From 39a33deff90d5fce37167eb5ddbb53d43fa8d540 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 26 Nov 2022 22:05:07 +0100 Subject: [PATCH] feat(aritmetica): aggiunge primi esempi di gruppi --- Aritmetica/1. Gruppi.tex | 56 ++++++++++++++++++++++++++++++++++---- Aritmetica/aritmetica.pdf | Bin 228263 -> 276630 bytes Aritmetica/aritmetica.tex | 1 + Aritmetica/evan.sty | 4 +-- 4 files changed, 54 insertions(+), 7 deletions(-) diff --git a/Aritmetica/1. Gruppi.tex b/Aritmetica/1. Gruppi.tex index 1877585..b690e70 100644 --- a/Aritmetica/1. Gruppi.tex +++ b/Aritmetica/1. Gruppi.tex @@ -22,12 +22,58 @@ teoremi e metodi che si fondano su una stessa logica. Come vedremo, esse condividono la natura di gruppo. \begin{definition} - Dato un insieme non vuoto $G$, esso si dice \textbf{gruppo} se data - un'operazione ben definita $\cdot : G \times G \to G$ è t.c: + Dato un insieme non vuoto $G$, $(G, \cdot)$ si dice \textbf{gruppo} se data + un'operazione ben definita $\cdot : G \times G \to G$ essa è t.c: \begin{itemize} \item (\vocab{associatività}) $\forall a, b, c \in G, \, (a \cdot b) \cdot c = a \cdot (b \cdot c)$ - \item (\vocab{esistenza dell'elem. neutro}) $\exists e \in G \mid a \cdot e = a = e \, \cdot a \forall a \in G$ - \item (\vocab{esistenza dell'elem. inverso}) $\forall a \in G, \, \exists a^{-1} \in G \mid a \cdot a^{-1} = a^{-1} \cdot a = e$ + \item (\vocab{esistenza dell'elem. neutro}) $\exists e \in G \mid a \cdot e = a = e \, \cdot a \,\, \forall a \in G$ + \item (\vocab{esistenza dell'elem. inverso}) $\forall a \in G, \, \exists a^{-1} \in G \mid a \cdot a^{-1} = e$ \end{itemize} -\end{definition} \ No newline at end of file +\end{definition} + +\begin{remark} + Nella definizione di gruppo si è chiaramente specificato che l'operazione dev'essere + ben definita e, soprattutto, che l'insieme $G$ dev'essere chiuso rispetto ad esso. + + Pertanto, non è sufficiente aver verificato le tre proprietà sopraelencate senza + aver prima verificato che l'operazione sia effettivamente un'operazione di + gruppo. +\end{remark} + +\begin{example}[Gruppo ciclico elementare] + L'insieme $\ZZ/n\ZZ$ (che talvolta indicheremo semplicemente come $\ZZ_n$) degli interi modulo $n$ è un gruppo con l'operazione di + somma $+$. Infatti: + + \begin{itemize} + \item $\forall \left[a\right]_n, \left[b\right]_n \in \ZZ/n\ZZ, \, \left[a\right]_n + \left[b\right]_n = \left[a+b\right]_n \in \ZZ/n\ZZ$ (\textit{chiusura rispetto all'operazione}) + \item $\forall \left[a\right]_n, \left[b\right]_n, \left[c\right]_n \in \ZZ/n\ZZ, \, \left(\left[a\right]_n + \left[b\right]_n\right) + \left[c\right]_n = \left[a+b\right]_n + + \left[c\right]_n = \left[a+b+c\right]_n = \left[a\right]_n + \left[b+c\right]_n = \left[a\right]_n + \left(\left[b\right]_n + \left[c\right]_n\right)$ (\textit{associatività}) + \item $\forall \left[a\right]_n \in \ZZ/n\ZZ, \, \left[a\right]_n + 0 = \left[a\right]_n$ (\textit{esistenza dell'elem. neutro}) + \item $\forall \left[a\right]_n \in \ZZ/n\ZZ, \, \exists \left[-a\right]_n \in \ZZ/n\ZZ \mid \left[a\right]_n + \left[-a\right]_n = 0$ (\textit{esistenza dell'elem. inverso}) + \end{itemize} +\end{example} + +\begin{example}[Gruppo simmetrico] + L'insieme $S_n$ delle funzioni bigettive da $X_n = \{1, 2, \ldots, n\}$ in sé stesso è un + gruppo rispetto all'operazione di composizione. Infatti: + + \begin{itemize} + \item $\forall f, g \in S_n, \, f \circ g \in S_n$ (\textit{chiusura rispetto all'operazione}) + \item $\forall f, g, h \in S_n, \, (f \circ g) \circ h = f \circ (g \circ h)$ (\textit{associatività}) + \item $\exists e = \Id \in S_n \mid f \circ e = f = e \circ f \forall f \in S_n$ (\textit{esistenza dell'elem. neutro}) + \item $\forall f \in S_n, \, \exists f^{-1} \in S_n \mid f \circ f^{-1} = e$ (\textit{esistenza dell'elem. inverso}) + \end{itemize} +\end{example} + +Le proprietà date dalla definizione di un gruppo ci permettono immediatamente di desumere +altre proprietà fondamentali, e che sulle quali faremo affidamento d'ora in poi. + +\begin{theorem} + L'inverso $a^{-1}$ di un elemento $a$ di un gruppo $G$ è unico. +\end{theorem} + +\begin{proof} + Supponiamo che $b$ e $c$ siano due elementi inversi distinti di $a$. Allora + $b=b\cdot e=b\cdot \underbrace{(a \cdot c)}_{=e}=\underbrace{(b \cdot a)}_{=e} \cdot c=c$, \Lightning. Pertanto l'inverso è unico. +\end{proof} diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index df2edb27fc4fb36ab48cee8aafea10e2c5a6a599..bdbfe6bb7e502b14194b3428833a9cd0cf079b88 100644 GIT binary patch delta 118175 zcmbTdWmH_twgyV$E(sF6ad&AfK+vGUCAbrU1a}th?!h5A!Civ826uOd1cJjOJNulo z?>*y=H{ScvHL7QQHRY>WwPvl+U2uh#c#i2^19D}D<$!RoCar@(w7@Z;PSjG}`GZ%$ ztugVDNjHX~F{)65m7V$f8e78};Uj@PCr7-z_xt3>y53-+R*w7cN_-z`Ki*j)&_-*# zNFo}&LN#-dav-D{pYF-j)AKmzlB;@)#grGUyDBFAGXFz7oy1wtZ0eqzM}l6xiK0$f zgZySv5@#6;Z34mZFDb$@x0jbm>UPe{Y8ot&g?ra)trteKt8InaEQgls<`O>o&G)F6 zvv@UTK{1AyzcWk3>X|VA_RZy|(F?1!dYc^oPo+6}Bg0md-q1i1Z;Sf)6<#^>i@uge zDRxR0GN}B3?6sf#V}{6D$ejVkWKrVM{_%-*^|Ho^Z~`d1rj#8!X*AZVJC7f~_ zB~E~|I=UAx8drCipLejla`o(>b)6wJ^#olDx;>SF_LWPF)bQi`7P-R0q1C~BW?@S9 zuo2PTMN{>aT_-?5`?_aVR-BWJ5A@j1e3mZ(E%QPIT;3(p8#7(h-Dvzcb2|mi z^kPs(JuR8H^yZF=MWd4cASk|GpDszH5{Ay(zSWMRYwk^#iVJsrz5aP#`Z~fXi!%OhDI9o%&-qoy zyF;1$jZC8Gx+8C5R&V())<;Jz&~`IP3>Lh4Jl}iD%d)?98j=CeM!TBMYli6G;KX*!rdU#%T+^wLq=9%U7ji#OmNF3LDM 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+\usepackage{marvosym} \setlength{\headsep}{0.3in} diff --git a/Aritmetica/evan.sty b/Aritmetica/evan.sty index 2857dfd..681a979 100644 --- a/Aritmetica/evan.sty +++ b/Aritmetica/evan.sty @@ -524,8 +524,8 @@ % Remark-style theorems %\theoremstyle{remark} \ifevanmdthm - \declaretheorem[style=thmblackbox,name=Osservazione,sibling=theorem]{remark} - \declaretheorem[style=thmblackbox,name=Osservazione,numbered=no]{remark*} + \declaretheorem[style=thmgreenbox,name=Osservazione,sibling=theorem]{remark} + \declaretheorem[style=thmgreenbox,name=Osservazione,numbered=no]{remark*} \else \declaretheorem[name=Osservazione,sibling=theorem]{remark} \declaretheorem[name=Osservazione,numbered=no]{remark*}