grows with it. For more than six years, the bestselling first edition of this classic text has helped them cross the mathematical bridge to more advanced studies in topics such as topology, abstract algebra, and real analysis. Carefully revised, brought thoroughly up to date, and expanded in essential areas, The Elements of Advanced Mathematics, Second Edition now does the job even better. It builds the background, tools, and skills students need to meet the challenges of mathematical rigor, axiomatics, and proofs."--BOOK JACKET. |
| Item Description: | Includes bibliographical references (p. 209-210) and index |
| Physical Description: | XVII, 214 S. Ill. |
| ISBN: | 1584883030 |
Staff View
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Record in the Search Index
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| adam_text | THE ELEMENTS OF ADVANCED MATHEMATICS / KRANTZ, STEVEN GEORGE : C2002
TABLE OF CONTENTS / INHALTSVERZEICHNIS
1 BASIC LOGIC 1
1.1 PRINCIPLES OF LOGIC1
1.2 TRUTH2
1.3 AND AND OR 3 3
1.4 NOT 6
1.5 IF- THEN 7
1.6 CONTRAPOSITIVE, CONVERSE, AND IFF 10
1.7 QUANTIFIERS13
1.8 TRUTH AND PROVABILITY17
EXERCISES21
2 METHODS OF PROOF 27
2.1 WHAT IS A PROOF?27
2.2 DIRECT PROOF28
2.3 PROOF BY CONTRADICTION33
2.4 PROOF BY INDUCTION36
2.5 OTHER METHODS OF PROOF40
EXERCISES43
3 SET THEORY 47
3.1 UNDEFINABLE TERMS47
3.2 ELEMENTS OF SET THEORY48
3.3 VENN DIAGRAMS52
3.4 FURTHER IDEAS IN ELEMENTARY SET THEORY53
3.5 INDEXING AND EXTENDED SET OPERATIONS55
EXERCISES57
4 RELATIONS AND FUNCTIONS 61
4.1 RELATIONS61
4.2 ORDER RELATIONS 64
4.3 FUNCTIONS 66
4.4 COMBINING FUNCTIONS 69
4.5 CANTOR S NOTION OF CARDINALITY 73
EXERCISES 85
5 AXIOMS OF SET THEORY, PARADOXES, AND RIGOR 91
5.1 AXIOMS OF SET THEORY91
5.2 THE AXIOM OF CHOICE 94
5.2.1 WELL ORDERING94
5.2.2 THE CONTINUUM HYPOTHESIS95
5.2.3 ZORN S LEMMA95
5.2.4 THE HAUSDORFF MAXIMALITY PRINCIPLE96
5.2.5 THE BANACH-TARSKI PARADOX97
5.3 INDEPENDENCE AND CONSISTENCY97
5.4 SET THEORY AND ARITHMETIC 99
EXERCISES101
6 NUMBER SYSTEMS 1056.0 PRELIMINARY REMARKS105
6.1 THE NATURAL NUMBER SYSTEM106
6.2 THE INTEGERS111
6.3 THE RATIONAL NUMBERS117
6.4 THE REAL NUMBER SYSTEM124
6.5 THE NON-STANDARD REAL NUMBER SYSTEM133
6.5.1 THE NEED FOR NON-STANDARD NUMBERS133
6.5.2 FILTERS AND ULTRAFILTERS133
6.5.3 A USEFUL MEASURE133
6.5.4 AN EQUIVALENCE RELATION134
6.5.5 AN EXTENSION OF THE REAL NUMBER SYSTEM134
6.6 THE COMPLEX NUMBERS135
6.7 THE QUATERNIONS, THE CAYLEY NUMBERS, AND BEYOND139
EXERCISES141
7 MORE ON THE REAL NUMBER SYSTEM 147
7.0 INTRODUCTION 147
7.1 SEQUENCES147
7.2 OPEN SETS AND CLOSED SETS149
7.3 COMPACT SETS151
7.4 THE CANTOR SET152
7.4.1 CONSTRUCTION OF A REMARKABLE COMPACT SET153
EXERCISES156
8 EXAMPLES OF AXIOMATIC THEORIES 159
8.0 INTRODUCTORY REMARKS159
8.1 GROUP THEORY159
8.2 EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY168
EXERCISES180.
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
|
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| dewey-ones | 510 - Mathematics |
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| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV014559645 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:03:32Z |
| institution | BVB |
| isbn | 1584883030 |
| language | English |
| lccn | 2001054765 |
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| owner | DE-739 DE-634 |
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| physical | XVII, 214 S. Ill. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Chapman & Hall |
| record_format | marc |
| series2 | Studies in advanced mathematics |
| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut The elements of advanced mathematics Steven G. Krantz 2. ed. Boca Raton, Fla. Chapman & Hall 2002 XVII, 214 S. Ill. txt rdacontent n rdamedia nc rdacarrier Studies in advanced mathematics Includes bibliographical references (p. 209-210) and index "The gap between the rote, calculational learning mode of calculus and ordinary differential equations and the more theoretical learning mode of analysis and abstract algebra grows ever wider and more distinct, and students' need for a well-guided transition grows with it. For more than six years, the bestselling first edition of this classic text has helped them cross the mathematical bridge to more advanced studies in topics such as topology, abstract algebra, and real analysis. Carefully revised, brought thoroughly up to date, and expanded in essential areas, The Elements of Advanced Mathematics, Second Edition now does the job even better. It builds the background, tools, and skills students need to meet the challenges of mathematical rigor, axiomatics, and proofs."--BOOK JACKET. Matemática (metodologia) larpcal Mathématiques Wiskunde gtt Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Mathematik (DE-588)4037944-9 s DE-604 Mathematische Logik (DE-588)4037951-6 s LoC Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009899013&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- The elements of advanced mathematics Matemática (metodologia) larpcal Mathématiques Wiskunde gtt Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd Mathematische Logik (DE-588)4037951-6 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4037951-6 |
| title | The elements of advanced mathematics |
| title_auth | The elements of advanced mathematics |
| title_exact_search | The elements of advanced mathematics |
| title_full | The elements of advanced mathematics Steven G. Krantz |
| title_fullStr | The elements of advanced mathematics Steven G. Krantz |
| title_full_unstemmed | The elements of advanced mathematics Steven G. Krantz |
| title_short | The elements of advanced mathematics |
| title_sort | the elements of advanced mathematics |
| topic | Matemática (metodologia) larpcal Mathématiques Wiskunde gtt Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd Mathematische Logik (DE-588)4037951-6 gnd |
| topic_facet | Matemática (metodologia) Mathématiques Wiskunde Mathematik Mathematics Mathematische Logik |
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