From 252213c8a8136701d1704c44dfe0549aebc00eb9 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 24 Sep 2022 19:03:41 +0200 Subject: [PATCH] aritmetica: insieme delle corrispondenze biunivoche --- ...elazioni di equivalenza e applicazioni.tex | 38 ++++++++++++++++++ Aritmetica/aritmetica.pdf | Bin 230049 -> 244640 bytes 2 files changed, 38 insertions(+) diff --git a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex index 4becb35..5307c65 100644 --- a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex +++ b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex @@ -149,6 +149,7 @@ essendo anch'essa un'applicazione. ossia che $(\alpha \circ \beta) \circ \gamma = \alpha \circ (\beta \circ \gamma)$. \begin{lemma}[Proprietà associativa della composizione] + \label{lemma:associativita_composizione} Date tre applicazioni $\alpha$, $\beta$, $\gamma$, $(\alpha \circ \beta) \circ \gamma = \alpha \circ (\beta \circ \gamma)$. \end{lemma} @@ -202,6 +203,7 @@ ereditate dalle applicazioni di cui è composta se tutte queste le rispettano, o \end{proof} \begin{lemma}[Bigettività della composizione] + \label{lemma:bigettivita_composizione} $(\tau \circ \sigma)$ è bigettiva se $\tau$ e $\sigma$ lo sono. \end{lemma} @@ -231,6 +233,7 @@ ossia l'identità di $\sigma$, per la quale ogni elemento viene associato a sé Banalmente, per ogni applicazione $\alpha$, $(\alpha \circ \Id) = (\Id \circ \alpha) = \alpha$. \begin{lemma} + \label{lemma:inversa_applicazione} $\sigma : S \to T$ è una corrispondenza biunivoca se e solo se esiste un'applicazione $\mu : T \to S$ tale per cui $(\sigma \circ \mu) = (\mu \circ \sigma) = \Id$. @@ -260,3 +263,38 @@ Banalmente, per ogni applicazione $\alpha$, $(\alpha \circ \Id) = (\Id \circ \al di $\sigma$. Allora $\alpha = \alpha \circ (\sigma \circ \beta) = (\alpha \circ \sigma) \circ \beta = \beta$, che è una contraddizione. \end{proof} + +\section{Il gruppo \texorpdfstring{$A(S)$}{A(S)} delle corrispondenze biunivoche} + +Si definisce $A(S)$ come l'insieme $\{\sigma : S \to S \mid \sigma \text{ sia biunivoca}\} = +\{\sigma : S \to S \mid \forall s \in S \existsone t \in S \mid t = \sigma(s)\}$. + +Prendendo in considerazione l'operazione di composizione $\circ$, si può dimostrare +che $(A(S), \circ)$ è un gruppo: + +\begin{itemize} + \item $\forall \alpha, \beta \in A(S), \alpha \circ \beta \in A(S)$ (vd. Lemma \ref{lemma:bigettivita_composizione}). + \item $\forall \alpha, \beta, \gamma \in A(S), (\alpha \circ \beta) \circ \gamma = \alpha \circ (\beta \circ \gamma)$ (vd. Lemma \ref{lemma:associativita_composizione}). + \item $\exists \Id \in A(S) \mid \forall \alpha \in A(S), (\Id \circ \alpha) = (\alpha \circ \Id) = \alpha$. + \item $\forall \alpha \in A(S), \exists \alpha^{-1} \in A(S) \mid (\alpha \circ \alpha^{-1}) = (\alpha^{-1} \circ \alpha) = \Id$ (vd. Lemma \ref{lemma:inversa_applicazione}). +\end{itemize} + +\begin{lemma} + Se $S$ consta di più di due elementi ($\nnorm{S} > 2$), allora esistono sicuramente + due applicazioni $\alpha, \beta \in A(S)$ tale per cui $(\alpha \circ \beta) \neq + (\beta \circ \alpha)$. +\end{lemma} + +\begin{proof} + Se $S$ consta di due elementi, $S$ possiede almeno tre elementi $s_1, s_2, s_3$, + possiamo definire due applicazioni $\sigma$ e $\tau$ come segue: + + \begin{itemize} + \item $\sigma(s_1) = s_2$, $\sigma(s_2) = s_3$, $\sigma(s_3) = s_1$. + \item $\tau(s_1) = s_1$, $\tau(s_2) = s_2$, $\tau(s_3) = s_3$. + \item $\sigma(a) = \tau(a) = a \forall a \notin \{s_1, s_2, s_3\}$. + \end{itemize} + + Allora $(\sigma \circ \tau)(s_1) = \sigma(s_1) = s_2$ e + $(\tau \circ \sigma)(s_1) = \tau(s_2) = s_3$, ma $s_2 \neq s_3$. +\end{proof} diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index 606edb61843f8105fc3f0a48b847758e636c7b8d..47a9e14cb05f3872772fb7212493fd3d89cde56a 100644 GIT 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