From 1878073d2baf60bb2e45d3f6205a80151d3ed1af Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 24 Sep 2022 18:08:10 +0200 Subject: [PATCH] aritmetica: applicazione inversa --- ...elazioni di equivalenza e applicazioni.tex | 67 ++++++++++++++++-- Aritmetica/aritmetica.pdf | Bin 205361 -> 230049 bytes Aritmetica/aritmetica.tex | 10 +++ 3 files changed, 72 insertions(+), 5 deletions(-) diff --git a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex index de57eae..4becb35 100644 --- a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex +++ b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex @@ -92,13 +92,19 @@ funzione. \begin{definition}[Applicazione] Dati due insiemi $S$ e $T$, si dice che $\sigma$ è un'applicazione - da $S$ a $T$, se $\sigma \subseteq S \times T \land \forall s \in S, \existsone + da $S$ a $T$, se $\sigma \subseteq (S \times T) \land \forall s \in S, \existsone t \in T \mid (s, t) \in \sigma$. Tale applicazione allora si scrive come $\sigma : S \to T$. \end{definition} Si scrive $\sigma : s \rightarrowtail \sigma(s)$ per sottintendere che -$\forall \, (s, t) \in \sigma, (s, t) = (s, \sigma(t))$. +$\forall \, (s, t) \in \sigma, (s, t) = (s, \sigma(t))$. Dato +$t=\sigma(s)$, si dice che $t$ è l'\textit{immagine} di $s$ appartenente +al \textit{codominio} $T$, enunciato come $\Codom(\sigma)$, mentre $s$ è +la \textit{preimmagine} di $t$, appartenente al \textit{dominio} $S$, detto +$\Dom(\sigma)$. L'insieme ${(s, t) \in \Dom(\sigma) \times \Codom(\sigma) \mid (s, t) \in \sigma}$ è +detto \textit{grafico} di $\sigma$, ossia $\Gr(\sigma)$. + \subsection{Proprietà delle applicazioni} @@ -187,9 +193,12 @@ ereditate dalle applicazioni di cui è composta se tutte queste le rispettano, o \end{lemma} \begin{proof} - Dal momento che $\sigma$ è surgettiva $\forall s \in \Dom(\sigma), - \exists t \in \Codom(\sigma) \mid t = \sigma(s)$. Tuttavia, essendo $t \in \Dom(\tau)$, - $\exists u \in \Codom(\tau) \mid u = \tau(t) = \tau(\sigma(s))$. + Dal momento che $\tau$ è surgettiva, allora $\forall u \in + \Codom(\tau), \exists t \in \Dom(\tau) \mid u = \tau(t)$. + Poiché $t \in \Codom(\sigma)$, allora, poiché anche + $\sigma$ è surgettiva, $\exists s \in \Dom(\sigma) \mid + t = \sigma(s)$. Pertanto $\exists s \in \Dom(\sigma) \mid + u = \tau(\sigma(s))$. \end{proof} \begin{lemma}[Bigettività della composizione] @@ -203,3 +212,51 @@ ereditate dalle applicazioni di cui è composta se tutte queste le rispettano, o \ref{lemma:surgettivita_composizione}. \end{proof} +\section{Applicazione inversa} + +Qualora un'applicazione $\sigma : S \to T$ sia bigettiva, si dice che essa +crea una \textit{corrispondenza biunivoca} tra $S$ e $T$, ossia che dato un +elemento qualsiasi appartenente a $S$ è possibile associarlo ad un unico elemento +di $T$, e viceversa. Questo è possibile dal momento che $\sigma$ è sia iniettiva +($\forall t \in T, \existsone \lor \nexists s \in S \mid t = \sigma(s)$) che +surgettiva ($\forall t \in T, \exists s \in S \mid t = \sigma(s)$), prescrivendo +che $\forall t \in T, \existsone s \in S \mid t = \sigma(s)$. + +Da questa conclusione è possibile definire l'\textit{applicazione inversa} di +$\sigma$, detta $\sigma^{-1}$, che è l'applicazione che associa ad ogni $t \in T$ +un unico $s \in S$. Quindi, $t = \sigma(s) \iff s = \sigma^{-1} (t)$. + +In particolare, $(\sigma \circ \sigma^{-1}) = (\sigma^{-1} \circ \sigma) = \Id$, +ossia l'identità di $\sigma$, per la quale ogni elemento viene associato a sé stesso. +Banalmente, per ogni applicazione $\alpha$, $(\alpha \circ \Id) = (\Id \circ \alpha) = \alpha$. + +\begin{lemma} + $\sigma : S \to T$ è una corrispondenza biunivoca se e solo se + esiste un'applicazione $\mu : T \to S$ tale per cui + $(\sigma \circ \mu) = (\mu \circ \sigma) = \Id$. +\end{lemma} + +\begin{proof} + Dal momento che $\sigma$ è bigettiva, $\sigma^{-1}$ esiste, e questa è + tale per cui $(\sigma \circ \mu) = (\mu \circ \sigma) = \Id$. + + In direzione opposta, se esiste una $\mu$ tale per cui $(\sigma \circ \mu) = + (\mu \circ \sigma) = \Id$, allora: + + \begin{itemize} + \item $\sigma$ è iniettiva: $\sigma(s_1) = \sigma(s_2) \implies + \mu(\sigma(s_1)) = \mu(\sigma(s_2)) \implies s_1 = s_2$. + \item $\sigma$ è surgettiva: $\forall t \in T, t = \sigma(\mu(t)) \implies + \exists s = \mu(t) \in S \mid t = \sigma(s)$. + \end{itemize} +\end{proof} + +\begin{lemma}[Unicità dell'applicazione inversa] + Per ogni applicazione bigettiva $\sigma$, $\sigma^{-1}$ è unica. +\end{lemma} + +\begin{proof} + Poniamo $\alpha \neq \beta$ come due applicazioni inverse distinte + di $\sigma$. Allora $\alpha = \alpha \circ (\sigma \circ \beta) = + (\alpha \circ \sigma) \circ \beta = \beta$, che è una contraddizione. +\end{proof} diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index b09983a249bcea4dd9038dff92d2f88c5e15d59f..606edb61843f8105fc3f0a48b847758e636c7b8d 100644 GIT binary patch delta 119419 zcmcG01z42Z_BSmc4Uz&whtv!?bazQ3qO^d7bn}wZAl(QENC^T82vRC37<6~Hpo9o~ z!*~oldhfaS{=f4)zK5CFvG!VP|Msf=R=mQ>3B~~!)TL$F!5qAp4Db3|hcS6TTp)WB zYfNEb5T_=HQ_fz?9>ggPVm7rjzT<4}$N^>niHc&H+nJp;;{U5L^L1{1KG;A0>s(x1 zT4xXQvxoiJ!}084b@nhmdtB$@f&Js+GCO;~S^->Kx6b~wIeWmHeQN>#V0ZSl)A!%t z!`i|>vV0>#;13i#nVULW+1qh|&(jPBpCcOhi)!VwL7C1TaH{W|{RN*5PCuM7_kSty|CnFESM@ljN9R8LH%f&G;rbKTzi$feuQ~;M-N_K{-!c6E&!&K@ z4xH&Q*}(Y!i-dlwPd{tvzhRH*SzGwBArJll&ME8Pg)RME=f7#L%J=obwHiJOd@I=fSj8VP z@%$=axB~n@bMEuV_%pDagKqdT;0%V(4POA9>E~L(KUkd63V*<9g?IYO_p=|)ngCzL zf3E8f3BuccTR40+_~hq!_G8Qa>vMwpw>S&H&bA*X1<%hw%l%uRRXC#@PQ^F6;2QR0 zeQ+6mQ}4gD8~*WI`yaf5b^50Na7FpC060J3tN)>!+<$`7Z)fwlV}kpiK`3l~uHWsr 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