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Introduction to modern mathematics
Introduction to Modern Mathematics.
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés Polaco |
| Publicado: |
Amsterdam,
North-Holland Pub. Co.,
1973.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Front Cover; Introduction to Modern Mathematics; Copyright Page; FOREWORD; Table of Contents; CHAPTER I. THE ALGEBRA OF SETS; 1. The concept of set; 2. The union of sets; 3. The intersection of sets. The laws of absorption and distributivity; 4. The difference of sets. Relationships between the difference of sets and the operations of union and intersection of sets; 5. The universe. The complement of a set; 6. Axioms of the algebra of sets; 7. Fields of sets; 8. Prepositional functions of one variable; 9. Note on axioms of set theory
- 10. Comments on the need of an axiomatic approach to set theory and on axiomatic theoriesCHAPTER II: NATURAL NUMBERS. PROOFS BY INDUCTION; 1. An axiomatic approach to natural numbers. The principle of induction; 2. Examples of proofs by induction; CHAPTER III. FUNCTIONS; 1. The concept of function; 2. One-to-one functions. Inverse function; 3. Composition of functions; 4. Groups of transformations; CHAPTER IV. GENERALIZED UNIONS AND INTERSECTIONS OF SETS; 1. The concept of generalized union and intersection; 2. The properties of generalized unions and intersections of sets
- CHAPTER V. CARTESIAN PRODUCTS OF SETS. RELATIONS. FUNCTIONS AS RELATIONS1. Cartesian products; 2. Binary relations; 3. Propositional functions of two variables; 4. Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, and transitive relations; 5. Functions as relations; CHAPTER VI. GENERALIZED PRODUCTS. m-ARY RELATIONS. FUNCTIONS OF SEVERAL VARIABLES. IMAGES AND INVERSE IMAGES UNDER A FUNCTION; 1. Generalized products; 2. m-ary relations; 3. Propositional functions of m variables; 4. Functions of several variables; 5. Images and inverse images under a function
- CHAPTER VII. EQUIVALENCE RELATIONS1. Definition of equivalence relations. Method of identification; 2. Application of the method of identification to the construction of integers; 3. Application of the method of identification to the construction of rational numbers; 4. Note on Cantor's theory of real numbers; CHAPTER VIII. POWERS OF SETS; 1. Equipotent sets. Power of a set; 2. Enumerable sets; 3. Examples of non-enumerable sets; 4. Inequalities for cardinal numbers. The Cantor-Bernstein theorem; 5. Sets of the power of the continuum
- 6. The power set. Cantor's theorem. Consequences of Cantor's theoremCHAPTER IX. ORDERED SETS; 1. Ordering relations; 2. Maximal and minimal elements; 3 Subsets of ordered sets. The Kuratowski-Zorn lemma; 4. Note on lattices; 5. Quasi-ordering relations; 6. Note on directed sets; CHAPTER X. LINEARLY ORDERED SETS; 1. Linear orderings; 2. Isomorphism of linearly ordered sets; 3. Dense linear ordering; 4. Continuous linear orderings; CHAPTER XI. WELL-ORDERED SETS; 1. Well-ordering relations. Ordinal numbers; 2. Comparison of ordinal numbers; 3. Sets of ordinal numbers