From 0c8798f26a71f906f44d1b51fb4a593fea29e13a Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 9 Jul 2025 13:06:22 +0200 Subject: [PATCH] feat(eti): aggiorna gli esercizi e aggiunge gli assiomi di ZFC --- Primo anno/Geometria 1/README.md | 2 +- .../Assiomi di ZFC/main.pdf | Bin 0 -> 121354 bytes .../Assiomi di ZFC/main.tex | 116 ++++++++++++++++++ .../{ => Esercizi}/commands.tex | 0 .../{ => Esercizi}/format.tex | 0 .../{ => Esercizi}/main.pdf | Bin .../{ => Esercizi}/main.tex | 0 .../{ => Esercizi}/preamble.tex | 0 .../{ => Esercizi}/quiver.sty | 0 .../README.md | 9 ++ 10 files changed, 126 insertions(+), 1 deletion(-) create mode 100644 Terzo anno/Elementi di Teoria degli Insiemi/Assiomi di ZFC/main.pdf create mode 100644 Terzo anno/Elementi di Teoria degli Insiemi/Assiomi di ZFC/main.tex rename Terzo anno/Elementi di Teoria degli Insiemi/{ => Esercizi}/commands.tex (100%) rename Terzo anno/Elementi di Teoria degli Insiemi/{ => Esercizi}/format.tex (100%) rename Terzo anno/Elementi di Teoria degli Insiemi/{ => Esercizi}/main.pdf (100%) rename Terzo anno/Elementi di Teoria degli Insiemi/{ => Esercizi}/main.tex (100%) rename Terzo anno/Elementi di Teoria degli Insiemi/{ => Esercizi}/preamble.tex (100%) rename Terzo anno/Elementi di Teoria degli Insiemi/{ => Esercizi}/quiver.sty (100%) create mode 100644 Terzo anno/Elementi di Teoria degli Insiemi/README.md diff --git a/Primo anno/Geometria 1/README.md b/Primo anno/Geometria 1/README.md index 4dbc733..7bb923f 100644 --- a/Primo anno/Geometria 1/README.md +++ b/Primo anno/Geometria 1/README.md @@ -4,7 +4,7 @@ - [Registro del corso 📑](https://unimap.unipi.it/registri/dettregistriNEW.php?re=7084691::::&ri=9631) - [Sito web 🔗](http://people.dm.unipi.it/salvetti/GeometriaI_Matematica/indice1.html) -Il corso di Geometria 1 è terminato, ma questa cartella vedrà ancora aggiornamenti per il momento. Gli appunti non sono stati presi con gli stessi strumenti sin dall'inizio: in particolar modo, +Il corso di Geometria 1 è terminato, quindi questa cartella non vedrà più aggiornamenti per il momento. Gli appunti non sono stati presi con gli stessi strumenti sin dall'inizio: in particolar modo, ho iniziato a prendere appunti con [Microsoft Journal](https://apps.microsoft.com/store/detail/microsoft-journal/9N318R854RHH?hl=it-it&gl=it), per poi passare definitivamente a prenderli in LaTeX con [TeXstudio](https://www.texstudio.org/). Alcuni appunti sono ancora in corso di revisione (troverete \[WIP\] nel titolo o un breve avviso all'inizio del documento), ma sono sostanzialmente completi. La cartella contiene in particolar modo una [Scheda riassuntiva](https://g1.hearot.it), che, come suggerisce il nome, vorrebbe essere un recap di diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/Assiomi di ZFC/main.pdf b/Terzo anno/Elementi di Teoria degli Insiemi/Assiomi di ZFC/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..db35b10ec6fabb79dd226d9da9955598099ef246 GIT binary patch literal 121354 zcmb@u1z40{yEjUgAl=;|H83-Pbc=L1NDL+2NF$Aegfs{Of|63wCDNUe0@B?LX8?Ww z?)|>sclJKl@w(<_JuB{Yue;W7p0!4LB?&1u06PecesHvX5{;XRlgiH63Qa%&jYAG% zYwm18#RUL?sqX)waY$KOJ3}0)IHatNoFNhr6FXA~nvf8hld~hl$Og@QYD`DQ?iZM# z@w8+Qon(z`(CnK$9d}Y2T^y2GjJV%f{JIQPq9xA|OP50dn$7Kn0J{#8Dt(s?a%TbE z+r{ao>iot!ufsm%cj-p{C@WiA2P<+h+;34OP=)I1g)Zk;3@;lB+*YmvR*apBdcKq< zp63l6^YwvG^H5~J3*So4?>cK6df9&tDi+FAtslyruPl1vTQc5xrS>=zs#CF9XbKD%r! z=@}SZ=zSz5QIb86x#|w@+LDUKPGZ7V$<~bcV#HfTok&ey(z8~MD~39%E6JaK*Wfh6 z-di`>lG&*=WDEH59qv#+w~4asDxF9Ijf}_E6VLKPTf@ZhcBbPw)F$N3!(8T19W?f# zCI@hONYzKT$;Qew3?6k$-n?^?_z*V8kct^O#NL~dII_5cemC<%KBRm80%lO=5px(5 z3K2>i>UImHPSUMDVPV)2h61JxndCS+rM7n5Qr(`ej4XvlQiX~c2@e-p@**XGJs4%P z6$^31aq|F8?uilT^yR#+HC?!W^kxGhDEo||mG5Qj8|y+p*ZE4=W|T`?D|7$Z7M5@9 z!HKPJd4xo$OkKgQ;S5_=ZXw4vE+_@*Wos+#X3EcSpAl*kmKv&jQV#v)^~-vn8S9wL zjruU=M7q6e5{`B5?Xro`H#$nXF6m$kDaC`9w=ddCVMz5RIx1|bx;=%jW1q@Xe?QPJ z*mLaLDk5}mGJZiUDrg$0S2H4y4Pj^Ww+F;SQYn_VW(B{-9|nTSTg$nw9_zE>x5s9^ ziN!|x9_Nm_5qec zc>#>{8_LWKrVD{@P}+N+)BRwYkD8T59U95ifG82e74xoQKX!v}jh`F9GR`_UOUXHln1bS^t`5J*`*5nsT9SlwpkVL&& z{Yk{~`ty@{doua`$!yT(zJU4}7SneVqd1%9!Ka57i4~b+8(lGo`p04C=i?Edo{*OC z!jo^OFBx}Mz-_FeAE&L~+3A6^BKUv9tT2kUN3y4RTOzW#_P zJTvXntVt7+pKZHNJf^@r>1{9D&;&bbvy@a zG*L0-L~quQ<$a(96VUY?Gkl`cE)mGA^z2KYVilHi5VeK(Bfs5+m9m$6uEmLiRhV{f zHoM%i7%`Y2JoHLvv2x1C4U*s376xTT%oEVv^=!h^n z6FtX+^!*bYs)j zS(Bs2I)LPjT;C~1IJ@Zehja&c*6hs7P9%x6v%S)geu>VO!^9Xm0>6anf^|FEV-TK; 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+\setlength{\parskip}{1ex} + +\usepackage{lmodern} +\renewcommand{\familydefault}{\sfdefault} + +\NewDocumentEnvironment{axiom}{m +b}{ + \par\medskip + \noindent\textbf{#1}\par + \begin{addmargin}[2em]{4em} + #2 + \end{addmargin} + \par\medskip +}{} + +\begin{document} + +\begin{LARGE} + \textsf{\textbf{Assiomi della teoria di Zermelo-Fraenkel con scelta (ZFC)}} +\end{LARGE} + +\vspace{1ex} + +\linespread{1.3} + +L'alfabeto del linguaggio consiste di una successione infinita di variabili +che rappresentano gli insiemi, dei connettivi logici $\lnot$ (``non''), $\lor$ (``o''), $\land$ (``e''), dei quantificatori $\forall$ (``per ogni''), $\exists$ (``esiste''), del simbolo di uguaglianza $=$, +del simbolo di appartenenza a un insieme $\in$ e delle parentesi tonde $($ e $)$. + + +Con questo alfabeto, si dicono formule ben formate le formule atomiche $x = y$ e $x \in y$, dove $x$ e $y$ sono metavariabili, e per ricorsione le formule $\exists x \, \varphi$, $\forall x \, \varphi$, $\lnot \varphi$, $(\varphi \land \phi)$, $(\varphi \lor \psi)$, dove $\varphi$ e $\psi$ sono a loro volta +formule ben formate. + + +Impieghiamo $(a \rightarrow b)$ (``$a$ implica $b$'') come abbreviazione per $(\lnot a \lor b)$, così come +$(a \leftrightarrow b)$ (``$a$ se e solo se $b$'') per $(a \rightarrow b \land b \rightarrow a)$. Per chiarezza ammettiamo come abbreviazione anche $(\varphi)$ per $\varphi$. + +\begin{axiom}{Assioma dell'estensionalità (ZF1)} + Se due insiemi $x$ e $y$ hanno gli stessi elementi, allora sono lo stesso insieme. + \[ \forall x \forall y(\forall z(z \in x \leftrightarrow z \in y) \rightarrow x = y). \] +\end{axiom} + +\begin{axiom}{Assioma dell'insieme vuoto (ZF2)} + Esiste un insieme privo di elementi. + \[ \exists x \lnot \exists y (y \in x). \] +\end{axiom} + +\begin{axiom}{Assioma della coppia (ZF3)} + Dati due insiemi $x$, $y$, esiste un insieme contenente esattamente $x$ e $y$, ovverosia $\{x, y\}$. + \[ \forall x \forall y \exists z\forall k (k \in z \leftrightarrow (k = x \lor k=y)). \] +\end{axiom} + +\begin{axiom}{Assioma delle parti (ZF4)} + Dato un insieme $x$, esiste l'insieme dei sottinsiemi di $x$, ovverosia l'insieme + delle parti $\mathcal{P}(x)$. + \[ \forall x \exists y \forall z(z \in y \leftrightarrow \forall k(k \in z \rightarrow k \in x)). \] +\end{axiom} + +\begin{axiom}{Assioma dell'unione (ZF5)} + Dato un insieme $x$, esiste l'insieme che contiene esattamente gli elementi degli elementi di $x$, ovverosia + l'insieme $\bigcup x$. + \[ \forall x \exists y \forall z(z \in y \leftrightarrow \exists k(k \in x \land z \in k)). \] +\end{axiom} + +\newpage + +\begin{axiom}{Assioma dell'infinito (ZF6)} + Esiste un insieme a cui appartiene l'insieme vuoto, e a cui appartiene $a \cup \{a\}$ se $a$ gli + appartiene. + \[ \exists x(\exists y(y \in x \land \lnot\exists z( z \in y)) \land \forall a (a \in x \rightarrow \exists b( b \in x \land \forall c( c \in b \leftrightarrow (c \in a \lor c = a))))). \] +\end{axiom} + +\begin{axiom}{Schema di assiomi di separazione (ZF7)} + Data una formula $\Psi(z, u_1, \ldots, u_n)$ dipendente dalla variabile $z$ libera + e da $u_1$, ..., $u_n$ eventualmente libere, esiste per ogni insieme $x$ il sottinsieme + $\{ z \in x \mid \Psi(z, u_1, \ldots, u_n) \}$. + \[ \forall u_1 \ldots \forall u_n \left[ \forall x \exists y\forall z(z \in y \leftrightarrow (z \in x \land \Psi(z, u_1, \ldots, u_n))) \right]. \] +\end{axiom} + +\begin{axiom}{Schema di assiomi di rimpiazzamento (ZF8)} + Data una formula funzionale $\Psi(x, y, u_1, \ldots, u_n)$ dipendente dalle variabili $x$ e $y$ + libere e da $u_1$, ..., $u_n$ eventualmente libere, esiste per ogni insieme $x$ l'insieme + $\{ y \mid \exists z\,(z \in x \land \Psi(z, y, u_1, \ldots, u_n)) \}$. + \begin{multline*} + \forall u_1 \ldots \forall u_n \bigl[ \forall x \forall y \forall z((\Psi(x, y, u_1, \ldots, u_n) \land \Psi(x, z, u_1, \ldots, u_n)) \rightarrow y = z) \rightarrow \\ + \forall a \exists b \forall c(c \in b \leftrightarrow \exists d (d \in a \land \Psi(d, c, u_1, \ldots, u_n))) \bigr]. + \end{multline*} +\end{axiom} + +\begin{axiom}{Assioma di buona fondazione (ZF9)} + Ogni insieme non vuoto $x$ contiene un elemento $y$ disgiunto da $x$. + \[ \forall x (\exists y (y \in x) \rightarrow \exists z (z \in x \land \forall a \lnot(a \in x \land a \in z))). \] +\end{axiom} + +\begin{axiom}{Assioma di scelta (AC)} + Data una famiglia $x$ di insiemi non vuoti a due a due disgiunti esiste un + insieme $e$ tale per cui l'intersezione $e \cap f$ contiene esattamente un elemento + per ogni $f \in x$. + \begin{multline*} + \forall x((\forall y(y \in x \rightarrow \exists z(z \in y)) \land \forall a \forall b((a \in x \land b \in x \land \lnot(a = b)) \rightarrow \\ + \lnot \exists d(d \in a \land d \in b))) \rightarrow \exists e \forall f (f \in x \rightarrow \\ + \exists g (g \in e \land g \in f \land \forall h ((h \in e \land h \in f) \rightarrow h = g)))). + \end{multline*} +\end{axiom} + +\end{document} diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/commands.tex b/Terzo anno/Elementi di Teoria degli Insiemi/Esercizi/commands.tex similarity index 100% rename from Terzo anno/Elementi di Teoria degli Insiemi/commands.tex rename to Terzo anno/Elementi di Teoria degli Insiemi/Esercizi/commands.tex diff --git a/Terzo 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similarity index 100% rename from Terzo anno/Elementi di Teoria degli Insiemi/preamble.tex rename to Terzo anno/Elementi di Teoria degli Insiemi/Esercizi/preamble.tex diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/quiver.sty b/Terzo anno/Elementi di Teoria degli Insiemi/Esercizi/quiver.sty similarity index 100% rename from Terzo anno/Elementi di Teoria degli Insiemi/quiver.sty rename to Terzo anno/Elementi di Teoria degli Insiemi/Esercizi/quiver.sty diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/README.md b/Terzo anno/Elementi di Teoria degli Insiemi/README.md new file mode 100644 index 0000000..25e860a --- /dev/null +++ b/Terzo anno/Elementi di Teoria degli Insiemi/README.md @@ -0,0 +1,9 @@ +# [Elementi di Teoria degli Insiemi](https://unipi.coursecatalogue.cineca.it/insegnamenti/2024/50253_686261_66435/2008/50253/10299?coorte=2022&schemaid=9099) + +- [Programma del corso 📘](https://unipi.coursecatalogue.cineca.it/insegnamenti/2024/50253_686261_66435/2008/50253/10299?coorte=2022&schemaid=9099) +- [Registro del corso 📑](https://unimap.unipi.it/registri/dettregistriNEW.php?re=11061889::::&ri=9783) +- [Home page del corso 📔](https://people.dm.unipi.it/dinasso/eti-2024b.html) + +La cartella contiene sotto la cartella *Esercizi* la raccolta delle soluzioni degli esercizi assegnati dal prof. Di Nasso durante +il corso nell'A.A 2024-2025. La cartella *Assiomi di ZFC* raccoglie ancora in modo compatto gli enunciati e le formule degli assiomi +della teoria assiomatica di Zermelo-Fraenkel con l'assioma di scelta. \ No newline at end of file