From 967fe0b68d4d7483fa43004e11d21059562f3551 Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Thu, 15 Sep 2022 18:37:55 +0200 Subject: [PATCH] aritmetica: insiemi e prime definizioni --- Aritmetica/1. Teoria degli insiemi.tex | 73 ++++++++++++++++++ Aritmetica/aritmetica.pdf | Bin 0 -> 169313 bytes Aritmetica/aritmetica.tex | 13 +++- .../1. Primo capitolo.tex | 0 Geometria I/geometria_i.tex | 48 ++++++++++++ 5 files changed, 133 insertions(+), 1 deletion(-) create mode 100644 Aritmetica/1. Teoria degli insiemi.tex create mode 100644 Aritmetica/aritmetica.pdf rename {Aritmetica => Geometria I}/1. Primo capitolo.tex (100%) create mode 100644 Geometria I/geometria_i.tex diff --git a/Aritmetica/1. Teoria degli insiemi.tex b/Aritmetica/1. Teoria degli insiemi.tex new file mode 100644 index 0000000..8137bd4 --- /dev/null +++ b/Aritmetica/1. Teoria degli insiemi.tex @@ -0,0 +1,73 @@ +\chapter{Teoria degli insiemi} + +Il concetto di insieme è primitivo e pertanto non definito formalmente +in questa sede. Viene tuttavia definita la terminologia che riguarda +le teoria dei suddetti insiemi. + +Quando si leggerà $a \in S$, s'intenderà che ``$a$ appartiene all'insieme $S$'', mentre +$a \notin S$ si legge ``$a$ non appartiene all'insieme $S$''. Un insieme $A$ si dice +sottoinsieme di $B$ ($A \subseteq B$) quando $a \in A \rightarrow a \in B$; in particolare +si dice sottoinsieme proprio di $B$ ($A \subset B$) quando +$A \subseteq B \land \exists b \in B \mid b \notin A$. + +Due insiemi $A$ e $B$ sono uguali se e solo se $A \subseteq B \land B \subseteq A$. +L'insieme vuoto è l'insieme che non ha elementi, ed è sottoinsieme di ogni insieme. + +\section{L'operazione di unione} + +L'unione di due insiemi $A$ e $B$ è un'operazione che restituisce un insieme +$A \cup B = \{x \mid x \in A \lor x \in B\}$. + +Tale operazione si può estendere a più insiemi mediante l'introduzione di +un \textit{insieme di indici} $T$ per una famiglia di insiemi. Un insieme di +indici $T$ rispetto a un famiglia $F=\{A_t\}$ ha la seguente proprietà: $\forall t \in + T, \exists A_t \in F$; ossia è in grado di enumerare gli insiemi della famiglia $F$. + +L'unione è pertanto definita su una famiglia $F$ come $\bigcup_{t \in T} A_t = + \{x \mid (\exists t \in T \mid x \in A_t)\}$. + +L'unione gode delle seguente proprietà: $A \subseteq B \rightarrow A \cup B = B$ +(in particolare, $A \cup \emptyset = A$). + +\section{L'operazione di intersezione} + +Analogamente a come è stata definita l'unione, l'intersezione è un'operazione che +resistuisce un insieme $A \cap B = \{x \mid x \in A \land x \in B\}$; ossia estesa a più +insiemi: $\bigcap_{t \in T} A_t = \{x \mid (\forall t \in T \mid x \in A_t)\}$. + +In modo opposto all'unione, l'intersezione è tale per cui $A \subseteq B \rightarrow + A \cap B = A$ (in particolare, $A \cap \emptyset = \emptyset$). + +\section{L'operazione di sottrazione} + +L'operazione di sottrazione su due insiemi $A$ e $B$ è definita come +$A \setminus B = \{x \mid x \in A \land x \notin B\}$. Si può facilmente +verificare che $A = (A \cap B) \cup (A \setminus B)$. + +\begin{proof} + Ogni elemento di $A$ può appartenere o non appartenere a $B$: nel primo caso, + appartiene anche a $A \cap B$, e quindi a $(A \cap B) \cup (A \setminus B)$; + altrimenti appartiene per definizione a $A \setminus B$, e quindi sempre + a $(A \cap B) \cup (A \setminus B)$. Pertanto $A \subseteq (A \cap B) \cup (A \setminus B)$. + + Ogni elemento di $(A \cap B) \cup (A \setminus B)$ appartiene ad almeno uno + dei due operandi dell'unione; in entrambi i casi deve appartenere ad $A$. Quindi + $(A \cap B) \cup (A \setminus B) \subseteq A$. +\end{proof} + +In particolare, se $B \subseteq A$, $A \setminus B$ si dice \textbf{complemento di $B$ in $A$}. + +\section{Il prodotto cartesiano} + +Il prodotto cartesiano di una famiglia ordinata di insiemi $F$ con un certo insieme +di indici $T$ è l'insieme +$\bigtimes_{t \in T} A_t = \{(a_{t_0}, a_{t_1}, \ldots) \mid a_{t_0} \in A_{t_0} \land + a_{t_1} \in A_{t_1} \land \ldots\}$. In particolare, il prodotto cartesiano di +due due insiemi $A$ e $B$ si indica con $A \times B = \{(a, b) \mid a \in A \land b \in B\}$. + +Una $n$-tupla ordinata, ossia la forma in cui è raccolto un certo elemento di un prodotto cartesiano, +è uguale ad una altra tupla se e solo se ogni elemento di una tupla è uguale a quello +corrispondente in ordine dell'altra: pertanto, in generale, $(a, b) \neq (b, a)$. + +Inoltre, il prodotto cartesiano $A \times A$ viene indicato con $A^2$ (analogamente, +$A^n = \bigtimes_{i=1}^{n} A$). \ No newline at end of file diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf new file mode 100644 index 0000000000000000000000000000000000000000..035dfa8a0e5116ea4fd5894a61d13c5ce7adb553 GIT binary patch literal 169313 zcma&MQ;;yg(x%zAZQHi(?yqgzwr$(CZQHhOn|pR=|En_*GkKMDSCyHS@x+@+svsgp z%SguxMLNGcvJS<>LBK#@XJ`q0KRhCygPm>^usD+CqOW%p5KFIqF@xsT`qwd@)R&U8mdOXL9%6f;MtyZg~x9ch4q1M z21I!APJ?z9eznUhYj^rX0U9iONs&@pCku!$iIlcIyt4EL%}fsvzLQGA>9A};wxPX! 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Primo capitolo.tex} +\include{1. Teoria degli insiemi.tex} \end{document} \ No newline at end of file diff --git a/Aritmetica/1. Primo capitolo.tex b/Geometria I/1. Primo capitolo.tex similarity index 100% rename from Aritmetica/1. Primo capitolo.tex rename to Geometria I/1. Primo capitolo.tex diff --git a/Geometria I/geometria_i.tex b/Geometria I/geometria_i.tex new file mode 100644 index 0000000..e254900 --- /dev/null +++ b/Geometria I/geometria_i.tex @@ -0,0 +1,48 @@ +\documentclass[oneside]{book} + +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{enumitem} +\usepackage[a4paper, total={6in, 8in}]{geometry} +\usepackage{hyperref} +\usepackage{mathtools} +\usepackage[italian]{babel} +\usepackage[utf8]{inputenc} +\usepackage[parfill]{parskip} +\usepackage{wrapfig} + +\usepackage{pgfplots} +\pgfplotsset{compat=1.15} +\usepackage{mathrsfs} +\usetikzlibrary{arrows,angles,quotes} + +\renewcommand\qedsymbol{$\blacksquare$} + +\newcommand{\gfrac}[2]{\displaystyle \frac{#1}{#2}} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\norm}[1]{\lVert \vec{#1} \rVert} +\newcommand{\nnorm}[1]{\lVert #1 \rVert} + +\newtheorem{axiom}{Assioma}[section] +\newtheorem{theorem}{Teorema}[section] +\newtheorem{corollary}{Corollario}[theorem] +\newtheorem{lemma}[theorem]{Lemma} + +\theoremstyle{definition} +\newtheorem{definition}{Definizione}[section] + +\begin{document} + +\author{Gabriel Antonio Videtta} +\title{Appunti di Geometria I} + +\maketitle +\newpage + +\tableofcontents +\newpage + +\include{1. Primo capitolo.tex} + +\end{document} \ No newline at end of file