Mathematical thinking and writing: a transition to abstract mathematics
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
San Diego, Calif. <<[u.a.]>>
Academic Press
2003
|
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of contents Inhaltsverzeichnis |
| Beschreibung: | Includes index |
| Beschreibung: | XVIII, 304 S. Ill., graph. Darst. 25 cm |
| ISBN: | 0124649769 |
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| 245 | 1 | 0 | |a Mathematical thinking and writing |b a transition to abstract mathematics |c Randall B. Maddox |
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| adam_text | RANDALL B.MADDOX :.--;** -INE UNIYERSITY, M&LIBU, CM MATHEMATICAL T H
I N K I N G AND WRITING A TRANSITION TO ABSTRACT MATHEMATICS IJJACADEMIC
PRESS A HARCOURT SCIENCE AND TECHNOLOGY COMPANY VIII CONTENTS 1.3
UNIVERSAL AND EXISTENTIAL QUANTIFIERS 27 1.3.1 THE UNIVERSAL QUANTIFIER
27 1.3.2 THE EXISTENTIAL QUANTIFIER 29 1.3.3 UNIQUE EXISTENCE 30 1.4
NEGATIONS OF STATEMENTS 31 1.4.1 NEGATIONS OF P A Q ANDP V Q 32 1.4.2
NEGATIONS OF P *** Q 33 1.4.3 NEGATIONS OF STATEMENTS WITH V AND 3 33
CHAPTER 2 BEGINNER-LEVEL PROOFS 38 2.1 PROOFS INVOLVING SETS 38 2.1.1
TERMS INVOLVING SETS 38 2.1.2 DIRECT PROOFS 41 2.1.3 PROOFS BY
CONTRAPOSITIVE 44 2.1.4 PROOFS BY CONTRADICTION 45 2.1.5 DISPROVING A
STATEMENT 45 2.2 INDEXED FAMILIES OF SETS 47 2.3 ALGEBRAIC AND ORDERING
PROPERTIES OF M. 53 2.3.1 BASIC ALGEBRAIC PROPERTIES OF REAL NUMBERS 53
2.3.2 ORDERING OF THE REAL NUMBERS 56 2.3.3 ABSOLUTE VALUE 57 2.4 THE
PRINCIPLE OF MATHEMATICAL INDUCTION 61 2.4.1 THE STANDARD PMI 62 2.4.2
VARIATION OF THE PMI 64 2.4.3 STRONG INDUCTION 65 2.5 EQUIVALENCE
RELATIONS: THE IDEA OF EQUALITY 68 2.5.1 ANALYZING EQUALITY 68 2.5.2
EQUIVALENCE CLASSES 72 2.6 EQUALITY, ADDITION, AND MULTIPLICATION IN Q
76 2.6.1 EQUALITY IN Q 77 2.6.2 WELL-DEFINED + AND X ON Q 78 2.7 THE
DIVISION ALGORITHM AND DIVISIBILITY 79 2.7.1 EVEN AND ODD INTEGERS; THE
DIVISION ALGORITHM 79 2.7.2 DIVISIBILITY IN Z 81 2.8 ROOTS AND
IRRATIONAL NUMBERS 85 2.8.1 ROOTS OF REAL NUMBERS 86 2.8.2 EXISTENCE OF
IRRATIONAL NUMBERS 87 2.9 RELATIONS IN GENERAL 90 CONTENTS IX CHAPTER 3
FUNCTIONS 97 3.1 DEFINITIONS AND TERMINOLOGY 97 3.1.1 DEFINITION AND
EXAMPLES 97 3.1.2 OTHER TERMINOLOGY AND NOTATION 101 3.1.3 THREE
IMPORTANT THEOREMS 103 3.2 COMPOSITION AND INVERSE FUNCTIONS 106 3.2.1
COMPOSITION OF FUNCTIONS 106 3.2.2 INVERSE FUNCTIONS 108 3.3 CARDINALITY
OF SETS 110 3.3.1 FINITE SETS 111 3.3.2 INFINITE SETS 113 3.4 COUNTING
METHODS AND THE BINOMIAL THEOREM 118 3.4.1 THE PRODUCT RULE 118 3.4.2
PERMUTATIONS 122 3.4.3 COMBINATIONS AND PARTITIONS 122 3.4.4 COUNTING
EXAMPLES 125 3.4.5 THE BINOMIAL THEOREM 126 PARTII BASIC PRINCIPLES OF
ANALYSIS 131 CHAPTER 4 THE REAL NUMBERS 133 4.1 THE LEAST UPPER BOUND
AXIOM 134 4.1.1 LEAST UPPER BOUNDS 134 4.1.2 THE ARCHIMEDEAN PROPERTY OF
IR 136 4.1.3 GREATEST LOWER BOUNDS 137 4.1.4 THE LUB AND GLB PROPERTIES
APPLIED TO FINITE SETS 137 4.2 SETS IN R 140 4.2.1 OPEN AND CLOSED SETS
140 4.2.2 INTERIOR, EXTERIOR, AND BOUNDARY 142 4.3 LIMIT POINTS AND
CLOSURE OF SETS 143 4.3.1 CLOSURE OF SETS 144 4.4 COMPACTNESS 146 4.5
SEQUENCES IN E 149 4.5.1 MONOTONE SEQUENCES 150 4.5.2 BOUNDED SEQUENCES
151 4.6 CONVERGENCE OF SEQUENCES 153 4.6.1 CONVERGENCE TO A REAL NUMBER
154 4.6.2 CONVERGENCE TO OO 158 4.7 THE NESTED INTERVAL PROPERTY 160
4.7.1 FROM LUB AXIOM TO NIP 161 CONTENTS 4.7.2 THE NIP APPLIED TO
SUBSEQUENCES 162 4.7.3 FROM NIP TO LUB AXIOM 164 4.8 CAUCHY SEQUENCES
165 4.8.1 CONVERGENCE OF CAUCHY SEQUENCES 166 4.8.2 FROM COMPLETENESS TO
THE NIP 168 CHAPTER 5 FUNCTIONS OF A REAL VARIABLE 170 5.1 BOUNDED AND
MONOTONE FUNCTIONS 170 5.1.1 BOUNDED FUNCTIONS 170 5.1.2 MONOTONE
FUNCTIONS 171 5.2 LIMITS AND THEIR BASIC PROPERTIES 173 5.2.1 DEFINITION
OF LIMIT 173 5.2.2 BASIC THEOREMS OF LIMITS 175 5.3 MORE ON LIMITS 180
5.3.1 ONE-SIDED LIMITS 180 5.3.2 SEQUENTIAL LIMIT OFF 181 5.4 LIMITS
INVOLVING INFINITY 182 5.4.1 LIMITS AT INFINITY 183 5.4.2 LIMITS OF
INFINITY 185 5.5 CONTINUITY 187 5.5.1 CONTINUITY AT A POINT 188 5.5.2
CONTINUITY ON A SET 190 5.5.3 ONE-SIDED CONTINUITY 194 5.6 IMPLICATIONS
OF CONTINUITY 195 5.6.1 THE INTERMEDIATE VALUE THEOREM 195 5.6.2
CONTINUITY AND OPEN SETS 197 5.7 UNIFORM CONTINUITY 200 5.7.1 DEFINITION
AND EXAMPLES 200 5.7.2 UNIFORM CONTINUITY AND COMPACT SETS 202 PART III
BASIC PRINCIPLES OF ALGEBRA 205 CHAPTER 6 GROUPS 207 6.1 INTRODUCTION TO
GROUPS 207 6.1.1 BASIC CHARACTERISTICS OF ALGEBRAIC STRUCTURES 208 6.1.2
GROUPS DEFINED 210 6.1.3 SUBGROUPS 213 6.2 GENERATED AND CYCLIC
SUBGROUPS 215 6.2.1 SUBGROUP GENERATED BY^ C 6 216 6.2.2 CYCLIC
SUBGROUPS 217 CONTENTS XI 6.3 INTEGERS MODULO N AND QUOTIENT GROUPS 220
6.3.1 INTEGERS MODULO N 220 6.3.2 QUOTIENT GROUPS 223 6.3.3 COSETS AND
LAGRANGE S THEOREM 225 6.4 PERMUTATION GROUPS AND NORMAL SUBGROUPS 227
6.4.1 PERMUTATION GROUPS 227 6.4.2 THE ALTERNATING GROUP 4 4 229 6.4.3
THE DIHEDRAL GROUP D 8 230 6.4.4 NORMAL SUBGROUPS 232 6.4.5 EQUIVALENCES
AND IMPLICATIONS OF NORMALITY 233 6.5 GROUP MORPHISMS 236 CHAPTER 7
RINGS 243 7.1 RINGS AND SUBRINGS 243 7.1.1 RINGS DEFINED 243 7.1.2
EXAMPLES OF RINGS 245 7.1.3 SUBRINGS 248 7.2 RING PROPERTIES AND FIELDS
249 7.2.1 RING PROPERTIES 249 7.2.2 FIELDS DEFINED 254 7.3 RING
EXTENSIONS 256 7.3.1 ADJOINING ROOTS OF RING ELEMENTS 256 7.3.2
POLYNOMIAL RINGS 258 7.3.3 DEGREE OF A POLYNOMIAL 259 7.4 IDEALS 260
7.4.1 DEFINITION AND EXAMPLES 260 7.4.2 GENERATED IDEALS 262 7.4.3 PRIME
IDEALS 264 7.4.4 MAXIMAL IDEALS 264 7.5 INTEGRAL DOMAINS 267 7.6
UFDSANDPIDS 273 7.6.1 UNIQUE FACTORIZATION DOMAINS 273 7.6.2 PRINCIPAL
IDEAL DOMAINS 274 7.7 EUCLIDEAN DOMAINS 279 7.7.1 DEFINITION AND
PROPERTIES 279 7.7.2 POLYNOMIALS OVERAFIELD 282 7.7.3 Z[T]ISAUFD 284 7.8
RING MORPHISMS 287 7.8.1 PROPERTIES OF RING MORPHISMS 288 7.9 QUOTIENT
RINGS 291 INDEX 299
|
| adam_txt |
RANDALL B.MADDOX ' :.--;** -INE UNIYERSITY, M&LIBU, CM MATHEMATICAL T H
I N K I N G AND WRITING A TRANSITION TO ABSTRACT MATHEMATICS IJJACADEMIC
PRESS A HARCOURT SCIENCE AND TECHNOLOGY COMPANY VIII CONTENTS 1.3
UNIVERSAL AND EXISTENTIAL QUANTIFIERS 27 1.3.1 THE UNIVERSAL QUANTIFIER
27 1.3.2 THE EXISTENTIAL QUANTIFIER 29 1.3.3 UNIQUE EXISTENCE 30 1.4
NEGATIONS OF STATEMENTS 31 1.4.1 NEGATIONS OF P A Q ANDP V Q 32 1.4.2
NEGATIONS OF P *** Q 33 1.4.3 NEGATIONS OF STATEMENTS WITH V AND 3 33
CHAPTER 2 BEGINNER-LEVEL PROOFS 38 2.1 PROOFS INVOLVING SETS 38 2.1.1
TERMS INVOLVING SETS 38 2.1.2 DIRECT PROOFS 41 2.1.3 PROOFS BY
CONTRAPOSITIVE 44 2.1.4 PROOFS BY CONTRADICTION 45 2.1.5 DISPROVING A
STATEMENT 45 2.2 INDEXED FAMILIES OF SETS 47 2.3 ALGEBRAIC AND ORDERING
PROPERTIES OF M. 53 2.3.1 BASIC ALGEBRAIC PROPERTIES OF REAL NUMBERS 53
2.3.2 ORDERING OF THE REAL NUMBERS 56 2.3.3 ABSOLUTE VALUE 57 2.4 THE
PRINCIPLE OF MATHEMATICAL INDUCTION 61 2.4.1 THE STANDARD PMI 62 2.4.2
VARIATION OF THE PMI 64 2.4.3 STRONG INDUCTION 65 2.5 EQUIVALENCE
RELATIONS: THE IDEA OF EQUALITY 68 2.5.1 ANALYZING EQUALITY 68 2.5.2
EQUIVALENCE CLASSES 72 2.6 EQUALITY, ADDITION, AND MULTIPLICATION IN Q
76 2.6.1 EQUALITY IN Q 77 2.6.2 WELL-DEFINED + AND X ON Q 78 2.7 THE
DIVISION ALGORITHM AND DIVISIBILITY 79 2.7.1 EVEN AND ODD INTEGERS; THE
DIVISION ALGORITHM 79 2.7.2 DIVISIBILITY IN Z 81 2.8 ROOTS AND
IRRATIONAL NUMBERS 85 2.8.1 ROOTS OF REAL NUMBERS 86 2.8.2 EXISTENCE OF
IRRATIONAL NUMBERS 87 2.9 RELATIONS IN GENERAL 90 CONTENTS IX CHAPTER 3
FUNCTIONS 97 3.1 DEFINITIONS AND TERMINOLOGY 97 3.1.1 DEFINITION AND
EXAMPLES 97 3.1.2 OTHER TERMINOLOGY AND NOTATION 101 3.1.3 THREE
IMPORTANT THEOREMS 103 3.2 COMPOSITION AND INVERSE FUNCTIONS 106 3.2.1
COMPOSITION OF FUNCTIONS 106 3.2.2 INVERSE FUNCTIONS 108 3.3 CARDINALITY
OF SETS 110 3.3.1 FINITE SETS 111 3.3.2 INFINITE SETS 113 3.4 COUNTING
METHODS AND THE BINOMIAL THEOREM 118 3.4.1 THE PRODUCT RULE 118 3.4.2
PERMUTATIONS 122 3.4.3 COMBINATIONS AND PARTITIONS 122 3.4.4 COUNTING
EXAMPLES 125 3.4.5 THE BINOMIAL THEOREM 126 PARTII BASIC PRINCIPLES OF
ANALYSIS 131 CHAPTER 4 THE REAL NUMBERS 133 4.1 THE LEAST UPPER BOUND
AXIOM 134 4.1.1 LEAST UPPER BOUNDS 134 4.1.2 THE ARCHIMEDEAN PROPERTY OF
IR 136 4.1.3 GREATEST LOWER BOUNDS 137 4.1.4 THE LUB AND GLB PROPERTIES
APPLIED TO FINITE SETS 137 4.2 SETS IN R 140 4.2.1 OPEN AND CLOSED SETS
140 4.2.2 INTERIOR, EXTERIOR, AND BOUNDARY 142 4.3 LIMIT POINTS AND
CLOSURE OF SETS 143 4.3.1 CLOSURE OF SETS 144 4.4 COMPACTNESS 146 4.5
SEQUENCES IN E 149 4.5.1 MONOTONE SEQUENCES 150 4.5.2 BOUNDED SEQUENCES
151 4.6 CONVERGENCE OF SEQUENCES 153 4.6.1 CONVERGENCE TO A REAL NUMBER
154 4.6.2 CONVERGENCE TO OO 158 4.7 THE NESTED INTERVAL PROPERTY 160
4.7.1 FROM LUB AXIOM TO NIP 161 CONTENTS 4.7.2 THE NIP APPLIED TO
SUBSEQUENCES 162 4.7.3 FROM NIP TO LUB AXIOM 164 4.8 CAUCHY SEQUENCES
165 4.8.1 CONVERGENCE OF CAUCHY SEQUENCES 166 4.8.2 FROM COMPLETENESS TO
THE NIP 168 CHAPTER 5 FUNCTIONS OF A REAL VARIABLE 170 5.1 BOUNDED AND
MONOTONE FUNCTIONS 170 5.1.1 BOUNDED FUNCTIONS 170 5.1.2 MONOTONE
FUNCTIONS 171 5.2 LIMITS AND THEIR BASIC PROPERTIES 173 5.2.1 DEFINITION
OF LIMIT 173 5.2.2 BASIC THEOREMS OF LIMITS 175 5.3 MORE ON LIMITS 180
5.3.1 ONE-SIDED LIMITS 180 5.3.2 SEQUENTIAL LIMIT OFF 181 5.4 LIMITS
INVOLVING INFINITY 182 5.4.1 LIMITS AT INFINITY 183 5.4.2 LIMITS OF
INFINITY 185 5.5 CONTINUITY 187 5.5.1 CONTINUITY AT A POINT 188 5.5.2
CONTINUITY ON A SET 190 5.5.3 ONE-SIDED CONTINUITY 194 5.6 IMPLICATIONS
OF CONTINUITY 195 5.6.1 THE INTERMEDIATE VALUE THEOREM 195 5.6.2
CONTINUITY AND OPEN SETS 197 5.7 UNIFORM CONTINUITY 200 5.7.1 DEFINITION
AND EXAMPLES 200 5.7.2 UNIFORM CONTINUITY AND COMPACT SETS 202 PART III
BASIC PRINCIPLES OF ALGEBRA 205 CHAPTER 6 GROUPS 207 6.1 INTRODUCTION TO
GROUPS 207 6.1.1 BASIC CHARACTERISTICS OF ALGEBRAIC STRUCTURES 208 6.1.2
GROUPS DEFINED 210 6.1.3 SUBGROUPS 213 6.2 GENERATED AND CYCLIC
SUBGROUPS 215 6.2.1 SUBGROUP GENERATED BY^ C 6 216 6.2.2 CYCLIC
SUBGROUPS 217 CONTENTS XI 6.3 INTEGERS MODULO N AND QUOTIENT GROUPS 220
6.3.1 INTEGERS MODULO N 220 6.3.2 QUOTIENT GROUPS 223 6.3.3 COSETS AND
LAGRANGE'S THEOREM 225 6.4 PERMUTATION GROUPS AND NORMAL SUBGROUPS 227
6.4.1 PERMUTATION GROUPS 227 6.4.2 THE ALTERNATING GROUP 4 4 229 6.4.3
THE DIHEDRAL GROUP D 8 230 6.4.4 NORMAL SUBGROUPS 232 6.4.5 EQUIVALENCES
AND IMPLICATIONS OF NORMALITY 233 6.5 GROUP MORPHISMS 236 CHAPTER 7
RINGS 243 7.1 RINGS AND SUBRINGS 243 7.1.1 RINGS DEFINED 243 7.1.2
EXAMPLES OF RINGS 245 7.1.3 SUBRINGS 248 7.2 RING PROPERTIES AND FIELDS
249 7.2.1 RING PROPERTIES 249 7.2.2 FIELDS DEFINED 254 7.3 RING
EXTENSIONS 256 7.3.1 ADJOINING ROOTS OF RING ELEMENTS 256 7.3.2
POLYNOMIAL RINGS 258 7.3.3 DEGREE OF A POLYNOMIAL 259 7.4 IDEALS 260
7.4.1 DEFINITION AND EXAMPLES 260 7.4.2 GENERATED IDEALS 262 7.4.3 PRIME
IDEALS 264 7.4.4 MAXIMAL IDEALS 264 7.5 INTEGRAL DOMAINS 267 7.6
UFDSANDPIDS 273 7.6.1 UNIQUE FACTORIZATION DOMAINS 273 7.6.2 PRINCIPAL
IDEAL DOMAINS 274 7.7 EUCLIDEAN DOMAINS 279 7.7.1 DEFINITION AND
PROPERTIES 279 7.7.2 POLYNOMIALS OVERAFIELD 282 7.7.3 Z[T]ISAUFD 284 7.8
RING MORPHISMS 287 7.8.1 PROPERTIES OF RING MORPHISMS 288 7.9 QUOTIENT
RINGS 291 INDEX 299 |
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| dewey-sort | 3511.3 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| index_date | 2024-07-02T22:34:39Z |
| indexdate | 2024-07-09T21:23:59Z |
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| spelling | Maddox, Randall B. Verfasser aut Mathematical thinking and writing a transition to abstract mathematics Randall B. Maddox San Diego, Calif. <<[u.a.]>> Academic Press 2003 XVIII, 304 S. Ill., graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Includes index Proof theory Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Beweissystem (DE-588)4711800-3 gnd rswk-swf Beweissystem (DE-588)4711800-3 s Mathematische Logik (DE-588)4037951-6 s DE-604 http://www.loc.gov/catdir/description/els031/2001091290.html Publisher description http://www.loc.gov/catdir/toc/els031/2001091290.html Table of contents GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016849052&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Maddox, Randall B. Mathematical thinking and writing a transition to abstract mathematics Proof theory Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd Beweissystem (DE-588)4711800-3 gnd |
| subject_GND | (DE-588)4037951-6 (DE-588)4711800-3 |
| title | Mathematical thinking and writing a transition to abstract mathematics |
| title_auth | Mathematical thinking and writing a transition to abstract mathematics |
| title_exact_search | Mathematical thinking and writing a transition to abstract mathematics |
| title_exact_search_txtP | Mathematical thinking and writing a transition to abstract mathematics |
| title_full | Mathematical thinking and writing a transition to abstract mathematics Randall B. Maddox |
| title_fullStr | Mathematical thinking and writing a transition to abstract mathematics Randall B. Maddox |
| title_full_unstemmed | Mathematical thinking and writing a transition to abstract mathematics Randall B. Maddox |
| title_short | Mathematical thinking and writing |
| title_sort | mathematical thinking and writing a transition to abstract mathematics |
| title_sub | a transition to abstract mathematics |
| topic | Proof theory Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd Beweissystem (DE-588)4711800-3 gnd |
| topic_facet | Proof theory Logic, Symbolic and mathematical Mathematische Logik Beweissystem |
| url | http://www.loc.gov/catdir/description/els031/2001091290.html http://www.loc.gov/catdir/toc/els031/2001091290.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016849052&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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