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b/MafIA1/mafia.tex @@ -254,6 +254,14 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \item Beweis durch vollstaendige Induktion \end{enumerate} + \paragraph{Induktionsbeweis:} + $ $\\Grundsätzliches Schema:\\ + Sei $ P: \N \rarr B $ ein Prädikat auf $ \N $ ($P(n)$ ist wahr oder falsch $ \forall n \in \N $). Dann ist folgendes zu zeigen: + \begin{enumerate}[1.] + \item Induktionsanfang: $ P(0) $ ist wahr + \item $ \forall n \in \N$ gilt: Aus $ P(n) $ ist wahr $ \rarr P(succ(n)) $ ist wahr. + \end{enumerate} + Dann folgt: $ P $ ist allgemeingültig über $ \N $, d.h. $ P(n) $ ist wahr $ \forall n\in \N $\footnote{Beispiele zur Induktion gibts auf den Übungsblättern oder im Skript} \subsection{Boolesche Funktionen} Sei $ B := \{0,1\} $. Dann existiert eine Abbildung $ f: B^n \rarr B $.\\ \ex $ f(A) = \lnot(A) $ @@ -313,7 +321,17 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \section{Zahlen}\label{zahlen} + \subsection{Allgemein} + Es wurden in der Vorlesung folgende Arten von Zahlen behandelt: + \begin{itemize} + \item natürliche Zahlen $\mathbb{N} := \{0, 1, 2, 3, ...\}$ + \item ganze Zahlen $\mathbb{Z} := \{..., -1, 0, 1, ...\}$ + \item rationale Zahlen $\mathbb{Q}:=\{x = \frac{a}{b} | a, b \in \mathbb{Z}\}$ + \item reele Zahlen $\mathbb{R}$: z.B. $\sqrt{2} \not\in \mathbb{Q}$ + \item komplexe Zahlen $\mathbb{C} := \{z = x + \sqrt{-1} \cdot y\ |\ x, y \in \mathbb{R}\}$ + \end{itemize} \subsection{Natürliche Zahlen} + \subsubsection{Peano-Axiome} Definition der natürlichen Zahlen durch Peano: \begin{enumerate} @@ -338,7 +356,17 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \end{enumerate} Gilt darüber hinaus die \emph{Kommutativität} \[a \circ b = b \circ a \forall a,b \in G \] - heißt die Gruppe abelsch oder kommutativ. + heißt die Gruppe abelsch oder kommutativ.\\ + \ex + \begin{itemize} + \item $(\Z,+),(\Q,+),(\R,+),(\C,+)$ sind Gruppen + \item $(\Z,\cdot) $ ist keine Gruppe (inverses Element ist nicht in $ \Z $) + \item $(\Q\backslash\{0\},\cdot) $ und $ (\R\backslash\{0\},\cdot)$ sind abelsche Gruppen + \end{itemize} + + \textbf{Fakt:} Seien $ (G, \cdot) $ und $ (H, *) $ Gruppen. Dann ist $ (G \times H, 0) $ mit $ (g,h) (\in (G\times H)) \circ (g',h') (\in (G\times H)) = (g \cdot g', h*h') $ eine Gruppe + + \subsubsection{Identitätsfunktion} \section{Vektorräume}\label{vektorraum} \subsection{Vektorraum}\label{vektorraum:def} diff --git a/env/commands.tex b/env/commands.tex index 0115b72..c695865 100644 --- a/env/commands.tex +++ b/env/commands.tex @@ -4,6 +4,7 @@ \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\K}{\mathbb{K}} +\newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Rq}{\R^2}