Understanding mathematical proof:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2014
|
| Series: | A Chapman & Hall book
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 385 |
| Physical Description: | XIX, 394 S. graph. Darst. |
| ISBN: | 9781466514904 1466514906 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804150650965065728 |
|---|---|
| adam_text | Contents
Preface
ix
List of Figures
xi
List of Tables
xv
List of Symbols
xvii
1
Introduction
1
1.1
The need for proof
........................ 1
1.2
The language of mathematics
.................. 3
1.3
Reasoning
............................ 6
1.4
Deductive reasoning and truth
................. 8
1.5
Example proofs
......................... 11
2
Logic and Reasoning
15
2.1
Introduction
........................... 15
2.2
Propositions, connectives, and truth tables
.......... 16
2.3
Logical equivalence and logical implication
.......... 34
2.4
Predicates and quantification
.................. 45
2.5
Logical reasoning
........................ 66
3
Sets and Functions
83
3.1
Introduction
........................... 83
3.2
Sets and membership
...................... 83
3.3
Operations on sets
........................ 92
vi
Contents
3.4
The Cartesian product
..................... 103
3.5
Functions and composite functions
............... 107
3.6
Properties of functions
..................... 117
4
The Structure of Mathematical Proofs
129
4.1
Introduction
........................... 129
4.2
Some proofs dissected
...................... 130
4.3
An informal framework for proofs
............... 135
4.4
Direct proof
........................... 141
4.5
A more formal framework
.................... 161
5
Finding Proofs
175
5.1
Direct proof route maps
..................... 175
5.2
Examples from sets and functions
............... 189
5.3
Examples from algebra
..................... 200
5.3.1
Group theory
....................... 200
5.3.2
Linear algebra
...................... 209
5.4
Examples from analysis
..................... 226
5.4.1
Sequences
......................... 227
5.4.2
Limits of functions
.................... 232
6
Direct Proof: Variations
239
6.1
Introduction
........................... 239
6.2
Proof using the
contrapositive
................. 240
6.3
Proof of biconditional statements
............... 246
6.4
Proof of conjunctions
...................... 251
6.5
Proof by contradiction
..................... 257
6.6
Further examples
........................ 260
6.6.1
Examples from algebra
.................. 260
6.6.2
Examples from analysis
................. 263
Contents
vii
7
Existence
and Uniqueness
269
7.1
Introduction
........................... 269
7.2
Constructive existence proofs
.................. 270
7.3
Non-constructive existence proofs
............... 276
7.4
Counter-examples
........................ 286
7.5
Uniqueness proofs
........................ 295
8
Mathematical Induction
303
8.1
Introduction
........................... 303
8.2
Proof by induction
........................ 305
8.3
Variations on proof by induction
................ 316
Hints and Solutions to Selected
Exercises
333
Bibliography
385
Index
387
Mathematics
The notion of proof is central to mathematics yet it is one of the most difficult
aspects of the subject to master. Understanding Mathematical Proof
describes the nature of mathematical proof, explores the various techniques
that mathematicians adopt to prove their results, and offers advice and
strategies for constructing proofs. It will improve your ability to understand
proofs and construct correct proofs of your own.
The first chapter of the text introduces the kind of reasoning that mathemati¬
cians use when writing their proofs and gives some example proofs to set
the scene. The book then describes basic logic to enable an understand¬
ing of the structure of both individual mathematical statements and whole
mathematical proofs. It also explains the notions of sets and functions and
dissects several proofs with a view to exposing some of the underlying fea¬
tures common to most mathematical proofs. The remainder of the book
delves further into different types of proof, including direct proof, proof using
contrapositive,
proof by contradiction, and mathematical induction. The au¬
thors also discuss existence and uniqueness proofs and the role of counter
examples.
Features
•
Presents a rigorous yet ac< sible introduction tc one of the defining
Bť
concepts of mathematics
•
Explores the structure of mathematical proof
•
Classifies different types of proof
•
Develops all the standard techniques for proving
r
theorems
•
Provides hints and guidance on finding the appropriate proof
•
Includes many elementary examples and. in addition, develops
proofs in linear algebra, group theory, and real analysis that will be of
particular interest to undergraduate mathematicians
CRC
Press
Taylor
&.
Francis Group
an
informa
Dusiness
6000
Broken Sound Parkwa
Suite i00r Boca Raton, FL
3 3-487
711
Third Avenue
York. NY
10017
2
Park Square. Muton Park
Abingdon. Oxon OX14 4RN. UK
K15D1Ô
ISBN:
17а-1-ЧЬЬ5-1ЧЧ0-Ч
90000
9
781466 514904
|
| any_adam_object | 1 |
| author | Taylor, John 1957- Garnier, Rowan |
| author_GND | (DE-588)141968737 (DE-588)141968257 |
| author_facet | Taylor, John 1957- Garnier, Rowan |
| author_role | aut aut |
| author_sort | Taylor, John 1957- |
| author_variant | j t jt r g rg |
| building | Verbundindex |
| bvnumber | BV041216270 |
| classification_rvk | SK 130 |
| ctrlnum | (OCoLC)881298533 (DE-599)BVBBV041216270 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041216270 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:42:18Z |
| institution | BVB |
| isbn | 9781466514904 1466514906 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026190896 |
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| physical | XIX, 394 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | CRC Press |
| record_format | marc |
| series2 | A Chapman & Hall book |
| spelling | Taylor, John 1957- Verfasser (DE-588)141968737 aut Understanding mathematical proof John Taylor ; Rowan Garnier Boca Raton, Fla. [u.a.] CRC Press 2014 XIX, 394 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Literaturverz. S. 385 Beweis (DE-588)4132532-1 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Beweistheorie (DE-588)4145177-6 gnd rswk-swf Beweis (DE-588)4132532-1 s Mathematik (DE-588)4037944-9 s DE-604 Beweistheorie (DE-588)4145177-6 s Garnier, Rowan Verfasser (DE-588)141968257 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026190896&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026190896&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Taylor, John 1957- Garnier, Rowan Understanding mathematical proof Beweis (DE-588)4132532-1 gnd Mathematik (DE-588)4037944-9 gnd Beweistheorie (DE-588)4145177-6 gnd |
| subject_GND | (DE-588)4132532-1 (DE-588)4037944-9 (DE-588)4145177-6 |
| title | Understanding mathematical proof |
| title_auth | Understanding mathematical proof |
| title_exact_search | Understanding mathematical proof |
| title_full | Understanding mathematical proof John Taylor ; Rowan Garnier |
| title_fullStr | Understanding mathematical proof John Taylor ; Rowan Garnier |
| title_full_unstemmed | Understanding mathematical proof John Taylor ; Rowan Garnier |
| title_short | Understanding mathematical proof |
| title_sort | understanding mathematical proof |
| topic | Beweis (DE-588)4132532-1 gnd Mathematik (DE-588)4037944-9 gnd Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Beweis Mathematik Beweistheorie |
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