From 87688d93535c4b371f239ba4e986f7d37efbcd89 Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Tue, 20 Sep 2022 09:22:28 +0200 Subject: [PATCH] aritmetica: relazioni di equivalenza e nozioni sulle applicazioni --- Aritmetica/1. Teoria degli insiemi.tex | 2 +- ...elazioni di equivalenza e applicazioni.tex | 99 ++++++++++++++++++ Aritmetica/aritmetica.pdf | Bin 184128 -> 197916 bytes Aritmetica/aritmetica.tex | 5 + 4 files changed, 105 insertions(+), 1 deletion(-) create mode 100644 Aritmetica/2. Relazioni di equivalenza e applicazioni.tex diff --git a/Aritmetica/1. Teoria degli insiemi.tex b/Aritmetica/1. Teoria degli insiemi.tex index adfd898..798fc27 100644 --- a/Aritmetica/1. Teoria degli insiemi.tex +++ b/Aritmetica/1. Teoria degli insiemi.tex @@ -140,4 +140,4 @@ Una $n$-tupla ordinata, ossia la forma in cui è raccolto un certo elemento di u corrispondente in ordine dell'altra: pertanto, in generale, $(a, b) \neq (b, a)$. Inoltre, il prodotto cartesiano $A \times A$ viene indicato con $A^2$ (analogamente, -$A^n = \bigtimes_{i=1}^{n} A$). \ No newline at end of file +$A^n = \bigtimes_{i=1}^{n} A$). diff --git a/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex new file mode 100644 index 0000000..69593c9 --- /dev/null +++ b/Aritmetica/2. Relazioni di equivalenza e applicazioni.tex @@ -0,0 +1,99 @@ +\chapter{Relazioni di equivalenza e applicazioni} + +\section{Le relazioni di equivalenza} + +Utilizzando le nozioni di base della teoria degli +insiemi è possibile definire formalmente il concetto +di relazione di equivalenza. + +Dato un sottoinsieme $R$ di $A \times A$, $R$ si +dice relazione di equivalenza se: + +\begin{itemize} + \item $(a,a) \in R$ (proprietà riflessiva) + \item $(a,b) \in R \implies (b,a) \in R$ (proprietà simmetrica) + \item $(a,b), (b,c) \in R \implies (a,c) \in R$ (proprietà transitiva) +\end{itemize} + +Tale definizione può essere semplificata +implementando l'operazione binaria $\sim$ tale per cui +$a\sim b \iff (a,b) \in R$. In questo modo, le condizioni +di una relazione di equivalenza $R$ diventano: + +\begin{itemize} + \item $a \sim a$ + \item $a \sim b \implies b \sim a$ + \item $a \sim b \land b \sim c \implies a \sim c$ +\end{itemize} + +\begin{theorem} + Definita una relazione di equivalenza $R$ con operazione + binaria $\sim$, $a \sim b \land c \sim b \implies a \sim c$. +\end{theorem} + +\begin{proof} + Dalla proprietà riflessiva di $R$, $c \sim b \implies b \sim c$. + Verificandosi sia $a \sim b$ che $b \sim c$, si applica la proprietà + transitiva di $R$, che implica $a \sim c$. +\end{proof} + +\subsection{Classi di equivalenza} + +Si definisce classe di equivalenza di $a$ per un certo insieme +$A$ e una certa relazione di equivalenza $R$ l'insieme +$\cl(a)=\{x \in A \mid a \sim x\}$, ossia l'insieme di tutti i punti che +si relazionano ad $a$ mediante tale relazione di equivalenza. + +\begin{theorem} + Le classi di equivalenza partizionano l'insieme di relazione + in insiemi a due a due disgiunti. +\end{theorem} + +\begin{proof} + Prima di tutto è necessario dimostrare che l'unione di tutte + le classi di equivalenza dà luogo all'insieme di relazione $A$. + + Per ogni elemento $a \in A$, $a$ appartiene a $\cl(a)$ per la proprietà + riflessiva di $R$, ossia della relazione di equivalenza su cui + $\cl$ è definita. Pertanto $\bigcup_{a \in A} \cl(a)$, che contiene solo + elementi di $A$, è uguale ad $A$. + + In secondo luogo, è necessario dimostrare che le classi di equivalenza + sono o disgiunte o identiche. Ponendo l'esistenza + di un $a \in \cl(x) \, \cap \, \cl(y)$, la dimostrazione deriva dalle proprietà + di $R$: sia $b \in cl(x)$, allora $b \sim a$; dunque, dal momento che $b \sim a$ e che + $a \sim y$, $b \sim y$, ossia $\cl(x) \subseteq \cl(y)$ (analogamente si ottiene + $\cl(y) \subseteq \cl(x)$, e quindi $\cl(x) = \cl(y)$). +\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 +$\sigma : S \rightarrow T$. + +Si scrive $\sigma : s \rightarrowtail \sigma(s)$ per sottintendere che +$\forall \, (s, t) \in \sigma, (s, t) = (s, \sigma(t))$. + +\subsection{Proprietà delle applicazioni} + +\begin{definition}[Iniettività] + Un'applicazione si dice iniettiva se ad ogni immagine + è corrisposto al più un elemento, ossia anche che + $s_1 \neq s_2 \implies \sigma(s_1) \neq \sigma(s_2)$. +\end{definition} + +\begin{definition}[Surgettività] + Un'applicazione si dice surgettiva se ad ogni immagine + è corrisposto almeno un elemento, ossia anche che + $\forall t \in T, \exists s \mid \sigma(s) = t$. +\end{definition} + +\begin{definition}[Bigettività] + Un'applicazione si dice bigettiva se è sia iniettiva che + suriettiva, ossia se $\forall t \in T, \existsone s \in S + \mid \sigma(s) = t$. +\end{definition} diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index 39025fbe80ed8dff7ac40d74c6cfe3c63eca81eb..d5522fa766e0781381e1f96903cd863fd525c9c1 100644 GIT binary patch delta 129791 zcmV(_K-9m$o(r6c3=kzyL`E$!E;R}z(A34)3O11t6#+MwQGWs{kx!Nx16iEC!^jRR z+buR$$PicT=f{&tz86hE9Fy`GCIdDzHJ4F;0x5sxm&=ZuFcgOO^Az6IETFz&<87~+ zYARI~yO}Pm;36b~DKK5+>1$(?gd|F-Mv*e<%ErezkdOcP{2)*Q1U}!PFuu$0-hN66 zAS@A?z##t#GMsRl0ZwI-Nec1;Ji_~?s8%}qk$vf27CjN7lgd`HPBcOkySgpR{cU_Mt1Hz$Go@sp!H62ll8EUtB zr1bC>IKM3SV-6V7`D%O`1YM@A-VDPma1dXabzAGT_5w6fED*Uet(=hzT_PE}mCS?o zf4bIXSrVVs?|2a{~1WO@c zy3jI_on;E=Mz`OMcE^f;fHd7c0JDFKueE8vtaj4BpVb!Y_N;zitG>5$c$z48L|xc< z(v6!a=Sul4yV3 zmo+rVLf3WhV)K(o(3>_ z0w{mYSxax*Mi9Q|SLhZBh&KCvH0U9Xo1#Tet=mh}0u)6jEGSYTsi5$$?<{xu^oo)l zyES|eyP|`6~#zODpuDwD@~AOdL=nWTJhEOZuLj> zECdBxtT%*E5&fJxs?}v2w4^Cs z8Du1to-!p!Q&0J~Cd;$i+ql?S@ui6J)cqD^M_zjJ@Vkv`|9Y3^xfM|IsDULCmDPWm zL|NRu{h0etvZBt?z5R#j=mVy7%@vD0mBV^NbyU`Mw$1XM@&`lB0z^Ov2@8J9Tfak&k3mw)Zo>zRX^0hg zxf|6WK{q}mT znXNkjJecXRhA&c-6dN=GHhD;w{^M);ATHc;wE-PGFVDme4?TTfE3^$|7HHN zbOP&+QlWk|+A z#bO>{p~ApU6^6(0zj%LEv&qfAO6`9uNN|uAznsi(kogf~ za}+V*#1<4b2oFRd)wU-=WwxicJPaI=h2l&wR4JkF>U%0XQrs+c)d~s9m#k{+6zH5) z?FdJoab&`mLIIA^b03~M7c=Ax8yikJqJm5-_d`>iz9b;@69mM}gd8zKhoU}qbZDZj zz)4Obin6FNe5)m$z!+=SBA7+za=g7zyZ3H`NNRqcqnmjqDUr_huV(z~SQzg2(RE7z29Y7hngatT9DCx?D2 z?r?7I;qL`)p9>0yjYDaJBW=#Y_|Yh|rJvY-?vo!~+8)nx7HFt!o}`ZpZ7%RZcV{pu zPZ~l>jv8~!hns~qJJV^(rfRs4XluH^VtALwb^VZg1p=yokZlS6YCoj1@38B3CJYOS zrS6}1e$#)vmFZrFY$<^Gxwi2{+k*7jY3PV8#dd}d0w(spBCXoqjCMB^?OLczlJV^6 zZ8()F((g#Sv%DAy9ONIzjogCtA*lh6VYTGJtyygbnN;HZWp`DvVbXx^Pc+qnB0dz( zkXk|QaytBmxRAlbZQX$O%-#f!gssk4=-gCBJewy0+b~cG%G&=<86 z2YV8Q!3413X^P37@2_9{3x4q(P?M2R6PFBA0TTrvGBq(amvL$WD1XgbS##XR5q{UN z*gO$ir5(+E*rf;BqEbw;EZU^~;4rCzmoN<%2+(`D68`%1T(~&c#YSQ(4-&AzV5YmL z`|GcJh!+nFy!h*zZ*Jeb`>5!`AS0<<-2S@I1WBeBN-!iCU)FU#R z#Td&dTCSAR*$)?2=YKz6e){gxKmOpZJiGY*$IG9Vgplm~>f*bf&;8pAe>uOtzPkK2 zeD;sqzrFiNac^->6~aV=Xl`9;hrGc-)_8sQQLx@OtN76hu0}Hd>Fv_UtY|jHV-70| z15y(NlJ_Tk_feYOL`ox~MO^!8DM(gWsdcul^6j>?_a?h9{eQE_Q|EQ#KdF`^J1gsLSv(fQNj7egD!0SKT4n8r`0-&}nErmbBH4Z!v3OhAIg2bD~Gzne=Ew6A%4_{^0C?OoH$;Xv84vvq%h)?M@wCA zN@rxIf?g`y?SFSWt2{Q@9j0{274tHytS;8Yc54;MHzlaj#4nuy`6dioFR9Lc3qPu} zeH|?{TqF*NU$q+*hk+B=O^TM zBUxC4Wt(;ry;JuclyHg|Rnc7M3BOl}LlA}m$csq0L4V2^xB<6j!L6^YS&EG{EzO#T zy7YHN1+G;uDHx?mNT4wirrobRjF9~i2*VO#08LC`5XWY=3U3oL2&7+#fCnN_)ka_b z^kl=RaC;64=Kiv-?y$g#$)#059hzPsDrFo1vk*Vy({k0bXcg9Z+B8p_ve*qg0-kTm zJ4^)luYdI@>7P0$NbEY*$f$P3ACe}MoOC9GxpdaR+;LhT=)~bi6&`MSXaNHGr57tk z4HBFU9XfcvjGn)uMr7N)z4C0mFT+g*S1DtlQoE*oeqVsDye@Y+;5}zF`ra2NZIV+oWTik6BMTl3HE?@ZmU6dkQIVZl7HcriQX^{VME7G*`G%sCJ#C#;GtMh zWVG>PCjAG@hP$a|`On+nk`bt!EgK(%-f``dZBsS=F2*RBuJl`x_8SITb+I@j)FAKi z7rVEA*ymNa&+k*=s9>kesi4Hc&nTlqP`b1!OQ~#Ifd2OuF@YU8@U%9Pcp#h|=yL$z ze}5QjJ-vLiuI_U{$Z+tcO9LvPyy~T66;K|oW0dy-H7*6iU0lc~L$?V^1ZcBlKcZX+K=3&&DF-djify^wxiGCk>}QwQ{5va}y~27ahr>u>dNL;CDNH`- z+AT8vE9u#?OAZbhU&*APLumPeghSCth2s=rtsfasB z-6TYX`m)xUAqQB8w|vMM%5tFzH|GR#E{^#*I|_4#_OOShP#*+8Q>tACqVVNl`; 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Teoria degli insiemi.tex} +\include{2. Relazioni di equivalenza e applicazioni.tex} \end{document} \ No newline at end of file