Elements of Mathematics - Theory of Sets

Elements of Mathematics - Theory of Sets......Page 2
Copyright......Page 3
To the Reader
......Page 4
Table of Contents
......Page 8
Introduction
......Page 14
1.1.1 Signs and Assemblies
......Page 22
1.1.2 Criteria of Substitution
......Page 25
1.1.3 Formative Constructions
......Page 26
1.1.4 Formative Criteria
......Page 28
1.2.1 Axioms ......Page 31
1.2.2 Proofs
......Page 32
1.2.4 Comparison of Theories
......Page 33
1.3.1 The Axioms
......Page 35
1.3.2 First Consequences
......Page 36
1.3.3 M
ethods of Proof......Page 37
1.3.4 Conjunction
......Page 40
1.3.5 Equivalence
......Page 41
1.4.1 Definition of Quantifiers
......Page 43
1.4.2 Axioms of Quantified Theories
......Page 44
1.4.3 Properties of Quantifiers
......Page 45
1.4.4 typical Quantifiers ......Page 48
1.5.1 The Axioms
......Page 51
1.5.2 Properties of Equality
......Page 52
1.5.3 Functional Relations
......Page 54
Appendix
......Page 57
Exercises
......Page 63
2.1.1 The Theory of Sets
......Page 72
2.1.2 Inclusion
......Page 73
2.1.4 Collectivizing Relations
......Page 74
2.1.6 The Scheme of Selection and Union
......Page 76
2.1.7 Complement of a Set, the Empty Set
......Page 78
2.2.1 The Axiom of the Ordered Pair
......Page 79
2.2.2 Product of Two Sets
......Page 81
2.3.1 Graphs and Correspondences
......Page 82
2.3.3 Composition of Two Correspondences ......Page 85
2.3.4 Functions
......Page 88
2.3.5 Restrictions and Extensions of Functions
......Page 89
2.3.6 Definition of a function by Means of a Term
......Page 90
2.3.7 Composition of Two Functions, Inverse Function
......Page 91
2.3.8 Retractions and Sections
......Page 93
2.3.9 Functions of Two Arguments
......Page 96
2.4.1 Definition of the Union and the Intersection of a Family of Sets
......Page 97
2.4.2 Properties of Union and Intersection ......Page 100
2.4.3 Images of a Union and an Intersection
......Page 101
2.4.5 Union and Intersection of Two Sets
......Page 103
2.4.6 Coverings
......Page 105
2.4.7 Partitions
......Page 106
2.4.8 Sum of a Family of Sets
......Page 107
2.5.1 The Axiom of the Set of Subsets
......Page 108
2.5.2 Set of Mappings of One Set into Another
......Page 109
2.5.3 Definitions of the Product of a Family of Sets
......Page 110
2.5.4 Partial Products
......Page 112
2.5.5 Associativity of Products of Sets
......Page 113
2.5.6 Distributivity Formulae
......Page 114
2.5.7 Extension of Mappings to Products
......Page 118
2.6 Equivalence Relations
......Page 119
2.6.1 Definition of an Equivalence Relation
......Page 120
2.6.2 Equivalence Classes, Quotient Set
......Page 121
2.6.3 Relations Compatible with an Equivalence Relation
......Page 123
2.6.4 Saturated Subsets
......Page 124
2.6.5 Mappings Compatible with Equivalence Relations
......Page 125
2.6.7 Quotients of Equivalence Relations
......Page 126
2.6.8 Product of Two Equivalence Relations
......Page 127
2.6.9 Classes of Equivalent Objects
......Page 128
Exercises
......Page 130
3.1.1 Definition of an Order Relation
......Page 138
3.1.2 Preorder Relations
......Page 140
3.1.3 Notation and Terminology
......Page 142
3.1.4 Ordered Subsets, Product of Ordered Sets
......Page 143
3.1.5 Increasing Mappings
......Page 145
3.1.6 Maximal and Minimal Elements
......Page 146
3.1.7 Greatest Element and Least Element
......Page 147
3.1.9 Least Upper Bound and Greatest Lower Bound
......Page 148
3.1.11 Lattices
......Page 152
3.1.12 Totally Ordered Sets
......Page 153
3.1.13 Intervals
......Page 154
3.2.1 Segments of a Well-Ordered Set
......Page 155
3.2.2 The Principle of Transfinite Induction
......Page 158
3.2.3 Zermelo's Theorem
......Page 159
3.2.4 Inductive Sets
......Page 161
3.2.5 Isomorphisms of Well-Ordered Sets
......Page 162
3.3.1 The Cardinal of a Set
......Page 164
3.3.2 Order Relation Between Cardinals
......Page 165
3.3.3 Operations on Cardinals
......Page 167
3.3.4 Properties of the Cardinals 0 and 1
......Page 169
3.3.5 Exponention of Cardinals
......Page 170
3.3.6 Order Relation and Operations on Cardinals
......Page 171
3.4.2 Inequalities Between Integers
......Page 173
3.4.3 The Principle of Induction
......Page 175
3.4.5 Properties of Finite Character
......Page 177
3.5.1 Operations on Integers and Finite Sets
......Page 178
3.5.2 Strict Inequalities Between Integers
......Page 180
3.5.3 Intervals in Sets of Integers
......Page 181
3.5.5 Characteristic Functions of Sets
......Page 182
3.5.6 Euclidean Division
......Page 183
3.5.7 Expansion to Base b
......Page 184
3.5.8 Combinatorial Analysis
......Page 186
3.6.1 The Set of Natural Integers
......Page 190
3.6.2 Definition of Mappings by Induction
......Page 191
3.6.3 Properties of Infinite Cardinals
......Page 193
3.6.4 Countable Sets
......Page 195
3.6.5 Stationary Sequences
......Page 196
3.7.1 Inverse Limits
......Page 198
3.7.2 Inverse Systems of Mappings
......Page 199
3.7.3 Double Inverse Limit
......Page 203
3.7.4 Conditions for an Inverse Limit to be Non-Empty
......Page 205
3.7.5 Direct Limits
......Page 209
3.7.6 Direct Systems of Mappings
......Page 211
3.7.7 Double Direct Limit, Product of Direct Limits
......Page 216
Exercises
......Page 219
Historical Note
......Page 264
4.1.1 Echelons
......Page 268
4.1.2 Cononical Extensions of Mappings
......Page 269
4.1.3 Transportable Relations
......Page 270
4.1.4 Species of Structures
......Page 271
4.1.5 Isomorphisims and Transport of Structures
......Page 273
4.1.6 Deduction of Structures
......Page 275
4.1.7 Equivalent Species of Structures
......Page 277
4.2.1 Morphisms
......Page 280
4.2.2 Finer Structures
......Page 282
4.2.3 Initial Structures
......Page 283
4.2.4 Examples of Initial Structures
......Page 285
4.2.5 Final Structures
......Page 289
4.2.6 Examples of Final Structures
......Page 290
4.3.1 Universal Sets and Mappings
......Page 292
4.3.2 Existence of Universal Mappings
......Page 293
4.3.3 Examples of Universal Mappings
......Page 295
Exercises
......Page 298
Historical Note
......Page 305
1 Elements and Subsets of a Set
......Page 356
2 Functions
......Page 360
3 Products of Sets
......Page 366
4 Union, Intersection, Product of a Family of Sets
......Page 372
5 Equivalence Relations and Quotient Sets
......Page 379
6 
Ordered Sets ......Page 382
7 Powers, Countable Sets
......Page 389
8 Scales of Sets, Structures
......Page 392
Index of Notation
......Page 396
Index of Terminology
......Page 401
Axioms and Schemes of the Theory of Sets
......Page 423