Examen de algèbre 1 MIP (SMIA) S1

Examen de algèbre 1 MIP (SMIA) S1

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Eléments de logique et méthodes de démonstration

Eléments de logique

Assertions Dénition

Une assertion p (ou proposition) est une phrase déclarative exclusivement vraie ou fausse.

Ensembles, applications et relations binaires

On rappelle qu'un ensemble est, intuitivement, une collection E d'objets. Ces objets s'appellent les éléments de l'ensemble E. Le long de ce chapitre, E, F, G et H désignent des ensembles quelconques

Opérations sur les ensembles

Parties d'un ensemble Dénition

on dit que F est inclus dans E ou que F est une partie de E si tout élément de F est élément de E. On dira aussi que F est contenu dans E ou que F est un sous-ensemble de E. On écrit F ⊂ E ou encore E ⊃ F.

Arithmétique dans Z

Ensemble des entiers naturels

Dénitions

On admet l'existence d'un ensemble ordonné non vide (N, ≤) vériant les trois propriétés suivantes : i) Toute partie non vide de N possède un plus petit élément. ii) Toute partie non vide majorée de N possède un plus grand élément. iii) N n'a pas de plus grand élément.

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