Mathematical proofs : a transition to advanced mathematics

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Detalles Bibliográficos
Autor principal: Chartrand, Gary
Otros Autores: Polimeni, Albert D., 1938-, Zhang, Ping, 1957-
Formato: Libro
Lenguaje:English
Publicado: Boston : Addison Wesley, [2003]
Materias:
Teoría de la prueba.
Tabla de Contenidos:
  • 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 Schröder-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.

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