les mots clés ! (en cours de rédaction !)

Ci-dessous les définitions des termes les plus couramment utilisés en mathématiques, essentiellement issues du Ramis Tome1  :

\textlinb{a}

Appartenance : On appelle appartenance la relation binaire que l’on écrit x \in y et que l’on lit « x est un élément de y « .

Application: On appelle application toute correspondance dont le graphe est fonctionnel et dont l’ensemble de définition coïncide avec l’ensemble de départ.

Application croissante : Soient (E,\preccurlyeq ) et (F,\leq) deux ensembles ordonnés, on appelle application croissante toute application f : E\rightarrow F telle que : \forall (x,y) \in E^2 \;\;x\preccurlyeq y \Rightarrow f(x)\leq f(y).

Assertion : On appelle assertion tout énoncé ne contenant pas de variable ne pouvant prendre que l’une des deux valeurs logiques « Vrai » ou « Faux ».

Classe d’équivalence : Soit \mathscr{R} une relation d’équivalence sur un ensemble E, on appelle classe d’équivalence de x l’ensemble \bar{x} =\{ y \in E/ x \mathscr{R} y\}.

Collectivisant : Se dit d’un prédicat \mathcal A si il existe un ensemble A tel que les objets x pour lesquels \mathcal A (x) est vrai sont les éléments de A.

Correspondance : Soient E et F des ensembles, on appelle correspondance de E vers F tout triplet de la forme (\Gamma , E,F)\Gamma est une partie de E\times F.

\textlinb{e}

Ensemble de définition : Soit \Gamma le graphe associé à une correspondance, on appelle ensemble de définition l’unique ensemble \mathcal{D} possédant la propriété \forall x\;\; (x\in \mathcal{D} ) \Longleftrightarrow (\exists y \;\; (x;y)\in \Gamma) .

Ensemble ordonné : On appelle ensemble ordonné tout ensemble muni d’une relation d’ordre.

Ensemble quotient : Soit \mathscr{R} une relation d’équivalence sur un ensemble E, on appelle ensemble quotient de E par \mathscr{R} et on note E / \mathscr{R} l’ensemble des classes d’équivalence, c’est une partition de E.

Ensemble vide : Soit E un ensemble, on appelle ensemble vide et on note O_E l’ensemble \{x \in E, x\not = x\}.

Fonction : On appelle fonction toute correspondance dont le graphe est fonctionnel.

Fonctionnel : Se dit d’un graphe \Gamma tel que pour tout x, il existe au plus un y vérifiant (x,y) \in \Gamma.

Graphe : On appelle graphe tout ensemble dont les éléments sont des couples.

\textlinb{i}

Injection: On appelle injection toute application vérifiant :f(x)=f(x')\;\; \Rightarrow \;\; x=x' .

Involution : on appelle involution toute application vérifiant f\circ f =I_d.

\textlinb{\Bma}

Maximum : Soit (E,\preccurlyeq ) un ensemble ordonné, si il existe un élément a tel que : \forall x \in E\;\; a\preccurlyeq x \;\Rightarrow \;x=a (resp.\forall x \in E\;\; x\preccurlyeq a \;\Rightarrow \;x=a) ,on dit que a est un élément maximal (resp. minimal) de E.

Exemple sur (\mathbb{N} \setminus 1,/) les élément minimaux sont les nombres premiers.

Morphisme : Soit f une application de E dans F, \mathscr{R} et \mathscr{S} deux relations binaires sur E et F. On dit que f est un morphisme de (E,\mathscr{R}) dans (F,\mathscr{S}) si et seulement si :

\forall (x,y)\in E^2\;\; x\mathscr{R}y \;\;\Rightarrow f(x) \mathscr{S} f(y) .

\textlinb{\Bpa}

Prédicat : On appelle prédicat tout énoncé contenant des variables tel qu’en subsistuant à chacune de ces lettres un objet d’un certain référentiel, on obtienne une assertion.

Préordre : On appelle préordre toute relation binaire réflexive et transitive.

\textlinb{\Bra}

Relation binaire : On appelle relation binaire tout prédicat à deux variables.

Relation d’équivalence : On appelle relation d’équivalence sur un ensemble E tout préordre symétrique sur E. On note x\;\equiv y \;(\text{modulo} \mathscr{R} ).

Exemple :  Soit f : E\rightarrow F une application, « f(x)=f(y) » est une relation d’équivalence.

Relation d’ordre : On appelle relation d’ordre sur un ensemble E tout préordre antisymétrique sur E.

Exemple :  sur \mathbb{N}, la relation « a/b » qui signifie il existe q \in \mathbb{N} tel que b=qa est une relation d’ordre, non totale.

 

\textlinb{\Bsa}

Surjection : On appelle surjection toute application telle que tout élément de l’ensemble d’arrivée possède au moins un antécédent.

\textlinb{\Bta}

Totalement ordonné : Se dit d’un ensemble ordonné dont deux élément quelconque sont comparables.

 

 

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