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Mathematical proofs : a transition to advanced mathematics

Author: Gary Chartrand; Albert D Polimeni; Ping Zhang
Publisher: Boston : Addison Wesley, ©2003.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own.  Read more...

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Additional Physical Format: Online version:
Chartrand, Gary.
Mathematical proofs.
Boston : Addison Wesley, ©2003
(OCoLC)681099726
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Gary Chartrand; Albert D Polimeni; Ping Zhang
ISBN: 0201710900 9780201710908
OCLC Number: 48263778
Description: xiii, 289 pages : illustrations ; 25 cm
Contents: Machine generated contents note: COMMUNICATING MATHEMATICS --
Learning Mathematics 01 --
What Others Have Said About Writing 03 --
Mathematical Writing 04 --
Using Symbols 05 --
Writing Mathematical Expressions 07 --
Common Words and Phrases in Mathematics 08 --
Some Closing Comments About Writing 11 --
SETS --
1.1 Describing a Set 13 --
1.2 Special Sets 15 --
1.3 Subsets 16 --
1.4 Set Operations 18 --
1.5 Indexed Collections of Sets 21 --
1.6 Partitions of Sets 23 --
1.7 Cartesian Products of Sets 24 --
Exercises for Chapter 1 24 --
LOGIC --
2.1 Statements 29 --
2.2 The Negation of a Statement 31 --
2.3 The Disjunction and Conjunction of Statements 32 --
2.4 The Implication 33 --
2.5 More On Implications 35 --
2.6 The Biconditional 36 --
2.7 Tautologies and Contradictions 38 --
2.8 Logical Equivalence 39 --
2.9 Some Fundamental Properties of Logical Equivalence 41 --
2.10 Characterizations of Statements 42 --
2.11 Quantified Statements and Their Negatiors 44 --
Exercises for Chapter 2 46 --
DIRECT PROOF AND PROOF BY CONTRAPOSITIVE --
3.1 Trivial and Vacuous Proofs 51 --
3.2 Direct Proofs 53 --
3.3 Proof by Contrapositive 56 --
3.4 Proof by Cases 60 --
3.5 Proof Evaluations 63 --
Exercises for Chapter 3 64 --
MORE ON DIRECT PROOF AND PROOF --
BY CONTRAPOSITIVE --
4.1 Proofs Involving Divisibility of Integers 67 --
4.2 Proofs Involving Congruence of Integers 70 --
4.3 Proofs Involving Real Numbers 73 --
4.4 Proofs Involving Sets 74 --
4.5 Fundamental Properties of Set Operations 77 --
4.6 Proofs Involving Cartesian Products of Sets 79 --
Exercises for Chapter 4 80 --
PROOF BY CONTRADICTION --
5.1 Proof by Contradiction 83 --
5.2 Examples of Proof by Contradiction 84 --
5.3 The Three Prisoners Problem 85 --
5.4 Other Examples of Proof by Contradiction 87 --
5.5 The Irrationality of /2 87 --
5.6 A Review of the Three Proof Techniques 88 --
Exercises for Chapter 5 90 --
PROVE OR DISPROVE --
6.1 Conjectures in Mathematics 93 --
6.2 A Review of Quantifiers 96 --
6.3. Existence Proofs 98 --
6.4 A Review of Negations of Quantified Statements 100 --
6.5 Counterexamples 101 --
6.6 Disproving Statements 103 --
6.7 Testing Statements 105 --
6.8 A Quiz of "Prove or Disprove" Problems 107 --
Exercises for Chapter 6 108 --
EQUIVALENCE RELATIONS --
7.1 Relations 113 --
7.2 Reflexive, Symmetric, and Transitive Relations 114 --
7.3 Equivalence Relations 116 --
7.4 Properties of Equivalence Classes 119 --
7.5 Congruence Modulo n 123 --
7.6 The Integers Modulo n 127 --
Exercises for Chapter 7 130 --
FUNCTIONS --
8.1 The Definition of Function 135 --
8.2 The Set of All Functions From A to B 138 --
8.3 One-to-one and Onto Functions 138 --
8.4 Bijective Functions 140 --
8.5 Composition of Functions 143 --
8.6 Inverse Functions 146 --
8.7 Permutations 149 --
Exercises for Chapter 8 150 --
MATHEMATICAL INDUCTION --
9.1 The Well-Ordering Principle 153 --
9.2 The Principle of Mathematical Induction 155 --
9.3 Mathematical Induction and Sums of Numbers 158 --
9.4 Mathematical Induction and Inequalities 162 --
9.5 Mathematical Induction and Divisibility 163 --
9.6 Other Examples of Induction Proofs 165 --
9.7 Proof By Minimum Counterexample 166 --
9.8 The Strong Form of Induction 168 --
Exercises for Chapter 9 171 --
CARDINALITIES OF SETS --
10.1 Numerically Equivalent Sets 176 --
10.2 Denumerable Sets 177 --
10.3 Uncountable Sets 183 --
10.4 Comparing Cardinalities of Sets 188 --
10.5 The Schr6der-Berstein Theorem 191 --
Exercises for Chapter 10 194 --
PROOFS IN NUMBER THEORY --
11.1 Divisibility Properties of Integers 197 --
11.2 The Division Algorithm 198 --
11.3 Greatest Common Divisors 202 --
11.4 The Euclidean Algorithm 204 --
11.5 Relatively Prime Integers 206 --
11.6 The Fundamental Theorem of Arithmetic 208 --
11.7 Concepts Involving Sums of Divisors 210 --
Exercises for Chapter 11 211 --
PROOFS IN CALCULUS --
12.1 Limits of Sequences 215 --
12.2 Infinite Series 220 --
12.3 Limits of Functions 224 --
12.4 Fundamental Properties of. Limits of Functions 230 --
12.5 Continuity 235 --
12.6 Differentiability 237 --
Exercises for Chapter 12 239 --
PROOFS IN GROUP THEORY --
13.1 Binary Operations 243 --
13.2 Groups 247 --
13.3 Permutation Groups 252 --
13.4 Fundamental Properties of Groups 255 --
13.5 Subgroups 257 --
13.6 Isomorphic Groups 260 --
Exercises for Chapter 13 263.
Responsibility: Gary Chartrand, Albert D. Polimeni, Ping Zhang.
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