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On a toujours~: + + - $P and Q equiv Q and P$ (commutativité de $and$) + - $P or Q equiv Q or P$ (commutativité de $or$) + - $(P and Q) and R equiv P and (Q and R)$ (associativité de $and$) + - $(P or Q) or R equiv P or (Q or R)$ (associativité de $or$) + - $not not P equiv P$ (double négation) + - $not (P and Q) equiv not P or not Q$ (lois de De Morgan) + - $not (P or Q) equiv not P and not Q$ (lois de De Morgan) + - $(P and Q) or R equiv (P or R) and (Q or R)$ (distributivité de $or$ par rapport à $and$) + - $(P or Q) and R equiv (P and R) or (Q and R)$ (distributivité de $and$ par rapport à $or$) +] + +#proof[ + Il suffit de dresser les tables de vérité. +] + + +#remark[Dualité][ + Remarquons que les formules présentées restent vraies si l'on change tous les $and$ par des $or$ et inversement. +] + + +== Implication et équivalence + + +#definition("Implication")[ + Étant données deux propositions P et Q, on appelle implication de P envers Q, et l'on note $P => Q$, la proposition $not P or Q$. +] + +#figure( +table( + columns: (auto, auto, auto), + [$P$], [$Q$], [$P => Q$], + [V], [V], [V], + [V], [F], [F], + [F], [V], [V], + [F], [F], [V] +), + +caption: "Table de vérité de l'implication") + + + +#definition[Soient $P$ et $Q$ deux propositions. On appelle *contraposée* de l'implication $P => Q$ l'implication $not Q => not P$, et *réciproque* de $P => Q$ l'implication $Q => P$.] + +#proposition[ + Une implication et sa contraposée sont équivalentes au sens de la vérité. +] + +#proof[ + Soient $P$ et $Q$ deux propositions. On a~: + + $ P => Q &equiv not P or Q \ + &equiv Q or not P quad &&"(commutativité)" \ + &equiv not not Q or not P &&"(double négation)" \ + &equiv not Q => not P $ +] + +#definition[ + On appelle équivalence de $P$ et $Q$, notée $P <=> Q$, la proposition + $ (P => Q) and (Q => P) $ +] + +#figure( +table( + columns: (auto, auto, auto), + [$P$], [$Q$], [$P <=> Q$], + [V], [V], [V], + [V], [F], [F], + [F], [V], [F], + [F], [F], [V] +), + +caption: "Table de vérité de l'équivalence") + + +#proposition[ + Soient P et Q deux propositions. Alors on a $P equiv Q$ si, et seulement si la proposition $P <=> Q$ est vraie. +] + +#proof[ + - Mq $==>$. Supposons $P equiv Q$. Alors $P$ et $Q$ ont la même valeur de vérité, donc on a~: + - Soit $P$ est vraie et $Q$ est vraie, donc $P <=> Q$ est vraie + - Soit $P$ est fausse et $Q$ est fausse, donc $P <=> Q$ est fausse + - Mq $<==$. Supposons que $P<=>Q$ soit vraie. Alors d'après la table de vérité de l'équivalence, on a soit $P$ vraie et $Q$ vraie, soit $P$ fausse et $Q$ fausse. Dans les deux cas, $P$ et $Q$ ont la même valeur de vérité donc $P equiv Q$. +] + + +== Raisonnements + +#rule[ + Pour montrer qu'une proposition de la forme $P => Q$ est vraie, on écrit~: + + "Supposons que $P$, montrons $Q$.", puis on démontre $Q$. + + Formellement, on dit que l'on démontre $Q$ dans la théorie obtenue en adjoignant $P$ aux axiomes de la théorie ambiante. +] + +#proposition(emph("Modus ponens"))[ + Soient P et Q deux propositions, alors + $ P and (P => Q) => Q $ est une tautologie. +] + +#proof[ + On a d'abord + $ P and (P => Q) &equiv P and (not P or Q) \ &equiv (P and not P) or (P and Q) \ &equiv P and Q, $ + puis $P and Q => Q$, car + + $ P and Q => Q equiv not(P and Q) or Q equiv not P or underbrace(not Q or Q, equiv top) equiv top. $ +] + + + + + +== Quantification + +Jusqu'ici nous avons travaillé à l'ordre zéro, c'est-à-dire que nous n'avons travaillé qu'avec des propositions, des relations d'arité zéro. Nous introduisons donc des notions de calcul des prédicats, essentielles. + + +=== Motivation + +Si l'on considère $P_1, ..., P_n$ $n$ propositions. On peut se demander s'il existe un $k$ entre $1$ et $n$ tel que $P_k$ soit vraie, et on peut se demander si toutes les propositions $P_k$ sont vraies. + + +En écrivant $ D = P_1 or ... or P_n = or.big_(i=1)^n P_i quad "et" quad C = P_1 and ... and P_n = and.big_(i=1)^n P_i, $ + +le problème se ramène à la vérité de $D$ ou de $C$. Maintenant on souhaite généraliser. On considère un prédicat unaire $P(x)$, d'unique argument $x$. On veut savoir s'il existe un $x$ tel que $P(x)$ ou si pour tout x, on ait $P(x)$. Cependant, on ne peut pas énumérer tous les termes $x$ pour les tester. On est donc limité par nos symboles actuels. On en introduit donc deux autres. + +#definition("Quantificateurs")[ + Soit $P(x)$ un prédicat unaire d'argument $x$. + + On note~: + - $exists x P(x)$ la proposition (il s'agit bien d'une proposition) vraie si, et seulement si il existe un terme $x$ tel que $P(x)$. + - $forall x P(x)$ la proposition vraie ssi pour tout terme $x$, $P(x)$ soit vraie +] + +#proposition("Négation d'une quantification")[ + Soit $P(x)$ un prédicat. On a~: + + - $ not( exists x space P(x) ) equiv forall x space not P(x) $ + - $ not( forall x space P(x) ) equiv exists x space not P(x) $ +] + +#remark[C'est une généralisation des lois de De Morgan.] + +#proof[ + + - Supposons $not (exists x space P(x))$. Montrons que $forall x space not P(x)$. + + Soit $x$ un terme, montrons que $P(x)$. Supposons par l'absurde $P(x)$, alors $P(x)$ serait exemplifiée par $x$, donc $exists x space P(x)$ ce qui contredit l'hypothèse. + + - Supposons $forall x space not P(x)$. Montrons que $not( exists x space P(x) )$. Supposons par l'absurde qu'il existe un $x$ tel que $P(x)$. Alors en applicant l'hypothèse à $x$, on aurait $not P(x)$, ce qui est absurde. + + - Pour le deuxième point, on utilise le premier. On a + + $ not( forall x space P(x) ) equiv not not (exists x space P(x)) equiv exists x space P(x). $ +] + +#proposition("Compatibilité des quantificateurs avec les connecteurs idoines")[ + Soient $P(x)$ et $Q(x)$ deux prédicats unaires. On a + + - $ exists x space (P(x) or Q(x)) equiv (exists x space P(x)) or (exists x space Q(x)) $ + - $ forall x space (P(x) and Q(x)) equiv (forall x space P(x)) and (forall x space Q(x)) $ +] + +#remark[C'est une généralisation de l'associativité de $or$ et $and$.] + +#proof[Triviale, laissée en exercice de rédaction au lecteur.] + + + += Module Ensembles, relations et applications (2h) + +== Vocabulaire ensembliste + +#hist[ + La théorie des ensembles s'est formalisée à la fin du XIX#super[e] siècle avec l'appui de mathématiciens tels que Georg Cantor (1845-1918), David Hilbert (1862-1943). WIP +] + +#definition("Ensemble")[ + On appelle ensemble tout terme. +] + +#remark[ + Cette définition paraît décevante au premier regard. On vous avait peut-être vendu précédemment une notion de "collection d'objets mathématiques", ou bien de "collection désordonnée d'objets mathématiques". Mais ces définitions n'en sont pas vraiment, elles utilisent elles-mêmes des mots non définis. + + Ces tentatives de définitions qui plaisent à l'intuition cachent le fait que le concept d'ensemble est une *notion primitive* de la théorie des ensembles. C'est un concept qui n'est pas défini car c'est une brique de base de la théorie. +] + + + +#notation[ + Soit $cal(R)$ une relation binaire. Quels que soient les termes $x$ et $y$, on note $x cal(R) y$ la proposition $cal(R)(x, y)$. +] + + + +#definition("Appartenance")[ + On considère sur la théorie des ensembles un prédicat binaire noté $in$. Pour tous termes $x, y$, on note $x in y$ au lieu de $op(in)(x, y)$ et l'on dit $x$ appartient à $y$. +] + + +#notation("Quantification restreinte ou typique")[ + Soit $P(x)$ un prédicat et $E$ un ensemble. + + On note~: + - $exists x in E quad P(x)$ au lieu de $exists x quad x in E and P(x)$~; + - $forall x in E quad P(x)$ au lieu de $forall x quad x in E => P(x)$. +] + +#remark[ + En particulier, *quel que soit* le prédicat $P$, + - $exists x in emptyset quad P(x)$ est une antilogie + - $forall x in emptyset quad P(x)$ est une tautologie +] + + +#definition("Égalité")[ + On définit une relation binaire notée $=$, et l'on note $x = y$ au lieu de $op(=)(x, y)$ et l'on dit que $x$ égale $y$ ou que $x$ et $y$. +] + +#rule("Critères de substitution")[ + Soit $P(x)$ un prédicat unaire, et $U(x)$, $V(x)$ deux termes à un argument. + + Alors~: + - $forall x forall y quad x=y => ( P(x) <=> P(y) )$~; + - $forall x quad x = y => U(x) = V(y)$. +] + +== Axiomes de la théorie des ensembles + +#axiom("Existence d'un ensemble vide")[ + Il existe un ensemble vide, c.-à-d.~: + $ exists E space forall x quad x in.not E $ +] + +#definition[ + Soient $E$ et $F$ deux ensembles. On dit que $E$ est inclus dans $F$ et l'on note $E subset F$ si, et seulement si + $forall x quad x in E => x in F$. +] + +#axiom("Axiome d'extensionnalité")[ + Les ensembles sont entièrement caractérisés par leurs éléments, c'est-à-dire~: + + $ forall E forall F quad E = F <=> (forall x quad x in E <=> x in F), $ + + ce qui est équivalent à + + $ forall E forall F quad E = F <=> (E subset F and F subset E) $ +] + + + + +#definition[ + On dit qu'une relation binaire est~: + - réflexive ssi $forall x quad x cal(R) x$ + - symétrique ssi $forall x forall y quad x cal(R) y => y cal(R) x$ + - transitive ssi $forall x forall y forall z quad x cal(R) y and y cal(R) z => x cal(R) z$ + - antisymétrique ssi $forall x forall y quad x cal(R) y and y cal(R) x => x = y $. +] + + + +#definition("Relation d'équivalence")[ + On dit qu'une relation binaire $cal(R)$ est une relation d'équivalence ssi elle est réflexive, symétrique et transitive. +] + +#definition("Relation d'ordre")[ + On dit qu'une relation binaire $cal(R)$ est une relation d'ordre ssi elle est réflexive, transitive et antisymétrique. +] + + +#proposition[ + L'égalité est une relation d'équivalence. +] + +#proof[ + Elle coïncide avec la relation binaire $cal(R)$ définie pour tous ensembles $E$, $F$ par $E cal(R) F <=> (forall x quad x in E <=> x in F)$ qui est évidemment une relation d'équivalence. +] + +#proposition[ + L'inclusion est une relation d'ordre. +] + +#proof[ + - La réflexivité et la transitivité découlent des propriétés de l'implication~; + - L'antisymétrie est l'axiome d'extensionnalité. +] + +#proposition[ + Il existe un unique ensemble vide. On le note $emptyset$. +] + +#proof[ + - Existence : par axiome + - Unicité : Soient $E$ et $F$ deux ensembles tels que $forall x space (x in.not E and x in.not F)$. On a alors $E subset F$ et $F subset E$, donc par l'axiome d'extensionnalité $E = F$. +] + + +#definition("Quantification existentielle avec unicité")[ + Soit $P(x)$ un prédicat. On note + $ exists! x quad P(x) $ + la proposition + $ ( exists x quad P(x) ) and (forall x forall y space P(x) and P(y) => x=y) $ +] + + +#definition[ + On dit qu'un prédicat $P$ de variable $x$ est collectivisant en $x$ si, et seulement si + + $ exists E space forall x quad (x in E <=> P(x)). $ +] + +#proposition[ + Dans ce cas, si un ensemble $E$ vérifie $ forall x quad x in E <=> P(x), $ alors il est unique et on le note ${ x | P(x) }$. +] + +#proof[ + Soient $E$ et $F$ deux ensembles, dont l'appartenance équivaut à vérifier $P$. Alors, pour tout $x$, $x in E <=> P(x) <=> x in F$ donc par l'axiome d'extensionnalité, $E=F$. +] + +#axiom("Schéma d'axiomes de compréhension")[ + Soit $P$ un prédicat unaire et $E$ un ensemble. Alors la relation $x in E and P(x)$ est collectivisante en $x$. +] + +#notation[ + On note ${x in E | P(x)} = {x | x in E and P(x)}$ +] + + +#proposition[Intersection binaire][ + Soient $A$ et $B$ deux ensembles. Alors la relation $x in A and x in B$ est collectivisante en $x$. L'ensemble associé est noté $A sect B$. +] + + +#proof[ + Il suffit de poser $A sect B = {x in A | x in B}$. +] + + +#proposition[Intersection quelconque][ + Soit $E$ un ensemble *non vide*. Alors la relation $forall F in E, x in F$ est collectivisante en $x$. L'ensemble associé se note alors $sect.big E$ ou encore $display(sect.big_(F in E) E)$. +] + +#proof[ + Comme $E$ est non vide, il possède un élément $F_0$. On pose alors $ I = { x in F_0 | forall F in E, x in F }. $ + + On a alors, pour tout $x$~: + + $ x in I &<=> x in F_0 and forall F in E quad x in F \ + &<=> forall F in E quad x in F quad "car" F_0 in E $ +] + +#proposition[Différence ensembliste][ + Soient $A$ et $B$ deux ensembles. Alors la relation $x in A and not (x in B)$ est collectivisante en $x$. L'ensemble associé est noté $A without B$. +] + + +#proof[ + Il suffit de poser $A without B = {x in A | x in.not B}$. +] + +#definition[Complémentaire][ + En particulier, quand $B$ est une partie de $A$, on note $complement_A B$ ou $overline(B)^A$, ou $overline(B)$ lorsqu'il n'y a pas d'ambiguïté sur la valeur de $A$, l'ensemble $A without B$, et on l'appelle le complémentaire de B dans A. +] + + +#axiom("Axiome de la paire")[ + Soient $a$ et $b$ deux termes (non nécessairement différents). Alors la relation $x = a or x = b$ est collectivisante en $x$. L'ensemble associé se note ${a; b}$. +] + +#remark[ + En particulier, cet axiome assure que pour tout $a$, le singleton ${a}$ existe. On a donc par définition + $ forall x quad x in {a} <=> x = a. $ +] + +#axiom("Axiome de l'union")[ + Soit $E$ un ensemble (comprendre ensemble d'ensembles). Alors la relation + $ exists F in E quad x in F $ + est collectivisante en $x$. L'ensemble associé se note $union.big E$ ou $display(union.big_(F in E) F)$. +] + + +#proposition[Union binaire][ + Soient $A$ et $B$ deux ensembles. La relation $x in A or x in B$ est collectivisante en $x$ et l'on note $A union B$ l'ensemble associé. +] + +#proof[ + Il suffit de poser $A union B = union.big {A; B}. $ +] + +// TODO +#proposition[Traductions ensemblistes ][ + Soit +] + + +#axiom[Axiome des parties][ + Soit $E$ un ensemble. Alors la relation $F subset E$ est collectivisante en $F$. L'ensemble associé se note $cal(P)(F)$ ou $frak(P)(F)$. +] + + +#axiom[Axiome de fondation][ + Soit $E$ un ensemble non vide. Alors il possède un élément $x$ tel que $x sect E = emptyset$. +] + + +#proposition[Irréflexivité de l'appartenance][ + Soit $E$ un ensemble. Alors $E in.not E$. +] + + +#proof[ + Soit $E$ un ensemble. Supposons par l'absurde $E in E$. On a donc ${E} subset E$. Comme ${E}$ est non vide, il possède un élément $x$ tel que $x sect E = emptyset$. Or $x in {E}$ donc $x = E$, puis $E sect E = emptyset$, donc $E = emptyset$, ce qui contredit le fait qu'il possède un élément. +] + + + +== Couples + +#definition("Couples au sens de Kuratowski")[ + Soient $x$ et $y$ deux termes. On appelle couple formé de $x$ et $y$ l'ensemble ${{x}; {x; y}}$. On le note $(x; y)$. +] + +#proposition("Unicité des couples")[ + Soient $a$, $b$, $c$, $d$ quatre termes. On a~: + + $ (a; b) = (c; d) <=> cases(a=c \ b=d) $ +] + +#proof[ + Par double implication. + + L'implication $<==$ est évidente. Montrons $==>$. + + Supposons $(a, b) = (c,d)$. Montrons $a=c$ et $b=d$. + + On a donc ${{a}; {a; b}} = {{c}; {c; d}}$, + + donc + + - Ou bien ${a} = {c; d}$ et ${a; b} = {c}$, ce qui implique $a=b$ et $c=d$, puis $a=c$ et $a=d$ + - Ou bien ${a} = {c}$ et ${b} = {d}$, donc $a=c$ et $b=d$. +] + + +#definition[ + Pour tout couple $(a, b)$, on note $op("pr"_1)((a, b)) = a$ et $op("pr"_2)((a, b)) = b$. +] + +Ces deux opérateurs sont bien définis par la propriété précédente. + +#notation[ + Si $P(x,y)$ est un prédicat binaire, au lieu de $forall x forall y quad P(x, y)$ on peut noter $forall (x; y) quad P(x; y)$. +] + +#definition[ + Soit $cal(R)$ une relation binaire. On dit que $cal(R)$ est collectivisante si, et seulement si la relation "$x$ est un couple de la forme $(a, b)$ et $cal(R)(a; b)$" est collectivisante en $x$. + + Dans le cas où $cal(R)$ est collectivisante, l'ensemble associé s'appelle *graphe* de $cal(R)$. +] + +#proposition("Produit cartésien")[ + Soient $E$ et $F$ deux ensembles. + + La relation $x in E and y in F$ est collectivisante en $(x, y)$. + + On appelle produit cartésien de $E$ et $F$, noté $E times F$ et lu "E croix F" l'ensemble associé. +] + +#proof[ + On cherche à trouver un surensemble de l'ensemble que l'on essaie de construire, pour en prendre une partie. + + Soit $x in E$ et $y in F$. On a $(x, y) = {{x}, {x, y}} = {{x}} union {{x, y}}.$ + + Or $x in E$ et $y in F$ donc ${x, y} subset E union F$ puis ${x, y} in cal(P)(E union F)$. De même ${x} in cal(P)(E union F)$, puis ${{x}, {x, y}} in cal(P)(cal(P)(E union F))$. + + Pour tout $a$, notons $P(a)$ la proposition "$a$ est un couple de la forme $(x, y)$ et $x in E and y in F$)". + + On a montré $forall a quad P(a) => a in cal(P)(cal(P)(E union F))$. + + Notons $G = { a in cal(P)(cal(P)(E union F)) | P(a)}.$ + + On a toujours $forall a quad P(a) => a in G$ mais on a en plus $forall a quad a in G => P(a)$, donc la relation $P(a)$ est collectivisante en $a$ d'ensemble associé $G$. +] + + + +== Correspondances et applications + + +#definition("Correspondance")[ + Soient $E$ et $F$ deux ensembles, et $Gamma subset E times F$. On appelle correspondance entre $E$ et $F$ le terme $f = (Gamma, (E, F))$. + + On note $op("Cor")(E, F)$ l'ensemble des correspondances de $E$ vers $F$. Il s'agit bien d'un ensemble car il s'identifie naturellement à $cal(P)(E times F)$. + + On note $x mapsto y$ la relation $(x, y) in Gamma$. + + On dit que $E$ est l'ensemble de départ de $f$ et que $F$ est l'ensemble d'arrivée de $f$. + + Pour tout $(x, y) in E times F$, si $x mapsto y$ on dit que $y$ est une image de $x$ et que $x$ est un antécédent de $x$. +] + +#definition("Fonction")[ + Soit $E$, $F$ deux ensembles et $f$ une correspondance de $E$ vers $F$. On dit que $f$ est une fonction de $E$ dans $F$ si, et seulement si + + $ forall x in E space forall(y, y') in F^2 quad ( x mapsto y and x mapsto y' ) => y=y' $ + + Cela revient à dire que si un élément $x$ de $E$ possède une image, alors elle est unique. On la note alors $f(x)$. +] + +#definition("Application")[ + Soit $E$, $F$ deux ensembles et $f$ une correspondance de $E$ vers $F$. On dit que $f$ est une application de $E$ dans $F$ si, et seulement si + + $ forall x in E space exists !y in F quad x mapsto y. $ + + Cela revient à dire que tout élément de $E$ possède une image et qu'elle est unique. On la note alors $f(x)$. + L'ensemble des applications de E dans F se note $F^E$ (ou plus rarement $cal(F)(E, F)$). +] + +#definition("Injectivité, surjectivité, bijectivité")[ + Soit $f$ une application de $E$ dans $F$. On dit qu'elle est~: + - injective ssi $forall (x, x') in E^2 quad f(x) = f(x') => x=x'$ + - surjective ssi $forall y in F space exists x in E quad f(x) = y$ + - bijective ssi elle est injective et surjective +] + +#proposition("Caractérisation des bijections")[ + Soit $f$ une application de $E$ dans $F$. Elle est bijective si, et seulement si + + $ forall y in F space exists! x in E quad f(x) = y. $ +] + +#definition("Permutation")[ + Soit $E$ un ensemble. Une bijection de $E$ dans $E$ s'appelle une permutation. L'ensemble des permutations de $E$ se note $S(E)$ ou $frak(S)(E)$ +] + +#definition[Identité][ + Pour tout ensemble $E$, on note $id_E$ l'application + + $ id_E : E &-> E \ + x &|->x $ +] + +#definition[Composition][ + Soient $E$, $F$, $G$ trois ensembles et $f: E -> F$, $g: F -> G$ deux applications. On note $ g compose f : E &-> G \ x &|-> g(f(x)). $ + + Attention, c'est la fonction $f$ qui est évaluée en premier. +] + +Attention, la proposition suivante est réservée pour une seconde lecture et fait appel à des notions encore non abordées. + +#proposition[Structure de $S_E$][ + Soit $E$ un ensemble. $(S_E, compose)$ est un groupe. +] + + +#definition[Image directe][ + Soit $E$, $F$ deux ensembles, $f$ une application de $E$ dans $F$ et $A$ une partie de $E$. On appelle image directe de $A$ par $f$, notée $f(A)$ l'ensemble + + $ {y in F | exists x in E space f(x)=y} $ +] + +#proposition[ + Soit $E$, $F$ deux ensembles, $f$ une application de $E$ dans $F$ et $A$, $B$ deux parties de $E$. On a~: + + - $f(A union B) subset f(A) union f(B)$ + - $f(A sect B) subset f(A) sect f(B)$ + - $overline(f(A)) subset f(overline(A))$ +] + +#notation[ + On le note aussi + ${f(x) : x in E}.$ Cette dernière notation permet de ne pas expliciter l'application $f$ mais seulement son comportement. +] + +#definition[Image réciproque][ + Soit $E$, $F$ deux ensembles, $f$ une application de $E$ dans $F$ et $B$ une partie de $F$. On appelle image réciproque de $B$ par $f$, notée $f^(-1)(B)$ l'ensemble + $ { x in E | f(x) in B }. $ +] + + +== Entiers naturels + +#definition[Opérateur "successeur"][ + Pour tout ensemble $E$, on appelle successeur de $E$, noté $s(E)$ l'ensemble $E union {E}$. +] + +#definition[ + On note~: $0 := emptyset$, $1 := s(emptyset)$, $2 := s(s(emptyset))$, $3 := s(s(s(emptyset)))$, ... +] + +#proposition[ + On a toujours $E subset s(E)$. +] +#proof[Triviale.] + +#proposition[ + Soit $E$ un ensemble. L'union $E union {E}$ est disjointe. +] + +#proof[ + Soit $E$ un ensemble. Supposons par l'absurde $E union {E} != emptyset$. Il existe donc $x in E union {E}$ puis $x = E$ donc $E in E$, ce qui est absurde. +] + +#proposition[Fondation][ + L'ensemble vide n'a pas de prédecesseur. +] + +#proof[ + Supposons qu'il existe un ensemble $E$ tel que $s(E) = emptyset$. Alors $E subset emptyset$ donc $E = emptyset$. +] + + +#definition[ + On dit qu'un ensemble $E$ est héréditaire si, et seulement si + $ forall x in E quad s(x) in E. $ +] + +#definition[ + Soit $E$ un ensemble héréditaire. On appelle support de $E$, noté $op("Supp")(E)$ l'ensemble $ { y in E | forall x in E space s(x) != y }. $ + + C'est le sous-ensemble des éléments de $E$ qui n'ont pas de prédecesseur. +] + + + +/* +#proposition[ + Soit $E$ un ensemble IH et $X in E$. Alors il existe $B in op("Supp")(E)$ tel que $B subset X$. +] + +#proof[ + Soit $E$ IH et $X in E$. Supposons par l'absurde que pour tout $B in op("Supp")(E), B subset.not X$. Ainsi, pour tout $B in op("Supp")(E)$, $B without X$ est non vide. +] + +#proposition[ + Soient $E$ et $F$ deux ensembles IH. Alors $E subset F$ si, et seulement si $op("Supp") E subset op("Supp") F$. +] + +#proof[ + Supposons $E subset F$. Montrons que +] + +#proposition[ + Soient $E$ et $F$ deux ensembles initalement héréditaires. Alors ils sont égaux si, et seulement si leurs supports sont égaux. +] + +#proof[ + Par antisymétrie~! +] + + +/* +#proposition[Unicité de l'ensemble des entiers naturels][ + Disons qu'un ensemble $E$ vérifie la propriété H si, et seulement si $emptyset in E and forall x in E, s(x) in E $. + + Alors il existe un unique ensemble $E$ tel que pour tout ensemble $F$, si $F$ vérifie la propriété $H$ alors $E subset F$. +] + +#proof[ + Par l'axiome de l'infini, il existe un ensemble qui vérifie la propriété H. Notons-le $H_0$. Notons $cal(H) = {H in cal(P)(H_0) | H "vérifie la propriété H" }$. Considérons + $ E = sect.big_(H in cal(H)) H. $ + + Montrons que E vérifie la propriété H. + + - On a $forall H in cal(H), emptyset in H$, donc $emptyset in E$~; + - Soit $x in E$. Montrons que $s(x) in E$. On a $forall x in cal(H), x in H$ donc en appliquant la propriété H à chacun des éléments de $cal(H)$, on obtient $forall H in cal(H), s(x) in H$, donc $s(x) in E$. + + Montrons que pour tout ensemble $F$, si $F$ vérifie la propriété $H$ alors $E subset F$. + + Soit $F$ un ensemble qui vérifie la propriété $H$. Montrons que $E subset F$. Soit $x in E$ et montrons que $x in F$. + + On considère +] +*/ +*/ + + +#proposition[Source d'un élément][ + Soit E un ensemble héréditaire, et $x in E$. +] + +#axiom[De l'infini][ + Il existe un ensemble héréditaire qui possède $emptyset$. +] + +Attention, un tel ensemble n'a pas de raison d'être unique~! En effet, on pourrait très bien imaginer un ensemble constitué de deux chaînes distinctes~: + +$ E_("hyp") = { emptyset, s(emptyset), s^2 (emptyset), ..., alpha, s(alpha), ...} $ + +On voudrait donc bien disposer d'un ensemble unique, qui ne possède qu'une seule chaîne~! + +#proposition[Ensemble des entiers naturels][ + Il existe un unique ensemble héréditaire de support ${emptyset}$. On le note $NN$. +] \ No newline at end of file