Logic, mathematics, and computer science: modern foundations with practical applications
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| Language: | English |
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New York
Springer
[2015]
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| Edition: | Second edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 391 Seiten Illustrationen, Diagramme |
| ISBN: | 9781493932221 |
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| 240 | 1 | 0 | |a Foundations of logic and mathematics (applications to computer science and cryptography, 2002) |
| 245 | 1 | 0 | |a Logic, mathematics, and computer science |b modern foundations with practical applications |c Yves Nievergelt |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York |b Springer |c [2015] | |
| 264 | 4 | |c © 2015 | |
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| adam_text | Titel: Logic, mathematics, and computer science
Autor: Nievergelt, Yves
Jahr: 2015
Contents
1 Propositional Logic: Proofs from Axioms and Inference Rules..................1
1.1 Introduction ..........................................................................................................................1
1.1.1 An Example Demonstrating the Use of Logic in
Real Life..............................................................................................................2
1.2 The Pure Propositional Calculus....................................... 4
1.2.1 Formulae, Axioms, Inference Rules, and Proofs..........................5
1.3 The Pure Positive Implications Propositional Calculus............................9
1.3.1 Examples of Proofs in the Implicational Calculus......................9
1.3.2 Derived Rules: Implications Subject to Hypotheses..................11
1.3.3 A Guide for Proofs: an Implicational Deduction Theorem.. 14
1.3.4 Example: Law of Assertion from the Deduction Theorem.. 18
1.3.5 More Examples to Design Proofs of
Implicational Theorems..............................................................................21
1.3.6 Another Guide for Proofs: Substitutivity of Equivalences .. 23
1.3.7 More Derived Rules of Inference..........................................................25
1.3.8 The Laws of Commutation and of Assertion..................................27
1.3.9 Exercises on the Classical Implicational Calculus......................28
1.3.10 Equivalent Implicational Axiom Systems........................................29
1.3.11 Exercises on Kleene s Axioms................................................................30
1.3.12 Exercises on Tarski s Axioms..................................................................31
1.4 Proofs by the Converse Law of Contraposition................................................32
1.4.1 Examples of Proofs in the Full Propositional Calculus............32
1.4.2 Guides for Proofs in the Propositional Calculus..........................34
1.4.3 Proofs by Reductio ad Absurdum........................................................35
1.4.4 Proofs by Cases................................................................................................36
1.4.5 Exercises on Frege s and Church s Axioms....................................37
1.5 Other Connectives..............................................................................................................38
1.5.1 Definitions of Other Connectives..........................................................38
1.5.2 Examples of Proofs of Theorems with Conjunctions................38
1.5.3 Examples of Proofs of Theorems with Equivalences................41
1.5.4 Examples of Proofs of Theorems with Disjunctions..................44
1.5.5 Examples of Proofs with Conjunctions and Disjunctions ... 46
1.5.6 Exercises on Other Connectives............................................................47
1.6 Patterns of Deduction with Other Connectives................................................48
1.6.1 Conjunctions of Implications..................................................................48
1.6.2 Proofs by Cases or by Contradiction..................................................53
1.6.3 Exercises on Patterns of Deduction......................................................54
1.6.4 Equivalent Classical Axiom Systems..................................................55
1.6.5 Exercises on Kleene s, Rosser s, and Tarski s Axioms............56
1.7 Completeness, Decidability, Independence, Provability,
and Soundness......................................................................................................................56
1.7.1 Multi-Valued Fuzzy Logics......................................................................56
1.7.2 Sound Multi-Valued Fuzzy Logics......................................................57
1.7.3 Independence and Unprovability..........................................................59
1.7.4 Complete Multi-Valued Fuzzy Logics................................................61
1.7.5 Peirce s Law as a Denial of the Antecedent....................................62
1.7.6 Exercises on Church s and Lukasiewicz s
Triadic Systems................................................................................................62
1.8 Boolean Logic......................................................................................................................63
1.8.1 The Truth Table of the Logical Implication....................................63
1.8.2 Boolean Logic on Earth and in Space................................................65
1.9 Automated Theorem Proving......................................................................................67
1.9.1 The Provability Theorem............................................................................67
1.9.2 The Completeness Theorem....................................................................69
1.9.3 Example: Peirce s Law from the Completeness Theorem... 69
1.9.4 Exercises on the Deduction Theorem ................................................72
2 First-Order Logic: Proofs with Quantifiers..............................................................75
2.1 Introduction..........................................................................................................................75
2.2 The Pure Predicate Calculus of First Order........................................................75
2.2.1 Logical Predicates..........................................................................................75
2.2.2 Variables, Quantifiers, and Formulae..................................................77
2.2.3 Proper Substitutions of Free or Bound Variables........................78
2.2.4 Axioms and Rules for the Pure Predicate Calculus....................80
2.2.5 Exercises on Quantifiers ............................................................................82
2.2.6 Examples with Implications and Predicate Calculi..................82
2.2.7 Examples with Pure Prepositional and Predicate Calculi ... 86
2.2.8 Other Axiomatic Systems for the Pure Predicate Calculus.. 87
2.2.9 Exercises on Kleene s, Margaris s, and Rosser s Axioms ... 89
2.3 Methods of Proof for the Pure Predicate Calculus ........................................90
2.3.1 Substituting Equivalent Formulae........................................................90
2.3.2 Discharging Hypotheses ............................................................................91
2.3.3 Prenex Normal Form....................................................................................95
2.3.4 Proofs with More than One Quantifier..............................................96
2.3.5 Exercises on the Substitutivity of Equivalence..............................97
2.4 Predicate Calculus with Other Connectives........................................................98
2.4.1 Universal Quantifiers and Conjunctions or Disjunctions________98
2.4.2 Existential Quantifiers and Conjunctions or Disjunctions... 100
2.4.3 Exercises on Quantifiers with Other Connectives........................101
2.5 Equality-Predicates ..........................................................................................................101
2.5.1 First-Order Predicate Calculi with an Equality-Predicate ... 102
2.5.2 Simple Applied Predicate Calculi
with an Equality-Predicate........................................................................103
2.5.3 Other Axiom Systems for the Equality-Predicate........................106
2.5.4 Defined Ranking-Predicates ....................................................................107
2.5.5 Exercises on Equality-Predicates..........................................................107
3 Set Theory: Proofs by Detachment, Contraposition, and
Contradiction..................................................................................................................................109
3.1 Introduction ..........................................................................................................................109
3.2 Sets and Subsets..................................................................................................................110
3.2.1 Equality and Extensionality......................................................................110
3.2.2 The Empty Set..................................................................................................114
3.2.3 Subsets and Supersets..................................................................................114
3.2.4 Exercises on Sets and Subsets................................................................118
3.3 Pairing, Power, and Separation..................................................................................119
3.3.1 Pairing ..................................................................................................................119
3.3.2 Power Sets..........................................................................................................122
3.3.3 Separation of Sets..........................................................................................124
3.3.4 Exercises on Pairing, Power, and Separation of Sets................126
3.4 Unions and Intersections of Sets..............................................................................127
3.4.1 Unions of Sets..................................................................................................127
3.4.2 Intersections of Sets......................................................................................132
3.4.3 Unions and Intersections of Sets............................................................135
3.4.4 Exercises on Unions and Intersections of Sets..............................139
3.5 Cartesian Products and Relations ............................................................................142
3.5.1 Cartesian Products of Sets........................................................................142
3.5.2 Cartesian Products of Unions and Intersections ..........................147
3.5.3 Mathematical Relations and Directed Graphs ..............................149
3.5.4 Exercises on Cartesian Products of Sets............................................153
3.6 Mathematical Functions................................................................................................154
3.6.1 Mathematical Functions..............................................................................154
3.6.2 Images and Inverse Images of Sets by Functions........................159
3.6.3 Exercises on Mathematical Functions................................................162
3.7 Composite and Inverse Functions............................................................................164
3.7.1 Compositions of Functions ......................................................................164
3.7.2 Injective, Surjective, Bijective, and Inverse Functions ............166
3.7.3 The Set of all Functions from a Set to a Set....................................171
3.7.4 Exercises on Injective, Surjective, and Inverse Functions ... 173
3.8 Equivalence Relations ..................................................174
3.8.1 Reflexive, Symmetric, Transitive, or
Anti-Symmetric Relations..............................................................174
3.8.2 Partitions and Equivalence Relations..................................................175
3.8.3 Exercises on Equivalence Relations.............................................179
3.9 Ordering Relations............................................................................................................180
3.9.1 Preorders and Partial Orders....................................................................180
3.9.2 Total Orders and Well-Orderings..........................................................182
3.9.3 Exercises on Ordering Relations............................................................185
4 Mathematical Induction: Definitions and Proofs by Induction 189
4.1 Introduction .........................................189
4.2 Mathematical Induction.....................................190
4.2.1 The Axiom of Infinity.........................................................190
4.2.2 The Principle of Mathematical Induction........................................193
4.2.3 Definitions by Mathematical Induction ............................................195
4.2.4 Exercises on Mathematical Induction................................................197
4.3 Arithmetic with Natural Numbers............................................................................198
4.3.1 Addition with Natural Numbers............................................................198
4.3.2 Multiplication with Natural Numbers................................................200
4.3.3 Exercises on Arithmetic by Induction................................................203
4.4 Orders and Cancellations..............................................................................................205
4.4.1 Orders on the Natural Numbers..............................................................205
4.4.2 Laws of Arithmetic Cancellations........................................................210
4.4.3 Exercises on Orders and Cancellations..............................................213
4.5 Integers....................................................................................................................................214
4.5.1 Negative Integers............................................................................................214
4.5.2 Arithmetic with Integers............................................................................217
4.5.3 Order on the Integers....................................................................................220
4.5.4 Nonnegative Integral Powers of Integers..........................................226
4.5.5 Exercises on Integers with Induction..................................................228
4.6 Rational Numbers..............................................................................................................229
4.6.1 Definition of Rational Numbers............................................................229
4.6.2 Arithmetic with Rational Numbers......................................................231
4.6.3 Notation for Sums and Products............................................................237
4.6.4 Order on the Rational Numbers..............................................................241
4.6.5 Exercises on Rational Numbers ............................................................243
4.7 Finite Cardinality ..............................................................................................................244
4.7.1 Equal Cardinalities........................................................................................244
4.7.2 Finite Sets............................................................................................................248
4.7.3 Exercises on Finite Sets..............................................................................252
4.8 Infinite Cardinality........................................................................................................252
4.8.1 Infinite Sets................................................................................................252
4.8.2 Denumerable Sets..........................................................................................254
4.8.3 The Bernstein-Cantor-Schroder Theorem......................................258
4.8.4 Denumerability of all Finite Sequences
of Natural Numbers......................................................................................260
4.8.5 Other Infinite Sets..........................................................................................262
4.8.6 Further Issues in Cardinality....................................................................263
4.8.7 Exercises on Infinite Sets..........................................................................265
5 Well-Formed Sets: Proofs by Transfinite Induction with
Already Well-Ordered Sets..................................................................................................267
5.1 Introduction ..........................................................................................................................267
5.2 Transfinite Methods..........................................................................................................267
5.2.1 Transfinite Induction....................................................................................267
5.2.2 Transfinite Construction..............................................................................269
5.2.3 Exercises on Transfinite Methods ........................................................271
5.3 Transfinite Sets and Ordinals......................................................................................271
5.3.1 Transitive Sets..................................................................................................271
5.3.2 Ordinals................................................................................................................272
5.3.3 Well-Ordered Sets of Ordinals................................................................274
5.3.4 Unions and Intersections of Sets of Ordinals................................275
5.3.5 Exercises on Ordinals..................................................................................276
5.4 Regularity of Well-Formed Sets................................................................................277
5.4.1 Well-Formed Sets ..........................................................................................277
5.4.2 Regularity............................................................................................................279
5.4.3 Exercises on Well-Formed Sets..............................................................281
6 The Axiom of Choice: Proofs by Transfinite Induction ..................................283
6.1 Introduction ..........................................................................................................................283
6.2 The Choice Principle........................................................................................................283
6.2.1 The Choice-Function Principle..............................................................284
6.2.2 The Choice-Set Principle ..........................................................................286
6.2.3 Exercises on Choice Principles..............................................................288
6.3 Maximality and Well-Ordering Principles..........................................................288
6.3.1 Zermelo s Well-Ordering Principle......................................................288
6.3.2 Zom s Maximal-Element Principle......................................................290
6.3.3 Exercises on Maximality and Well-Orderings..............................292
6.4 Unions, Intersections, and Products of Families of Sets............................292
6.4.1 The Multiplicative Principle....................................................................292
6.4.2 The Distributive Principle..........................................................................293
6.4.3 Exercises on the Distributive and Multiplicative
Principles ............................................................................................................295
6.5 Equivalence of the Choice, Zorn s, and Zermelo s Principles................295
6.5.1 Towers of Sets..................................................................................................296
6.5.2 Zorn s Maximality from the Choice Principle..............................297
6.5.3 Exercises on Towers of Sets ....................................................................299
6.6 Yet Other Principles Related to the Axiom of Choice..................................299
6.6.1 Yet Other Principles Equivalent to the Axiom of Choice.... 299
6.6.2 Consequences of the Axiom of Choice..............................................300
6.6.3 Exercises on Related Principles ............................................................301
7 Applications: Nobel-Prize Winning Applications of Sets,
Functions, and Relations ......................................................................................................303
7.1 Introduction ..........................................................................................................................303
7.2 Game Theory........................................................................................................................304
7.2.1 Introduction........................................................................................................304
7.2.2 Mathematical Models for The Prisoner s Dilemma....................305
7.2.3 Dominant Strategies......................................................................................307
7.2.4 Mixed Strategies..............................................................................................309
7.2.5 Existence of Nash Equilibria for Two Players
with Two Mixed Strategies........................................................................310
7.2.6 Exercises on Mathematical Games......................................................313
7.3 Match Making......................................................................................................................315
7.3.1 Introduction........................................................................................................315
7.3.2 A Mathematical Model for Optimal Match Making..................316
7.3.3 An Algorithm for Optimal Match Making
with a Match Maker......................................................................................317
7.3.4 An Algorithm for Optimal Match Making
Without a Match Maker..............................................................................319
7.3.5 Exercises on Gale Shapley s Algorithms....................................320
7.3.6 Projects..................................................................................................................321
7.4 Arrow s Impossibility Theorem................................................................................321
7.4.1 Introduction........................................................................................................321
7.4.2 A Mathematical Model for Arrow s
Impossibility Theorem................................................................................324
7.4.3 Statement and Proof of Arrow s Impossibility Theorem________326
7.4.4 Exercises on Arrow s Impossibility Theorem................................330
Solutions to Some Odd-Numbered Exercises..................................................................331
References........................373
Index 381
|
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| spelling | Nievergelt, Yves 1954- Verfasser (DE-588)121171108 aut Foundations of logic and mathematics (applications to computer science and cryptography, 2002) Logic, mathematics, and computer science modern foundations with practical applications Yves Nievergelt Second edition New York Springer [2015] © 2015 xii, 391 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 s Mengenlehre (DE-588)4074715-3 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4939-3223-8 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028304331&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nievergelt, Yves 1954- Logic, mathematics, and computer science modern foundations with practical applications Mengenlehre (DE-588)4074715-3 gnd Mathematische Logik (DE-588)4037951-6 gnd |
| subject_GND | (DE-588)4074715-3 (DE-588)4037951-6 |
| title | Logic, mathematics, and computer science modern foundations with practical applications |
| title_alt | Foundations of logic and mathematics (applications to computer science and cryptography, 2002) |
| title_auth | Logic, mathematics, and computer science modern foundations with practical applications |
| title_exact_search | Logic, mathematics, and computer science modern foundations with practical applications |
| title_full | Logic, mathematics, and computer science modern foundations with practical applications Yves Nievergelt |
| title_fullStr | Logic, mathematics, and computer science modern foundations with practical applications Yves Nievergelt |
| title_full_unstemmed | Logic, mathematics, and computer science modern foundations with practical applications Yves Nievergelt |
| title_short | Logic, mathematics, and computer science |
| title_sort | logic mathematics and computer science modern foundations with practical applications |
| title_sub | modern foundations with practical applications |
| topic | Mengenlehre (DE-588)4074715-3 gnd Mathematische Logik (DE-588)4037951-6 gnd |
| topic_facet | Mengenlehre Mathematische Logik |
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