Discrete mathematics using a computer
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| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV022269978 | ||
| 003 | DE-604 | ||
| 005 | 20150506 | ||
| 007 | t| | ||
| 008 | 070214s2006 xx ad|| |||| 00||| eng d | ||
| 015 | |a 05,N43,0437 |2 dnb | ||
| 016 | 7 | |a 976543451 |2 DE-101 | |
| 020 | |a 9781846282416 |9 978-1-84628-241-6 | ||
| 020 | |a 1846282411 |c Pb. (Pr. in Vorb.) |9 1-84628-241-1 | ||
| 024 | 3 | |a 9781846282416 | |
| 028 | 5 | 2 | |a 11011002 |
| 035 | |a (OCoLC)255037636 | ||
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| 084 | |a 510 |2 sdnb | ||
| 084 | |a MAT 050f |2 stub | ||
| 100 | 1 | |a O'Donnell, John |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Discrete mathematics using a computer |c John O'Donnell, Cordelia Hall and Rex Page |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a London |b Springer |c 2006 | |
| 300 | |a XIX, 441 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Diskrete Mathematik | |
| 650 | 4 | |a HASKELL | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Informatik | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Computer science |x Mathematics | |
| 650 | 4 | |a Mathematics |x Data processing | |
| 650 | 0 | 7 | |a Mengenlehre |0 (DE-588)4074715-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematische Logik |0 (DE-588)4037951-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Diskrete Mathematik |0 (DE-588)4129143-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a HASKELL |0 (DE-588)4318275-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Programmierung |0 (DE-588)4076370-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a HASKELL |0 (DE-588)4318275-6 |D s |
| 689 | 0 | 1 | |a Mathematische Logik |0 (DE-588)4037951-6 |D s |
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| 689 | 1 | 1 | |a HASKELL |0 (DE-588)4318275-6 |D s |
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| 689 | 2 | 0 | |a Mengenlehre |0 (DE-588)4074715-3 |D s |
| 689 | 2 | |8 3\p |5 DE-604 | |
| 700 | 1 | |a Hall, Cordelia |d 1955- |e Verfasser |0 (DE-588)120908972 |4 aut | |
| 700 | 1 | |a Page, Rex |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015480481&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| 883 | 1 | |8 3\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-015480481 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 050f 2007 A 2240(2) |
|---|---|
| DE-BY-TUM_katkey | 1583655 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010252194 |
| _version_ | 1820897707180425216 |
| adam_text | Contents
I
Programming
and Reasoning with Equations
1
1
Introduction to Haskell
....................... 3
1.1
Obtaining and Running Haskell
................. 4
1.2
Expressions
............................. 6
1.2.1
Integer and
Int
..................... 6
1.2.2
Rational and Floating Point Numbers
......... 8
1.2.3
Booleans
.......................... 9
1.2.4
Characters
......................... 10
1.2.5
Strings
........................... 10
1.3
Basic Data Structures: Tuples and Lists
............. 11
1.3.1
Tuples
........................... 11
1.3.2
Lists
............................ 11
1.3.3
List Notation and
(:) .................. 12
1.3.4
List Comprehensions
................... 13
1.4
Functions
.............................. 15
1.4.1
Function Application
................... 15
1.4.2
Function Types
...................... 15
1.4.3
Operators and Functions
................. 16
1.4.4
Function Definitions
................... 16
1.4.5
Pattern Matching
..................... 17
1.4.6
Higher Order Functions
................. 19
1.5
Conditional Expressions
...................... 20
1.6
Local Variables: let Expressions
................. 21
1.7
Type Variables
........................... 21
1.8
Common Functions on Lists
................... 22
1.9
Data Type Definitions
....................... 28
1.10
Type Classes and Overloading
.................. 31
1.11
Suggestions for Further Reading
................. 33
1.12
Review Exercises
.......................... 33
ХШ
xiv
CONTENTS
2
Equational
Reasoning
........................ 37
2.1
Equations and Substitutions
................... 37
2.2
Equational Reasoning as Hand-execution
............ 38
2.2.1
Conditionals
........................ 41
2.3
Equational Reasoning with Lists
................. 42
2.4
The Role of the Language
..................... 43
2.5
Rigor and Formality in Proofs
.................. 44
3
Recursion
................................ 47
3.1
Recursion Over Lists
....................... 48
3.2
Higher Order Recursive Functions
................ 54
3.3
Peano Arithmetic
......................... 57
3.4
Data Recursion
........................... 58
3.5
Suggestions for Further Reading
................. 59
3.6
Review Exercises
.......................... 59
4
Induction
................................ 61
4.1
The Principle of Mathematical Induction
............ 62
4.2
Examples of Induction on Natural Numbers
........... 63
4.3
Induction and Recursion
..................... 66
4.4
Induction on Peano Naturals
................... 67
4.5
Induction on Lists
......................... 70
4.6
Functional Equality
........................ 76
4.7
Pitfalls and Common Mistakes
.................. 78
4.7.1
A Horse of Another Colour
............... 78
4.8
Limitations of Induction
..................... 78
4.9
Suggestions for Further Reading
................. 80
4.10
Review Exercises
.......................... 81
5
Trees
................................... 83
5.1
Components of a Tree
....................... 83
5.2
Representing Trees in Haskell
................... 86
5.3
Processing Trees with Recursion
................. 88
5.3.1
Tree Traversal
....................... 89
5.3.2
Processing Tree Structure
................ 90
5.3.3
Evaluating Expression Trees
............... 92
5.3.4
Binary Search Trees
................... 92
5.4
Induction on Trees
......................... 96
5.4.1
Repeated Reflection Theorem
.............. 96
5.4.2
Reflection and Reversing
................. 97
5.4.3
The Height of a Balanced Tree
............. 98
5.4.4
Length of a Flattened Tree
................ 99
5.5
Improving Execution Time
.................... 100
5.6
Flattening Trees in Linear Time
................. 103
CONTENTS xv
II
Logic 107
6 Propositional Logic.......................... 109
6.1
The Need for Formalism
.....................
Ill
6.2
The Basic Logical Operators
................... 112
6.2.1
Logical And
(Л)
...................... 113
6.2.2
Inclusive Logical Or (V)
................. 114
6.2.3
Exclusive Logical Or (®)
................. 115
6.2.4
Logical Not
(->)...................... 115
6.2.5
Logical Implication
(—>·)................. 115
6.2.6
Logical Equivalence
(<->)................. 117
6.3
The Language of Propositional Logic
.............. 118
6.3.1
The Syntax of Well-Formed Formulas
.......... 118
6.3.2
Precedence of Logical Operators
............. 120
6.3.3
Object Language and Meta-Language
......... 120
6.3.4
Computing with Boolean Expressions
.......... 121
6.4
Truth Tables: Semantic Reasoning
................ 122
6.4.1
Truth Table Calculations and Proofs
.......... 123
6.4.2
Limitations of Truth Tables
............... 124
6.4.3
Computing Truth Tables
................. 125
6.5
Natural Deduction: Inference Reasoning
............. 126
6.5.1
Definitions of True,
->,
and
Ό
.............. 128
6.5.2
And Introduction {M}
.................. 129
6.5.3
And Elimination
{AEL},
{aEr}
............ 131
6.5.4
Imply Elimination
{->
E}
................ 132
6.5.5
Imply Introduction
{->■ 1}................ 133
6.5.6
Or Introduction {Vh},
{V/ň}
.............. 136
6.5.7
Or Elimination {v£}
................... 137
6.5.8
Identity {ID}
....................... 138
6.5.9
Contradiction {CTR}
.................. 138
6.5.10
Reductio
ad Absurdum {RAA}
............. 140
6.5.11
Inferring the Operator Truth Tables
.......... 141
6.6
Proof Checking by Computer
................... 142
6.6.1
Example of Proof Checking
............... 143
6.6.2
Representation of WFFs
................. 146
6.6.3
Representing Proofs
................... 147
6.7
Boolean Algebra: Equational Reasoning
............. 149
6.7.1
The Laws of Boolean Algebra
.............. 150
6.7.2
Operations with Constants
................ 151
6.7.3
Basic Properties of
Λ
and V
............... 153
6.7.4
Distributive and DeMorgan s Laws
........... 154
6.7.5
Laws on Negation
..................... 154
6.7.6
Laws on Implication
................... 155
6.7.7
Equivalence
........................ 156
6.8
Logic in Computer Science
.................... 156
xvi
CONTENTS
6.9
Meta-Logic
............................. 158
6.10
Suggestions
for Further Reading
................. 159
6.11
Review Exercises
.......................... 161
7
Predicate Logic
............................ 163
7.1
The Language of Predicate Logic
................. 163
7.1.1
Predicates
......................... 163
7.1.2
Quantifiers
......................... 164
7.1.3
Expanding Quantified Expressions
........... 166
7.1.4
The Scope of Variable Bindings
............. 168
7.1.5
Translating Between English and Logic
......... 169
7.2
Computing with Quantifiers
................... 172
7.3
Logical Inference with Predicates
................. 174
7.3.1
Universal Introduction {V/}
............... 175
7.3.2
Universal Elimination
{VE}
............... 177
7.3.3
Existential Introduction
{31}.............. 179
7.3.4
Existential Elimination {3E}
.............. 180
7.4
Algebraic Laws of Predicate Logic
................ 181
7.5
Suggestions for Further Reading
................. 183
7.6
Review Exercises
.......................... 184
III Set Theory
187
8
Set Theory
............................... 189
8.1
Notations for Describing Sets
................... 189
8.2
Basic Operations on Sets
..................... 192
8.2.1
Subsets and Set Equality
................. 192
8.2.2
Union, Intersection, and Difference
........... 192
8.2.3
Complement and Power
................. 194
8.3
Finite Sets with Equality
..................... 196
8.3.1
Computing with Sets
................... 198
8.4
Set Laws
.............................. 200
8.4.1
Associative and Commutative Set Operations
..... 201
8.4.2
Distributive Laws
..................... 202
8.4.3
DeMorgan s Laws for Sets
................ 202
8.5
Summary
.............................. 203
8.6
Suggestions for Further Reading
................. 205
8.7
Review Exercises
.......................... 205
9
Inductively Defined Sets
....................... 207
9.1
The Idea Behind Induction
.................... 207
9.1.1
The Induction Rule
.................... 210
9.2
How to Define a Set Using Induction
.............. 212
9.2.1
Inductive Definition of the Set of Natural Numbers
. . 213
CONTENTS xvii
9.2.2
The Set of Binary Machine Words
........... 214
9.3
Defining the Set of Integers
.................... 215
9.3.1
First Attempt
....................... 215
9.3.2
Second Attempt
...................... 216
9.3.3
Third Attempt
...................... 216
9.3.4
Fourth Attempt
...................... 218
9.3.5
Fifth Attempt
....................... 219
9.4
Suggestions for Further Reading
................. 219
9.5
Review Exercises
.......................... 219
10
Relations
................................. 223
10.1
Binary Relations
.......................... 223
10.2
Representing Relations with Digraphs
.............. 225
10.3
Computing with Binary Relations
................ 226
10.4
Properties of Relations
...................... 228
10.4.1
Reflexive Relations
.................... 228
10.4.2 Irreflexive
Relations
................... 229
10.4.3
Symmetric Relations
................... 231
10.4.4
Antisymmetric Relations
................. 233
10.4.5
Transitive Relations
................... 235
10.5
Relational Composition
...................... 237
10.6
Powers of Relations
........................ 240
10.7
Closure Properties of Relations
.................. 245
10.7.1
Reflexive Closure
..................... 246
10.7.2
Symmetric Closure
.................... 248
10.7.3
Transitive Closure
.................... 249
10.8
Order Relations
.......................... 252
10.8.1
Partial Order
....................... 252
10.8.2
Quasi Order
........................ 257
10.8.3
Linear Order
....................... 258
10.8.4
Well Order
......................... 259
10.8.5
Topological Sort
...................... 260
10.9
Equivalence Relations
....................... 261
10.10
Suggestions for Further Reading
................. 264
10.11
Review Exercises
.......................... 264
11
Functions
................................ 267
11.1
The Graph of a Function
..................... 268
11.2
Functions in Programming
.................... 271
11.2.1
Inductively Defined Functions
.............. 272
11.2.2
Primitive Recursion
.................... 273
11.2.3
Computational Complexity
............... 274
11.2.4
State
............................ 275
11.3
Higher Order Functions
...................... 276
11.3.1
Functions That Take Functions as Arguments
..... 277
xviii CONTENTS
11.3.2
Functions That Return Functions
............ 278
11.3.3
Multiple Arguments as Tuples
.............. 280
11.3.4
Multiple Results as a Tuple
............... 281
11.3.5
Multiple Arguments with Higher Order Functions
... 281
11.4
Total and Partial Functions
.................... 282
11.5
Function Composition
....................... 287
11.6
Properties of Functions
...................... 291
11.6.1
Surjective Functions
................... 291
11.6.2
Injective Functions
.................... 293
11.6.3
The Pigeonhole Principle
................. 296
11.7
Bijective Functions
........................ 296
11.7.1
Permutations
....................... 297
11.7.2
Inverse Functions
..................... 299
11.8
Cardinality of Sets
......................... 299
11.8.1
The Rational Numbers Are Countable
......... 302
11.8.2
The Real Numbers Are Uncountable
.......... 302
11.9
Suggestions for Further Reading
................. 304
11.10
Review Exercises
.......................... 304
IV Applications
311
12
The AVL Tree Miracle
........................ 313
12.1
How to Find a Folder
....................... 313
12.2
The Filing Cabinet Advantage
.................. 314
12.3
The New-File Problem
...................... 315
12.4
The AVL Miracle
......................... 316
12.5
Search Trees and Occurrence of Keys
.............. 317
12.5.1
Ordered Search Trees and Tree Induction
....... 319
12.5.2
Retrieving Data from a Search Tree
........... 324
12.5.3
Search Time in the Equational Model
......... 326
12.6
Balanced Trees
........................... 329
12.6.1
Rebalancing in the Easy Cases
............. 332
12.6.2
Rebalancing in the Hard Cases
............. 336
12.6.3
Rebalancing Left-Heavy and Right-Heavy Trees
.... 338
12.6.4
Inductive Equations for Insertion
............ 339
12.6.5
Insertion in Logarithmic Time
.............. 342
12.6.6
Deletion
.......................... 344
12.6.7
Shrinking the Spine
.................... 347
12.6.8
Equations for Deleting Root
............... 349
12.6.9
Equations for Deletion
..................
35O
12.6.10
Deletion in Logarithmic Time
.............. 351
12.7
Things We Didn t Tell You
.................. 352
CONTENTS xix
13
Discrete
Mathematics in Circuit Design
............. 355
13.1
Boolean Logic Gates
........................ 356
13.2
Functional Circuit Specification
................. 357
13.2.1
Circuit Simulation
.................... 358
13.2.2
Circuit Synthesis from Truth Tables
.......... 359
13.2.3
Multiplexors
........................ 362
13.2.4
Bit Arithmetic
...................... 363
13.2.5
Binary Representation
.................. 366
13.3
Ripple Carry Addition
...................... 367
13.3.1
Circuit Patterns
...................... 369
13.3.2
The
η
-Bit
Ripple Carry Adder
............. 370
13.3.3
Correctness of the Ripple Carry Adder
......... 371
13.3.4
Binary Comparison
.................... 372
13.4
Suggestions for Further Reading
................. 374
13.5
Review Exercises
.......................... 374
A Software Tools
............................. 377
В
Resources on the Web
........................ 379
С
Solutions to Selected Exercises
................... 381
C.I Introduction to Haskell
...................... 381
C.3 Recursion
.............................. 384
C.4 Induction
.............................. 387
C.5 Trees
................................ 397
C.6 Propositional Logic
........................ 398
C.7 Predicate Logic
.......................... 411
C.8 Set Theory
............................. 417
C.9 Inductively Defined Sets
...................... 420
CIO Relations
.............................. 422
C.ll Functions
.............................. 424
C.13 Discrete Mathematics in Circuit Design
............. 428
Bibliography
................................ 431
Index
..................................... 434
|
| any_adam_object | 1 |
| author | O'Donnell, John Hall, Cordelia 1955- Page, Rex |
| author_GND | (DE-588)120908972 |
| author_facet | O'Donnell, John Hall, Cordelia 1955- Page, Rex |
| author_role | aut aut aut |
| author_sort | O'Donnell, John |
| author_variant | j o jo c h ch r p rp |
| building | Verbundindex |
| bvnumber | BV022269978 |
| callnumber-first | Q - Science |
| callnumber-label | QA76 |
| callnumber-raw | QA76.95 |
| callnumber-search | QA76.95 |
| callnumber-sort | QA 276.95 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 890 ST 600 |
| classification_tum | DAT 362f MAT 050f |
| ctrlnum | (OCoLC)255037636 (DE-599)BVBBV022269978 |
| dewey-full | 004.0151 510.285 |
| dewey-hundreds | 000 - Computer science, information, general works 500 - Natural sciences and mathematics |
| dewey-ones | 004 - Computer science 510 - Mathematics |
| dewey-raw | 004.0151 510.285 |
| dewey-search | 004.0151 510.285 |
| dewey-sort | 14.0151 |
| dewey-tens | 000 - Computer science, information, general works 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV022269978 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T19:57:04Z |
| institution | BVB |
| isbn | 9781846282416 1846282411 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015480481 |
| oclc_num | 255037636 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-706 DE-703 DE-1046 DE-573 |
| owner_facet | DE-91G DE-BY-TUM DE-706 DE-703 DE-1046 DE-573 |
| physical | XIX, 441 S. Ill., graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| spellingShingle | O'Donnell, John Hall, Cordelia 1955- Page, Rex Discrete mathematics using a computer Diskrete Mathematik HASKELL Datenverarbeitung Informatik Mathematik Computer science Mathematics Mathematics Data processing Mengenlehre (DE-588)4074715-3 gnd Mathematische Logik (DE-588)4037951-6 gnd Diskrete Mathematik (DE-588)4129143-8 gnd HASKELL (DE-588)4318275-6 gnd Programmierung (DE-588)4076370-5 gnd |
| subject_GND | (DE-588)4074715-3 (DE-588)4037951-6 (DE-588)4129143-8 (DE-588)4318275-6 (DE-588)4076370-5 |
| title | Discrete mathematics using a computer |
| title_auth | Discrete mathematics using a computer |
| title_exact_search | Discrete mathematics using a computer |
| title_full | Discrete mathematics using a computer John O'Donnell, Cordelia Hall and Rex Page |
| title_fullStr | Discrete mathematics using a computer John O'Donnell, Cordelia Hall and Rex Page |
| title_full_unstemmed | Discrete mathematics using a computer John O'Donnell, Cordelia Hall and Rex Page |
| title_short | Discrete mathematics using a computer |
| title_sort | discrete mathematics using a computer |
| topic | Diskrete Mathematik HASKELL Datenverarbeitung Informatik Mathematik Computer science Mathematics Mathematics Data processing Mengenlehre (DE-588)4074715-3 gnd Mathematische Logik (DE-588)4037951-6 gnd Diskrete Mathematik (DE-588)4129143-8 gnd HASKELL (DE-588)4318275-6 gnd Programmierung (DE-588)4076370-5 gnd |
| topic_facet | Diskrete Mathematik HASKELL Datenverarbeitung Informatik Mathematik Computer science Mathematics Mathematics Data processing Mengenlehre Mathematische Logik Programmierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015480481&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT odonnelljohn discretemathematicsusingacomputer AT hallcordelia discretemathematicsusingacomputer AT pagerex discretemathematicsusingacomputer |