Sets, logic and maths for computing:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
London [u.a.]
Springer
2012
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Undergraduate Topics in Computer Science
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024956189&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXI, 283 S. graph. Darst. |
| ISBN: | 9781447124993 9781447125006 |
Internformat
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| 100 | 1 | |a Makinson, David |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Sets, logic and maths for computing |c David Makinson |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a London [u.a.] |b Springer |c 2012 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: Sets, logic and maths for computing
Autor: Makinson, David
Jahr: 2012
1 Collecting Things Together: Sets 1
1.1 The Intuitive Concept of a Set 1
1.2 Basic Relations Between Sets 2
1.2.1 Inclusion 2
1.2.2 Identity and Proper Inclusion 3
1.2.3 Diagrams 7
1.2.4 Ways of Defining a Set 9
1.3 The Empty Set 10
1.3.1 Emptiness 10
1.3.2 Disjoint Sets 11
1.4 Boolean Operations on Sets 11
1.4.1 Intersection 11
1.4.2 Union 13
1.4.3 Difference and Complement 16
1.5 Generalized Union and Intersection 18
1.6 Power Sets 20
Selected Reading 25
2 Comparing Things: Relations 27
2.1 Ordered Tuples, Cartesian Products and Relations 27
2.1.1 Ordered Tuples 28
2.1.2 Cartesian Products 28
2.1.3 Relations 30
2.2 Tables and Digraphs for Relations 32
2.2.1 Tables 33
2.2.2 Digraphs 33
2.3 Operations on Relations 34
2.3.1 Converse 35
2.3.2 Join, Projection, Selection 36
2.3.3 Composition 38
2.3.4 Image 40
2.4 Reflexivity and Transitivity 41
2.4.1 Reflexivity 42
2.4.2 Transitivity 43
2.5 Equivalence Relations and Partitions 44
2.5.1 Symmetry 44
2.5.2 Equivalence Relations 44
2.5.3 Partitions 46
2.5.4 The Partition/Equivalence Correspondence 47
2.6 Relations for Ordering 49
2.6.1 Partial Order 49
2.6.2 Linear Orderings 50
2.6.3 Strict Orderings 50
2.7 Closing with Relations 52
2.7.1 Transitive Closure of a Relation 52
2.7.2 Closure of a Set Under a Relation 54
Selected Reading 56
3 Associating One Item with Another:
Functions 57
3.1 What Is a Function? 57
3.2 Operations on Functions 60
3.2.1 Domain and Range 60
3.2.2 Restriction, Image, Closure 61
3.2.3 Composition 62
3.2.4 Inverse 63
3.3 Injections, Surjections, Bijections 64
3.3.1 Injectivity 64
3.3.2 Surjectivity 65
3.3.3 Bijective Functions 67
3.4 Using Functions to Compare Size 68
3.4.1 Equinumerosity 68
3.4.2 Cardinal Comparison 70
3.4.3 The Pigeonhole Principle 70
3.5 Some Handy Functions 72
3.5.1 Identity Functions 72
3.5.2 Constant Functions 72
3.5.3 Projection Functions 73
3.5.4 Characteristic Functions 73
3.6 Families and Sequences 74
3.6.1 Families of Sets 74
3.6.2 Sequences and Suchlike 75
Selected Reading 7g
4 Recycling Outputs as Inputs: Induction and Recursion 79
4.1 What Are Induction and Recursion? 79
4.2 Proof by Simple Induction on the Positive Integers 80
4.2.1 An Example gj
4.2.2 The Principle Behind the Example 82
4.3 Definition by Simple Recursion on the Positive Integers. !! 85
4.4 Evaluating Functions Defined by Recursion 87
4.5 Cumulative Induction and Recursion 89
4.5.1 Cumulative Recursive Definitions 89
4.5.2 Proof by Cumulative Induction 90
4.5.3 Simultaneous Recursion and Induction 92
4.6 Structural Recursion and Induction 93
4.6.1 Defining Sets by Structural Recursion 94
4.6.2 Proof by Structural Induction 97
4.6.3 Defining Functions by Structural Recursion
on Their Domains 98
4.7 Recursion and Induction on Weil-Founded Sets* 102
4.7.1 Well-Founded Sets 102
4.7.2 Proof by Weil-Founded Induction 104
4.7.3 Defining Functions by Weil-Founded
Recursion on Their Domains 107
4.7.4 Recursive Programs 108
Selected Reading Ill
5 Counting Things: Combinatorics 113
5.1 Basic Principles for Addition and Multiplication 113
5.1.1 Principles Considered Separately 114
5.1.2 Using the Two Principles Together 117
5.2 Four Ways of Selecting k Items Out of n 117
5.2.1 Order and Repetition 118
5.2.2 Connections with Functions 119
5.3 Counting Formulae: Permutations and Combinations 122
5.3.1 The Formula for Permutations 122
5.3.2 Counting Combinations 124
5.4 Selections Allowing Repetition 126
5.4.1 Permutations with Repetition Allowed 127
5.4.2 Combinations with Repetition Allowed 128
5.5 Rearrangements and Partitions* 130
5.5.1 Rearrangements 130
5.5.2 Counting Configured Partitions 132
Selected Reading 136
6 Weighing the Odds: Probability 137
6.1 Finite Probability Spaces 137
6.1.1 Basic Definitions 138
6.1.2 Properties of Probability Functions 139
6.2 Philosophy and Applications 141
6.2.1 Philosophical Interpretations 141
6.2.2 The Art of Applying Probability Theory 143
6.2.3 Digression: A Glimpse of the Infinite Case* 143
6.3 Some Simple Problems 144
6.4 Conditional Probability 147
6.4.1 The Ratio Definition 147
6.4.2 Applying Conditional Probability 149
6.5 Interlude: Simpson s Paradox* 153
6.6 Independence 155
6.7 Bayes Theorem 157
6.8 Expectation* 159
Selected Reading 164
7 Squirrel Math: Trees 165
7.1 My First Tree 165
7.2 Rooted Trees 167
7.2.1 Explicit Definition 167
7.2.2 Recursive Definition 170
7.3 Working with Trees 171
7.3.1 Trees Grow Every where 171
7.3.2 Labelled and Ordered Trees 172
7.4 Interlude: Parenthesis-Free Notation 175
7.5 Binary Trees 176
7.6 Unrooted Trees 180
7.6.1 Definition 180
7.6.2 Properties 181
7.6.3 Spanning Trees 185
Selected Reading 188
8 Yea and Nay: Prepositional Logic 189
8.1 What Is Logic? 189
8.2 Truth-Functional Connectives 190
8.3 Tautological Relations 193
8.3.1 The Language of Propositional Logic 194
8.3.2 Tautological Implication 195
8.3.3 Tautological Equivalence 197
8.3.4 Tautologies and Contradictions 200
8.4 Normal Forms 202
8.4.1 Disjunctive Normal Form 202
8.4.2 Conjunctive Normal Form* 204
8.4.3 Eliminating Redundant Letters 206
8.4.4 Most Modular Version 207
8.5 Semantic Decomposition Trees 209
Selected Reading 215
9 Something About Everything: Quantificational Logic 217
9.1 The Language of Quantifiers 217
9.1.1 Some Examples 218
9.1.2 Systematic Presentation of the Language 219
9.1.3 Freedom and Bondage 223
9.2 Some Basic Logical Equivalences 224
9.2.1 Quantifier Interchange 224
9.2.2 Distribution 225
9.2.3 Vacuity and Relettering 226
9.3 Two Semantics for Quantificational Logic 227
9.3.1 The Common Part of the Two Semantics 227
9.3.2 Substitutional Reading 229
9.3.3 The x-Variant Reading 231
9.4 Semantic Analysis 233
9.4.1 Logical Implication 233
9.4.2 Clean Substitutions 236
9.4.3 Fundamental Rules 236
9.4.4 Identity 238
Selected Reading 241
10 Just Supposing: Proof and Consequence 243
10.1 Elementary Derivations 243
10.1.1 My First Derivation 243
10.1.2 The Logic Behind Chaining 245
10.2 Consequence Relations 247
10.2.1 The Tarski Conditions 247
10.2.2 Consequence and Chaining 249
10.2.3 Consequence as an Operation* 251
10.3 A Higher-Level Proof Strategy 252
10.3.1 Informal Conditional Proof 252
10.3.2 Conditional Proof as a Formal Rule 254
10.3.3 Flattening Split-Level Proofs 256
10.4 Other Higher-Level Proof Strategies 257
10.4.1 Disjunctive Proof and Proof by Cases 257
10.4.2 Proof by Contradiction 260
10.4.3 Rules with Quantifiers 263
10.5 Proofs as Recursive Structures* 266
10.5.1 Second-Level Proofs 266
10.5.2 Split-Level Proofs 268
10.5.3 Four Views of Mount Fuji 269
Selected Reading 274
Index 275
|
| any_adam_object | 1 |
| author | Makinson, David |
| author_facet | Makinson, David |
| author_role | aut |
| author_sort | Makinson, David |
| author_variant | d m dm |
| building | Verbundindex |
| bvnumber | BV040099607 |
| classification_rvk | ST 120 ST 130 ST 150 |
| ctrlnum | (OCoLC)802316068 (DE-599)BSZ362214476 |
| dewey-full | 004.0151 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 004 - Computer science |
| dewey-raw | 004.0151 |
| dewey-search | 004.0151 |
| dewey-sort | 14.0151 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik |
| edition | 2. ed. |
| format | Book |
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| genre_facet | Einführung |
| id | DE-604.BV040099607 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:08:01Z |
| institution | BVB |
| isbn | 9781447124993 9781447125006 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024956189 |
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| owner_facet | DE-11 DE-573 DE-29T |
| physical | XXI, 283 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate Topics in Computer Science |
| spellingShingle | Makinson, David Sets, logic and maths for computing Logik (DE-588)4036202-4 gnd Diskrete Mathematik (DE-588)4129143-8 gnd Informatik (DE-588)4026894-9 gnd Mengenlehre (DE-588)4074715-3 gnd Mathematische Logik (DE-588)4037951-6 gnd Kombinatorik (DE-588)4031824-2 gnd |
| subject_GND | (DE-588)4036202-4 (DE-588)4129143-8 (DE-588)4026894-9 (DE-588)4074715-3 (DE-588)4037951-6 (DE-588)4031824-2 (DE-588)4151278-9 |
| title | Sets, logic and maths for computing |
| title_auth | Sets, logic and maths for computing |
| title_exact_search | Sets, logic and maths for computing |
| title_full | Sets, logic and maths for computing David Makinson |
| title_fullStr | Sets, logic and maths for computing David Makinson |
| title_full_unstemmed | Sets, logic and maths for computing David Makinson |
| title_short | Sets, logic and maths for computing |
| title_sort | sets logic and maths for computing |
| topic | Logik (DE-588)4036202-4 gnd Diskrete Mathematik (DE-588)4129143-8 gnd Informatik (DE-588)4026894-9 gnd Mengenlehre (DE-588)4074715-3 gnd Mathematische Logik (DE-588)4037951-6 gnd Kombinatorik (DE-588)4031824-2 gnd |
| topic_facet | Logik Diskrete Mathematik Informatik Mengenlehre Mathematische Logik Kombinatorik Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024956189&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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