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Paysages Mathématiques Éléments de mathématiques pour le XXIe siècle, volume 2: Fondements des mathématiques 2 (théorie des ensembles, mathématiques discrètes, structures algébriques de base)

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Description

Ce livre est le deuxième volume d’une série qui doit, à terme, couvrir l’ensemble des notions du premier cycle universitaire en mathématiques, tout en débordant largement sur le deuxième cycle. De manière plus générale, cette série d’ouvrages pourra être utile à toute personne s’intéressant aux mathématiques actuelles. Elle devrait, en théorie, être accessible même sans connaissance préalable. En effet, les mathématiques sont prises à leur début et les différents concepts progressivement construits, chaque définition, théorème et démonstration ne faisant appel qu’à ce qui a été défini précédemment. Chaque ouvrage se veut à la fois didactique, avec des preuves très détaillées, des explications informelles, et de nombreux exemples et contre-exemples ; complet, voire encyclopédique, avec un exposé de nombreuses notions, des théorèmes tous démontrés, et de nombreux détails historiques ; synthétique, avec en particulier la volonté de multiplier les points de vue. Les cinq premiers volumes traitent des fondements modernes des mathématiques. Ce deuxième volume expose la théorie des ensembles de Zermelo-Fraenkel, qui est le fondement formel des mathématiques le plus classique : Le premier chapitre présente les axiomes principaux de la théorie, et quelques conséquences. Le deuxième chapitre reprend et généralise les structures algébriques de base vues dans le volume 1. Le troisième chapitre présente l'axiome dit de l'infini et la construction de l'ensemble N des entiers naturels. Le quatrième chapitre est consacré à la construction de l'ensemble Z des entiers relatifs, et à la présentation d'applications diverses dans le domaine de ce qu'on appelle les mathématiques discrètes : éléments de théorie des nombres, et introduction à l'analyse combinatoire. Le cinquième chapitre traite de l'axiome dit du choix et de ses conséquences. Le sixième chapitre donne les autres axiomes de la théorie, dont les conséquences sont principalement utilisées dans la théorie elle-même, mais assez peu en dehors.

Product Specifications

Format
paperback
Domain
Amazon UK
Release Date
29 June 2019
Listed Since
18 September 2019

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