From a5cec75563148e9057bb2f842b38ebdca1f6b49c Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Fri, 23 Sep 2022 15:52:10 +0200 Subject: [PATCH] aritmetica: composizione di applicazioni --- ...elazioni di equivalenza e applicazioni.tex | 120 +++++++++++++++++- Aritmetica/aritmetica.pdf | Bin 197916 -> 205361 bytes Aritmetica/aritmetica.tex | 2 + 3 files changed, 115 insertions(+), 7 deletions(-) diff --git a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex index 69593c9..de57eae 100644 --- a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex +++ b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex @@ -26,10 +26,10 @@ di una relazione di equivalenza $R$ diventano: \item $a \sim b \land b \sim c \implies a \sim c$ \end{itemize} -\begin{theorem} +\begin{lemma} Definita una relazione di equivalenza $R$ con operazione binaria $\sim$, $a \sim b \land c \sim b \implies a \sim c$. -\end{theorem} +\end{lemma} \begin{proof} Dalla proprietà riflessiva di $R$, $c \sim b \implies b \sim c$. @@ -66,14 +66,36 @@ si relazionano ad $a$ mediante tale relazione di equivalenza. $\cl(y) \subseteq \cl(x)$, e quindi $\cl(x) = \cl(y)$). \end{proof} +\begin{theorem} + Data una partizione di un insieme che lo compone in insiemi a due + a due disgiunti, è sempre possibile costruire delle classi di equivalenza. +\end{theorem} + +\begin{proof} + Vogliamo dimostrare che, data la stessa appartenenza ad un insieme come relazione, + essa è una relazione di equivalenza. + + Sicuramente $a \sim a$ (proprietà riflessiva). + Inoltre, $a \sim b \implies a, b \in A_\alpha \implies b \sim a$ + (proprietà simmetrica). + Infine, $a \sim b, \, b \sim c \implies a, b, c \in A_\alpha \implies a \sim c$ + (proprietà transitiva). + + In particolare, dato $a \in A_\alpha$, $\cl(a) = A_\alpha$. +\end{proof} + \section{Le applicazioni} La nozione di applicazione di un insieme in un altro ci permette di generalizzare, ma soprattutto di definire, il concetto di -funzione. Dati due insiemi $S$ e $T$, si dice che $\sigma$ è un'applicazione -da $S$ a $T$, se $\sigma \subseteq S \times T \land \forall s \in S, \existsone - t \in T \mid (s, t) \in \sigma$. Tale applicazione allora si scrive come -$\sigma : S \rightarrow T$. +funzione. + +\begin{definition}[Applicazione] + Dati due insiemi $S$ e $T$, si dice che $\sigma$ è un'applicazione + da $S$ a $T$, se $\sigma \subseteq S \times T \land \forall s \in S, \existsone + t \in T \mid (s, t) \in \sigma$. Tale applicazione allora si scrive come + $\sigma : S \to T$. +\end{definition} Si scrive $\sigma : s \rightarrowtail \sigma(s)$ per sottintendere che $\forall \, (s, t) \in \sigma, (s, t) = (s, \sigma(t))$. @@ -87,7 +109,7 @@ $\forall \, (s, t) \in \sigma, (s, t) = (s, \sigma(t))$. \end{definition} \begin{definition}[Surgettività] - Un'applicazione si dice surgettiva se ad ogni immagine + Un'applicazione si dice surgettiva (o talvolta \textit{su $T$}) se ad ogni immagine è corrisposto almeno un elemento, ossia anche che $\forall t \in T, \exists s \mid \sigma(s) = t$. \end{definition} @@ -97,3 +119,87 @@ $\forall \, (s, t) \in \sigma, (s, t) = (s, \sigma(t))$. suriettiva, ossia se $\forall t \in T, \existsone s \in S \mid \sigma(s) = t$. \end{definition} + +\subsection{Composizione di applicazioni} + +\begin{definition}[Composizione] + Date due applicazioni $\sigma : S \to T$ e + $\tau : T \to U$, si può definire + un'applicazione detta composizione + $(\tau \circ \sigma) : S \to U$, tale per cui + $(\tau \circ \sigma) : s \mapsto \tau(\sigma(s))$. +\end{definition} + +Dobbiamo tuttavia assicurarci che tale applicazione +possa esistere, ossia verificare che $\forall s \in S \existsone + u \in U \mid (s, u) \in S \times U$; quindi che $\tau(\sigma(s))$ +sia unico. Tuttavia questa proprietà è banale: $\sigma(s)$ è Sicuramente +unico poiché $\sigma$ è un'applicazione, e pertanto $\tau(\sigma(s))$ lo è, +essendo anch'essa un'applicazione. + +\subsubsection{Proprietà associativa della composizione} + +È inoltre interessante dimostrare che la composizione rispetta la proprietà associativa, +ossia che $(\alpha \circ \beta) \circ \gamma = \alpha \circ (\beta \circ \gamma)$. + +\begin{lemma}[Proprietà associativa della composizione] + Date tre applicazioni $\alpha$, $\beta$, $\gamma$, + $(\alpha \circ \beta) \circ \gamma = \alpha \circ (\beta \circ \gamma)$. +\end{lemma} + +\begin{proof} + Preso un $a$ appartenente al dominio di $\gamma$, per il primo membro abbiamo: + + $$((\alpha \circ \beta) \circ \gamma)(a) = (\alpha \circ \beta)(\gamma(a)) = + \alpha(\beta(\gamma(a)))$$ + + Analogamente per il secondo membro abbiamo: + + $$(\alpha \circ (\beta \circ \gamma))(a) = \alpha((\beta \circ \gamma)(a)) = + \alpha(\beta(\gamma(a)))$$ +\end{proof} + +\subsubsection{Iniettività, surgettività e bigettività della composizione} + +L'iniettività, la surgettività e la bigettività di una composizione sono +ereditate dalle applicazioni di cui è composta se tutte queste le rispettano, ossia: + +\begin{itemize} + \item $(\tau \circ \sigma)$ è iniettiva se $\tau$ e $\sigma$ lo sono. + \item $(\tau \circ \sigma)$ è surgettiva se $\tau$ e $\sigma$ lo sono. + \item $(\tau \circ \sigma)$ è bigettiva se $\tau$ e $\sigma$ lo sono. +\end{itemize} + +\begin{lemma}[Iniettività della composizione] + \label{lemma:iniettivita_composizione} + $(\tau \circ \sigma)$ è iniettiva se $\tau$ e $\sigma$ lo sono. +\end{lemma} + +\begin{proof} + Dal momento che $\sigma$ è iniettiva $s_1 \neq s_2 \implies \sigma(s_1) \neq \sigma(s_2)$, + ma a sua volta, essendo $\tau$ iniettiva, $\sigma(s_1) \neq \sigma(s_2) \implies + \tau(\sigma(s_1)) \neq \tau(\sigma(s_2))$. +\end{proof} + +\begin{lemma}[Surgettività della composizione] + \label{lemma:surgettivita_composizione} + $(\tau \circ \sigma)$ è surgettiva se $\tau$ e $\sigma$ lo sono. +\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 $t \in \Dom(\tau)$, + $\exists u \in \Codom(\tau) \mid u = \tau(t) = \tau(\sigma(s))$. +\end{proof} + +\begin{lemma}[Bigettività della composizione] + $(\tau \circ \sigma)$ è bigettiva se $\tau$ e $\sigma$ lo sono. +\end{lemma} + +\begin{proof} + Se $\tau$ e $\sigma$ sono bigettive, sono sia iniettive che surgettive; + pertanto $(\tau \circ \sigma)$ è sia iniettiva che bigettiva per i + lemmi \ref{lemma:iniettivita_composizione} e + \ref{lemma:surgettivita_composizione}. +\end{proof} + diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index d5522fa766e0781381e1f96903cd863fd525c9c1..b09983a249bcea4dd9038dff92d2f88c5e15d59f 100644 GIT binary patch delta 105827 zcmZUYQ*@pU5N>0;vF*l2W83^<+i38Hjcwa#oW@CG+je7H=RdqTH?!tu?q;oL_MYrW zr1}9QbSh$0 z9n4(aT+EH_|7YiDVuO&Tyas^@$;HKyrm_q{2OOr4$F8wp{(2G!^>oLU@rxpI2!9!<_&EAuO}81Jp9G5`$ha1pdC+SYz|O@P3_osFAXd!arr9OSg?c?HiLQs>-{s};Ae!R^^{aHfqIsGon+SyP_1BXHcNhv6d1-_P|Gw@SQ32uH32E0P^0cfuj 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@@ \DeclareMathOperator{\existsone}{\exists !} \DeclareMathOperator{\cl}{cl} +\DeclareMathOperator{\Dom}{Dom} +\DeclareMathOperator{\Codom}{Cod} \let\oldemptyset\emptyset \let\emptyset\varnothing