From 39ff170dcb04852d3b5e3005e95185bbacb728f5 Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Fri, 16 Sep 2022 16:36:54 +0200 Subject: [PATCH] aritmetica: complementi e leggi di de morgan --- Aritmetica/1. Teoria degli insiemi.tex | 72 ++++++++++++++++++++++++- Aritmetica/aritmetica.pdf | Bin 169313 -> 184128 bytes Aritmetica/aritmetica.tex | 4 ++ 3 files changed, 75 insertions(+), 1 deletion(-) diff --git a/Aritmetica/1. Teoria degli insiemi.tex b/Aritmetica/1. Teoria degli insiemi.tex index 8137bd4..adfd898 100644 --- a/Aritmetica/1. Teoria degli insiemi.tex +++ b/Aritmetica/1. Teoria degli insiemi.tex @@ -38,7 +38,30 @@ insiemi: $\bigcap_{t \in T} A_t = \{x \mid (\forall t \in T \mid x \in A_t)\}$. In modo opposto all'unione, l'intersezione è tale per cui $A \subseteq B \rightarrow A \cap B = A$ (in particolare, $A \cap \emptyset = \emptyset$). -\section{L'operazione di sottrazione} +\subsection{Relazioni tra l'operazione di intersezione e di unione} + +Si può facilmente dimostrare la seguente relazione, valida per qualunque scelta +di insiemi $A$, $B$ e $C$: $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$. + +\begin{proof} + Prima di tutto, un elemento di entrambi i due insiemi appartiene obbligatoriamente a $C$: + nel caso del primo membro, il motivo è banale; riguardo al secondo membro, invece, ci accorgiamo + che esso appartiene almeno a uno dei due insiemi dell'unione, riconducendoci a un'intersezione + con l'insieme $C$. + + Ogni elemento di $(A \cup B) \cap C$ appartiene inoltre ad almeno $A$ o $B$, e quindi, + appartenendo anche a $C$, appartiene a $A \cap C$ o $B \cap C$, e quindi a $(A \cap C) \cup (B \cap C)$. + Pertanto $(A \cup B) \cap C \subseteq (A \cap C) \cup (B \cap C)$. + + In direzione opposta, ogni elemento di $(A \cap C) \cup (B \cap C)$ appartiene almeno + ad uno di dei due insiemi dell'unione. Per appartenere all'intersezione, tale elemento + appartiene ad almeno $A$ o $B$; e quindi appartiene ad $A \cup B$. Appartenendo anche a $C$, + appartiene anche $(A \cup B) \cap C$. Quindi $(A \cap C) \cup (B \cap C) \subseteq (A \cup B) \cap C$. + + Valendo l'inclusione in entrambe le direzioni, $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$. +\end{proof} + +\section{L'operazione di sottrazione e di complemento} L'operazione di sottrazione su due insiemi $A$ e $B$ è definita come $A \setminus B = \{x \mid x \in A \land x \notin B\}$. Si può facilmente @@ -57,6 +80,53 @@ verificare che $A = (A \cap B) \cup (A \setminus B)$. In particolare, se $B \subseteq A$, $A \setminus B$ si dice \textbf{complemento di $B$ in $A$}. +L'operazione di complemento viene indicata con $A'$ qualora sia noto l'universo di riferimento $U$ +per cui $A' = U \setminus A$. + +\subsection{Le leggi di De Morgan} + +Si possono dimostrare le seguenti proprietà: + +\begin{itemize} + \item $(A \cup B)' = A' \cap B'$ + \item $(A \cap B)' = A' \cup B'$ +\end{itemize} + +\begin{proof}[Prima legge di De Morgan] + Un elemento che appartiene a $(A \cup B)'$ non appartiene né a $A$ né a $B$, e quindi + appartiene sia a $A'$ che a $B'$, pertanto anche alla loro intersezione $A' \cap B'$ + [$(A \cup B)' \subseteq A' \cap B'$]. + + Allo stesso modo, un elemento di $A' \cap B'$ non appartiene né ad $A$ né a $B$, e quindi + non appartiene ad $A \cup B$, appartenendo dunque a $(A \cup B)'$ + [$A' \cap B' \subseteq (A \cup B)'$]. Pertanto $(A \cup B)' = A' \cap B'$. +\end{proof} + +\begin{proof}[Seconda legge di De Morgan] + Un elemento che appartiene a $(A \cap B)'$ può appartenere al più ad $A$ o esclusivamente + a $B$; pertanto appartiene ad almeno $A'$ o $B'$, e qunidi alla loro unione + [$(A \cap B)' \subseteq A' \cup B'$]. + + Allo stesso modo, un elemento di $A' \cup B'$ appartiene ad almeno $A'$ o $B'$, e quindi + non può appartenere a entrambi $A$ e $B$, appartenendo dunque a $(A \cap B)'$ + [$A' \cup B' \subseteq (A \cap B)'$]. Pertanto $(A \cap B)' = A' \cup B'$. +\end{proof} + +\subsection{La logica affrontata con gli insiemi} + +In modo veramente interessante, ogni operatore logico segue la logica +dell'insiemistica (e viceversa); laddove l'operatore $\cup$ (o $\cap$) ha una certa proprietà, +la soddisfa anche $\lor$ (o $\land$). + +Quindi valgono tutte le leggi sopracitate: + +\begin{itemize} + \item $(a \lor b) \land c = (a \land c) \lor (b \land c)$ + \item $(a \land b) \lor c = (a \lor c) \land (b \lor c)$ + \item $\lnot (a \land b) = \lnot a \lor \lnot b$ + \item $\lnot (a \lor b) = \lnot a \land \lnot b$ +\end{itemize} + \section{Il prodotto cartesiano} Il prodotto cartesiano di una famiglia ordinata di insiemi $F$ con un certo insieme diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index 035dfa8a0e5116ea4fd5894a61d13c5ce7adb553..39025fbe80ed8dff7ac40d74c6cfe3c63eca81eb 100644 GIT binary patch delta 97832 zcmZU4LvUb$qHJv2w(XfDlT2(*tch*s#I|kQwrx8T+t$0ce*ML(x9RFtZ+g{T-FxM5 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