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Produktbild: Theory of Sets
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Theory of Sets

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Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

20.10.2004

Verlag

Springer Berlin

Seitenzahl

414

Maße (L/B/H)

23,6/15,7/2,5 cm

Gewicht

640 g

Auflage

2nd pr.

Originaltitel

Éléments de Mathématique, Théorie des Ensembles

Sprache

Englisch

ISBN

978-3-540-22525-6

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

20.10.2004

Verlag

Springer Berlin

Seitenzahl

414

Maße (L/B/H)

23,6/15,7/2,5 cm

Gewicht

640 g

Auflage

2nd pr.

Originaltitel

Éléments de Mathématique, Théorie des Ensembles

Sprache

Englisch

ISBN

978-3-540-22525-6

Herstelleradresse

Springer-Verlag GmbH
Heidelberger Platz 3
14197 Berlin
Deutschland
Email: sdc-bookservice@springer.com
Url: www.springer.com
Telephone: +49 6221 3454301
Fax: +49 30 8214091

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  • Produktbild: Theory of Sets
  • I. Description of Formal Mathematics.-
    1. Terms and relations.- 1. Signs and assemblies.- 2. Criteria of substitution.- 3. Formative constructions.- 4. Formative criteria.-
    2. Theorems.- 1. The axioms.- 2. Proofs.- 3. Substitutions in a theory.- 4. Comparison of theories.-
    3. Logical theories.- 1. Axioms.- 2. First consequences.- 3. Methods of proof.- 4. Conjunction.- 5. Equivalence.-
    4. Quantified theories.- 1. Definition of quantifiers.- 2. Axioms of quantified theories.- 3. Properties of quantifiers.- 4. Typical quantifiers.-
    5. Equalitarian theories.- 1. The axioms.- 2. Properties of equality.- 3. Functional relations.- Appendix. Characterization of terms and relations.- 1. Signs and words.- 2. Significant words.- 3. Characterization of significant words.- 4. Application to assemblies in a mathematical theory.- Exercises for
    1.- Exercises for
    2.- Exercises for
    3.- Exercises for
    4.- Exercises for
    5.- Exercises for the Appendix.- II. Theory of Sets.-
    1. Collectivizing relations.- 1. The theory of sets.- 2. Inclusion.- 3. The axiom of extent.- 4. Collectivizing relations.- 5. The axiom of the set of two elements.- 6. The scheme of selection and union.- 7. Complement of a set. The empty set.-
    2. Ordered pairs.- 1. The axiom of the ordered pair.- 2. Product of two sets.-
    3. Correspondences.- 1. Graphs and correspondences.- 2. Inverse of a correspondence.- 3. Composition of two correspondences.- 4. Functions.- 5. Restrictions and extensions of functions.- 6. Definition of a function by means of a term.- 7. Composition of two functions. Inverse function.- 8. Retractions and sections.- 9. Functions of two arguments.-
    4. Union and intersection of a family of sets.- 1. Definition of the union and the intersection of a family of sets.- 2. Properties of union and intersection.- 3. Images of a union and an intersection.- 4. Complements of unions and intersections.- 5. Union and intersection of two sets.- 6. Coverings.- 7. Partitions.- 8. Sum of a family of sets.-
    5. Product of a family of sets.- 1. The axiom of the set of subsets.- 2. Set of mappings of one set into another.- 3. Definitions of the product of a family of sets.- 4. Partial products.- 5. Associativity of products of sets.- 6. Distributivity formulae.- 7. Extension of mappings to products.-
    6. Equivalence relations.- 1. Definition of an equivalence relation.- 2. Equivalence classes; quotient set.- 3. Relations compatible with an equivalence relation.- 4. Saturated subsets.- 5. Mappings compatible with equivalence relations.- 6. Inverse image of an equivalence relation; induced equivalence relation.- 7. Quotients of equivalence relations.- 8. Product of two equivalence relations.- 9. Classes of equivalent objects.- Exercises for
    1.- Exercises for
    2.- Exercises for
    3.- Exercises for
    4.- Exercises for
    5.- Exercises for
    6.- III. Ordered Sets, Cardinals, Integers.-
    1. Order relations. Ordered sets.- 1. Definition of an order relation.- 2. Preorder relations.- 3. Notation and terminology.- 4. Ordered subsets. Product of ordered sets.- 5. Increasing mappings.- 6. Maximal and minimal elements.- 7. Greatest element and least element.- 8. Upper and lower bounds.- 9. Least upper bound and greatest lower bound.- 10. Directed sets.- 11. Lattices.- 12. Totally ordered sets.- 13. Intervals.-
    2. Well-ordered sets.- 1. Segments of a well-ordered set.- 2. The principle of transfinite induction.- 3. Zermelo’s theorem.- 4. Inductive sets.- 5. Isomorphisms of well-ordered sets.- 6. Lexicographic products.-
    3. Equipotent sets. Cardinals.- 1. The cardinal of a set.- 2. Order relation between cardinals.- 3. Operations on cardinals.- 4. Properties of the cardinals 0 and 1.- 5. Exponentiation of cardinals.- 6. Order relation and operations on cardinals.-
    4. Natural integers. Finite sets.- 1. Definition of integers.- 2. Inequalities between integers.- 3. The principle of induction.- 4. Finite subsets of ordered sets.- 5. Properties of finite character.-
    5. Properties of integers.- 1. Operations on integers and finite sets.- 2. Strict inequalities between integers.- 3. Intervals in sets of integers.- 4. Finite sequences.- 5. Characteristic functions of sets.- 6. Euclidean division.- 7. Expansion to base b.- 8. Combinatorial analysis.-
    6. Infinite sets.- 1. The set of natural integers.- 2. Definition of mappings by induction.- 3. Properties of infinite cardinals.- 4. Countable sets.- 5. Stationary sequences.-
    7. Inverse limits and direct limits.- 1. Inverse limits.- 2. Inverse systems of mappings.- 3. Double inverse limit.- 4. Conditions for an inverse limit to be non-empty.- 5. Direct limits.- 6. Direct systems of mappings.- 7. Double direct limit. Product of direct limits.- Exercises for
    1.- Exercises for
    2.- Exercises for
    3.- Exercises for
    4.- Exercises for
    5.- Exercises for
    6.- Exercises for
    7.- Historical Note on
    5.- IV. Structures.-
    1. Structures and isomorphisms.- 1. Echelons.- 2. Canonical extensions of mappings.- 3. Transportable relations.- 4. Species of structures.- 5. Isomorphisms and transport of structures.- 6. Deduction of structures.- 7. Equivalent species of structures.-
    2. Morphisms and derived structures.- 1. Morphisms.- 2. Finer structures.- 3. Initial structures.- 4. Examples of initial structures.- 5. Final structures.- 6. Examples of final structures.-
    3. Universal mappings.- 1. Universal sets and mappings.- 2. Existence of universal mappings.- 3. Examples of universal mappings.- Exercises for
    1.- Exercises for
    2.- Exercises for
    3.- Historical Note on Chapters I-IV.- Summary of Results.-
    1. Elements and subsets of a set.-
    2. Functions.-
    3. Products of sets.-
    4. Union, intersection, product of a family of sets.-
    5. Equivalence relations and quotient sets.-
    6. Ordered sets.-
    7. Powers. Countable sets.-
    8. Scales of sets. Structures.- Index of notation.- Index of terminology.- Axioms and schemes of the theory of sets.