| |
| |
| Volume I | |
| |
| |
| Prefaces | |
| |
| |
| Preface to the English edition | |
| |
| |
| Prefaces to the fourth and third editions | |
| |
| |
| Preface to the second edition | |
| |
| |
| From the preface to the first edition | |
| |
| |
| |
| Some General Mathematical Concepts and Notation | |
| |
| |
| |
| Logical symbolism | |
| |
| |
| |
| Connectives and brackets | |
| |
| |
| |
| Remarks on proofs | |
| |
| |
| |
| Some special notation | |
| |
| |
| |
| Concluding remarks | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Sets and elementary operations on them | |
| |
| |
| |
| The concept of a set | |
| |
| |
| |
| The inclusion relation | |
| |
| |
| |
| Elementary operations on sets | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Functions | |
| |
| |
| |
| The concept of a function (mapping) | |
| |
| |
| |
| Elementary classification of mappings | |
| |
| |
| |
| Composition of functions. Inverse mappings | |
| |
| |
| |
| Functions as relations. The graph of a function | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Supplementary material | |
| |
| |
| |
| The cardinality of a set (cardinal numbers) | |
| |
| |
| |
| Axioms for set theory | |
| |
| |
| |
| Set-theoretic language for propositions | |
| |
| |
| |
| Exercises2. The Real Numbers | |
| |
| |
| |
| Axioms and properties of real numbers | |
| |
| |
| |
| Definition of the set of real numbers | |
| |
| |
| |
| Some general algebraic properties of real numbers | |
| |
| |
| |
| Consequences of the addition axioms | |
| |
| |
| |
| Consequences of the multiplication axioms | |
| |
| |
| |
| Consequences of the axiom connecting addition and multiplication | |
| |
| |
| |
| Consequences of the order axioms | |
| |
| |
| |
| Consequences of the axioms connecting order with addition and multiplication | |
| |
| |
| |
| The completeness axiom. Least upper bound | |
| |
| |
| |
| Classes of real numbers and computations | |
| |
| |
| |
| |
| |
| |
| Definition of the set of natural numbers | |
| |
| |
| |
| The principle of mathematical induction | |
| |
| |
| |
| Rational and irrational numbers | |
| |
| |
| |
| The integers | |
| |
| |
| |
| The rational numbers | |
| |
| |
| |
| The irrational numbers | |
| |
| |
| |
| The principle of Archimedes Corollaries | |
| |
| |
| |
| Geometric interpretation. Computational aspects | |
| |
| |
| |
| The real line | |
| |
| |
| |
| Defining a number by successive approximations | |
| |
| |
| |
| The positional computation system | |
| |
| |
| |
| Problems and exercises | |
| |
| |
| |
| Basic lemmas on completeness | |
| |
| |
| |
| The nested interval lemma | |
| |
| |
| |
| The finite covering lemma | |
| |
| |
| |
| The limit point lemma | |
| |
| |
| |
| Problems and exercises | |
| |
| |
| |
| Countable and uncountable sets | |