Mathematical proofs: a transition to advanced mathematics
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston ; Munich [u.a.]
Pearson/Addison Wesley
2008
|
| Edition: | 2. ed., international ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 365 S. graph. Darst. |
| ISBN: | 9780321526731 0321390539 |
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MARC
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| 245 | 1 | 0 | |a Mathematical proofs |b a transition to advanced mathematics |c Gary Chartrand ; Albert D. Polimeni ; Ping Zhang |
| 250 | |a 2. ed., international ed. | ||
| 264 | 1 | |a Boston ; Munich [u.a.] |b Pearson/Addison Wesley |c 2008 | |
| 300 | |a XV, 365 S. |b graph. Darst. | ||
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| 700 | 1 | |a Polimeni, Albert D. |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804138514601738240 |
|---|---|
| adam_text | Contents
О
Communicating
Mathematics
Learning Mathematics
1
What Others Have Said about Writing
3
Mathematical Writing
5
Using Symbols
6
Writing Mathematical Expressions
8
Common Words and Phrases in Mathematics
Some Closing Comments about Writing
12
1
Sets
1.1
Describing a Set
13
1.2
Subsets
16
1.3
Set Operations
19
1.4
Indexed Collections of Sets
1.5
Partitions of Sets
25
1.6
Cartesian Products of Sets
Exercises for Chapter
1 27
Additional Exercises for Chapter
1
13
22
26
31
Logic
33
2.1
Statements
33
2.2
The Negation of a Statement
35
2.3
The Disjunction and Conjunction of Statements
37
2.4
The Implication
38
2.5
More on Implications
40
2.6
The Biconditional
42
2.7
Tautologies and Contradictions
45
2.8
Logical Equivalence
47
2.9
Some Fundamental Properties of Logical Equivalence
48
2.10
Quantified Statements
50
2.11
Characterizations of Statements
56
Exercises for Chapter
2 57
Additional Exercises for Chapter
2 64
Direct Proof and Proof by
Contrapositive
67
3.1
Trivial and Vacuous Proofs
68
3.2
Direct Proofs
70
3.3
Proof by
Contrapositive
74
3.4
Proof by Cases
78
3.5
Proof Evaluations
81
Exercises for Chapter
3 83
Additional Exercises for Chapter
3 85
More on Direct Proof and Proof by
Contrapositive
87
4.1
Proofs Involving Divisibility of Integers
87
4.2
Proofs Involving Congruence of Integers
91
4.3
Proofs Involving Real Numbers
93
4.4
Proofs Involving Sets
96
4.5
Fundamental Properties of Set Operations
99
4.6
Proofs Involving Cartesian Products of Sets
100
Exercises for Chapter
4 101
Additional Exercises for Chapter
4 104
Existence and Proof by Contradiction
107
5.1
Counterexamples
107
5.2
Proof by Contradiction 111
5.3
A Review of Three Proof Techniques
116
5.4
Existence Proofs
118
5.5
Disproving Existence Statements
122
Exercises for Chapter
5 124
Additional Exercises for Chapter
5 125
6
7
Mathematical Induction
129
6.1
The Principle of Mathematical Induction
129
6.2
A More General Principle of Mathematical Induction
138
6.3
Proof by Minimum Counterexample
144
6.4
The Strong Principle of Mathematical Induction
146
Exercises for Chapter
6 150
Additional Exercises for Chapter
6 152
Prove or Disprove
155
7.1
Conjectures in Mathematics
155
7.2
Revisiting Quantified Statements
158
7.3
Testing Statements
163
7.4
A Quiz of Prove or Disprove Problems
167
Exercises for Chapter
7 169
Additional Exercises for Chapter
7 172
Equivalence Relations
175
8.1
Relations
175
8.2
Properties of Relations
176
8.3
Equivalence Relations
178
8.4
Properties of Equivalence Classes
181
8.5
Congruence Modulo
η
185
8.6
The Integers Modulo
и
189
Exercises for Chapter
8 192
Additional Exercises for Chapter
8 195
Functions
197
9.1
The Definition of Function
197
9.2
The Set of All Functions from A to
В
200
9.3
One-to-One and Onto Functions
200
9.4
Bijective Functions
203
9.5
Composition of Functions
205
9.6
Inverse Functions
209
9.7
Permutations
212
Exercises for Chapter
9 213
Additional Exercises for Chapter
9 216
10
11
13
Cardinalities of Sets
221
10.1
Numerically Equivalent Sets
222
10.2
Denumerable Sets
223
10.3
Uncountable Sets
229
10.4
Comparing Cardinalities of Sets
234
10.5
The
Schröder-Bernstein
Theorem
237
Exercises for Chapter
10 241
Additional Exercises for Chapter
10 243
Proofs in Number Theory
245
11.1
Divisibility Properties of Integers
245
11.2
The Division Algorithm
246
11.3
Greatest Common Divisors
250
11.4
The Euclidean Algorithm
252
11.5
Relatively Prime Integers
254
11.6
The Fundamental Theorem of Arithmetic
256
11.7
Concepts Involving Sums of Divisors
259
Exercises for Chapter
11 260
Additional Exercises for Chapter
11 263
267
284
12
Proofs in Calculus
12.1
Limits of Sequences
267
12.2
Infinite Series
273
12.3
Limits of Functions
277
12.4
Fundamental Properties of Limits of Functions
12.5
Continuity
289
12.6
Differentiability
291
Exercises for Chapter
12 293
Additional Exercises for Chapter
12 295
Proofs in Group Theory
297
13.1
Binary Operations
297
13.2
Groups
301
13.3
Permutation Groups
305
13.4
Fundamental Properties of Groups
308
13.5
Subgroups
311
13.6
Isomorphic Groups
313
Exercises for Chapter
13 317
Additional Exercises for Chapter
13 321
Solutions to Odd-Numbered Section Exercises
323
References
357
Index of Symbols
359
Index
361
|
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| author | Chartrand, Gary 1936- Polimeni, Albert D. Zhang, Ping |
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| callnumber-raw | QA9.54 |
| callnumber-search | QA9.54 |
| callnumber-sort | QA 19.54 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 110 SK 130 |
| ctrlnum | (OCoLC)71006820 (DE-599)BVBBV035241765 |
| dewey-full | 511.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/6 |
| dewey-search | 511.3/6 |
| dewey-sort | 3511.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed., international ed. |
| format | Book |
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| language | English |
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| spelling | Chartrand, Gary 1936- Verfasser (DE-588)140142266 aut Mathematical proofs a transition to advanced mathematics Gary Chartrand ; Albert D. Polimeni ; Ping Zhang 2. ed., international ed. Boston ; Munich [u.a.] Pearson/Addison Wesley 2008 XV, 365 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Proof theory Textbooks Mathematik (DE-588)4037944-9 gnd rswk-swf Beweistheorie (DE-588)4145177-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Mathematik (DE-588)4037944-9 s Beweistheorie (DE-588)4145177-6 s b DE-604 Polimeni, Albert D. Verfasser aut Zhang, Ping Verfasser aut Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017047540&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chartrand, Gary 1936- Polimeni, Albert D. Zhang, Ping Mathematical proofs a transition to advanced mathematics Proof theory Textbooks Mathematik (DE-588)4037944-9 gnd Beweistheorie (DE-588)4145177-6 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4145177-6 (DE-588)4123623-3 |
| title | Mathematical proofs a transition to advanced mathematics |
| title_auth | Mathematical proofs a transition to advanced mathematics |
| title_exact_search | Mathematical proofs a transition to advanced mathematics |
| title_full | Mathematical proofs a transition to advanced mathematics Gary Chartrand ; Albert D. Polimeni ; Ping Zhang |
| title_fullStr | Mathematical proofs a transition to advanced mathematics Gary Chartrand ; Albert D. Polimeni ; Ping Zhang |
| title_full_unstemmed | Mathematical proofs a transition to advanced mathematics Gary Chartrand ; Albert D. Polimeni ; Ping Zhang |
| title_short | Mathematical proofs |
| title_sort | mathematical proofs a transition to advanced mathematics |
| title_sub | a transition to advanced mathematics |
| topic | Proof theory Textbooks Mathematik (DE-588)4037944-9 gnd Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Proof theory Textbooks Mathematik Beweistheorie Lehrbuch |
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