Introduction to proofs and proof strategies:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge, United Kingdom
Cambridge University Press
2023
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| Edition: | First published |
| Series: | Cambridge mathematical textbooks
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 342 Seiten Illustrationen, Diagramme |
| ISBN: | 9781009096287 |
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CONTENTS List of Symbols and Notation Preface Part I 1 2 3 page x xiii Core Material 3 Numbers, Quadratics and Inequalities 1.1 The Quadratic Formula 1.2 Working with Inequalities - Settingthe Stage 1.3 The Arithmetic-Geometric Mean andthe Triangle Inequalities 1.4 Types of Numbers 1.5 Problems 1.6 Solutions to Exercises 9 16 20 23 Sets, Functions and the Field Axioms 2.1 Sets 2.2 Functions 2.3 The Field Axioms 2.4 Appendix: Infinite Unions and Intersections 2.5 Appendix: Defining Functions 2.6 Problems 2.7 Solutions to Exercises 25 35 43 49 52 55 63 Informal Logic and Proof Strategies 3.1 Mathematical Statements and their Building Blocks 3.2 The Logic Symbols 3.3 Truth and Falsity of Compound Statements 3.4 Truth Tables and Logical Equivalences 3.5 Negation 3.6 Proof Strategies 3.7 Problems 3.8 Solutions to Exercises 3 6 25 66 67 70 72 77 82 86 92 98
viii ) Contents 4 Mathematical Induction 4.1 The Principle of Mathematical Induction 4.2 Summation and Product Notation 4.3 Variations 4.4 Additional Examples 4.5 Strong Mathematical Induction 4.6 Problems 4.7 Solutions to Exercises 99 99 106 109 111 115 121 126 5 Bijections and Cardinality 5.1 Injections, Surjections and Bijections 5.2 Compositions 5.3 Cardinality 5.4 Cardinality Theorems 5.5 More Cardinality and the Schröder-Bernstein Theorem 5.6 Problems 5.7 Solutions to Exercises 128 128 135 139 144 151 153 162 6 Integers and Divisibility 6.1 Divisibility and the Division Algorithm 6.2 Greatest Common Divisors and the Euclidean Algorithm 6.3 The Fundamental Theorem of Arithmetic 6.4 Problems 6.5 Solutions to Exercises 164 164 168 175 179 183 7 Relations 7.1 The Definition of a Relation 7.2 Equivalence Relations 7.3 Equivalence Classes 7.4 Congruence Modulo n 7.5 Problems 7.6 Solutions to Exercises 184 184 187 191 196 199 203 Part II Additional Topics 8 Elementary Combinatorics 8.1 Counting Arguments: Selections, Arrangements and Permutations 8.2 The Binomial Theorem and Pascal’s Triangle 8.3 The Pigeonhole Principle 8.4 The Inclusion-Exclusion Principle 8.5 Problems Solutions to Exercises 8.6 207 207 215 219 221 228 231
Contents 9 Preview of Real Analysis - Limits and Continuity 9.1 9.2 9.3 9.4 9.5 9.6 10 Complex Numbers 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11 The Limit of a Sequence The Limit of a Function The Relation between Limits of Functions and Sequences Continuity and Differentiability Problems Solutions to Exercises Background The Field of Complex Numbers The Complex Plane and the Triangle Inequality Square Roots and Quadratic Equations Polar Representation of Complex Numbers De Moivre’s Theorem and Roots The Exponential Function Problems Solutions to Exercises Preview of Linear Algebra 11.1 The Spaces R" and their Properties 11.2 11.3 11.4 11.5 11.6 11.7 Index Geometric Vectors Abstract Vector Spaces Subspaces Linear Maps and Isomorphisms Problems Solutions to Exercises 234 234 242 252 254 260 264 266 266 267 276 280 284 288 293 295 299 306 306 309 317 321 325 331 335 339 |
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| spelling | Fuchs, Shay Verfasser (DE-588)1305699394 aut Introduction to proofs and proof strategies Shay Fuchs, University of Toronto First published Cambridge, United Kingdom Cambridge University Press 2023 © 2023 xiv, 342 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical textbooks Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Beweisführung (DE-588)4227233-6 gnd rswk-swf Beweis (DE-588)4132532-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Mathematische Logik (DE-588)4037951-6 s Beweis (DE-588)4132532-1 s Beweisführung (DE-588)4227233-6 s DE-604 Erscheint auch als Online-Ausgabe, EPUB 10.1017/9781009089005 978-1-009-08900-5 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034772506&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fuchs, Shay Introduction to proofs and proof strategies Mathematische Logik (DE-588)4037951-6 gnd Beweisführung (DE-588)4227233-6 gnd Beweis (DE-588)4132532-1 gnd |
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| title | Introduction to proofs and proof strategies |
| title_auth | Introduction to proofs and proof strategies |
| title_exact_search | Introduction to proofs and proof strategies |
| title_exact_search_txtP | Introduction to proofs and proof strategies |
| title_full | Introduction to proofs and proof strategies Shay Fuchs, University of Toronto |
| title_fullStr | Introduction to proofs and proof strategies Shay Fuchs, University of Toronto |
| title_full_unstemmed | Introduction to proofs and proof strategies Shay Fuchs, University of Toronto |
| title_short | Introduction to proofs and proof strategies |
| title_sort | introduction to proofs and proof strategies |
| topic | Mathematische Logik (DE-588)4037951-6 gnd Beweisführung (DE-588)4227233-6 gnd Beweis (DE-588)4132532-1 gnd |
| topic_facet | Mathematische Logik Beweisführung Beweis Lehrbuch |
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