From e13f5bf45b4579759bf05df7938eb6415ce3a6c0 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 27 Nov 2022 13:36:38 +0100 Subject: [PATCH] feat(aritmetica): aggiunge primi teoremi importanti --- Aritmetica/1. Gruppi.tex | 44 +++++++++++++++++++++++++++++++++++++- Aritmetica/aritmetica.pdf | Bin 276630 -> 284976 bytes Aritmetica/aritmetica.tex | 2 ++ 3 files changed, 45 insertions(+), 1 deletion(-) diff --git a/Aritmetica/1. Gruppi.tex b/Aritmetica/1. Gruppi.tex index b690e70..6d60bb7 100644 --- a/Aritmetica/1. Gruppi.tex +++ b/Aritmetica/1. Gruppi.tex @@ -56,7 +56,7 @@ esse condividono la natura di gruppo. \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: + gruppo rispetto all'operazione di composizione, detto \vocab{gruppo simmetrico}. Infatti: \begin{itemize} \item $\forall f, g \in S_n, \, f \circ g \in S_n$ (\textit{chiusura rispetto all'operazione}) @@ -70,6 +70,7 @@ Le proprietà date dalla definizione di un gruppo ci permettono immediatamente d altre proprietà fondamentali, e che sulle quali faremo affidamento d'ora in poi. \begin{theorem} + \label{th:gruppo:inverso_unico} L'inverso $a^{-1}$ di un elemento $a$ di un gruppo $G$ è unico. \end{theorem} @@ -77,3 +78,44 @@ altre proprietà fondamentali, e che sulle quali faremo affidamento d'ora in poi 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} + +\begin{theorem} + L'inverso dell'inverso $\left(a^{-1}\right)^{-1}$ è pari a $a$. +\end{theorem} + +\begin{proof} + Dal momento che l'inverso è unico (per il \vocab{Teorema~\ref{th:gruppo:inverso_unico}}), + $\left(a^{-1}\right)^{-1} a^{-1} = e \implies \left(a^{-1}\right)^{-1} = a$. +\end{proof} + +\begin{theorem} + L'inverso di $ab$ è $b^{-1}a^{-1}$. +\end{theorem} + +\begin{proof} + Si verifica facilmente che $ab b^{-1}a^{-1}= a e a^{-1} = a a^{-1} = e$. Poiché + l'inverso è unico (per il \vocab{Teorema~\ref{th:gruppo:inverso_unico}}), allora + $\left(ab\right)^{-1} = b^{-1}a^{-1}$. +\end{proof} + +\begin{remark} + In realtà, sebbene a prima vista potrebbe sembrare inusuale l'inversione dei due + fattori nell'ultima identità, essa è una conseguenza del modo in cui operiamo + naturalmente. Si prenda per esempio la composizione $f \circ g$, per ottenere + l'identità è necessario prima decomporre $f$, l'ultima funzione aggiunta, ed infine + $g$, ossia seguendo l'ordine da sinistra a destra. + + Nel corso di Geometria vi sarà spiegato come anche la matrici si comportano in + questo modo (non è un caso, dal momento che anch'esse, sotto talune condizioni, + formano un gruppo, il cosiddetto \vocab{gruppo lineare} $\GL_n(\KK)$). +\end{remark} + +\begin{theorem} + Un'equazione della forma $ax=bx$ è vera se e solo se $a=b$. +\end{theorem} + +\begin{proof} + Infatti, moltiplicando per l'inverso di $x$, $ax=bx \iff axx^{-1}=bxx^{-1} \iff a=b$. +\end{proof} + + diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index bdbfe6bb7e502b14194b3428833a9cd0cf079b88..35d29dbc089705f878fc24acfda2b5016ecd28a4 100644 GIT binary patch delta 74803 zcmbTd1z23m(k{#l?(P8w3+^xs?u0;a5}e=?+?~M*5;QQl2SRX%5Zr>hI|N9AyTc{P z-uLWt&i(E$JkQixU8}mP>+N1$)vKGNBbl!tg)}Te8GwoY!pzRX#S+BJ%?p98 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\let\exists\undefined \DeclareMathOperator{\exists}{\oldexists}