From c9cb73b61546405200a2bb9b9ed3d686c3bf8df3 Mon Sep 17 00:00:00 2001 From: Marcel Schneider Date: Wed, 3 Jan 2018 15:11:38 +0100 Subject: [PATCH] Kombinatorik und Relationen --- DiMa/dima.pdf | Bin 27968 -> 40927 bytes DiMa/dima.tex | 28 +++++++++++++++++++++++++++- MafIA1/mafia.pdf | Bin 55013 -> 60069 bytes MafIA1/mafia.tex | 27 ++++++++++++++++++++++++--- 4 files changed, 51 insertions(+), 4 deletions(-) diff --git a/DiMa/dima.pdf b/DiMa/dima.pdf index 67045d9737dd7f97038d74cc8accc478b9678a53..6636e90dc7b1d7ad80b3ebaf472ff9e433975c1a 100644 GIT binary patch delta 33116 zcmZshQ;;t0maSLWwyjmRZQHACyH@!t+qP}nwr$&Xoqu=tMeo!5A|o;)XGBKiGcn$g zvv(b6rw15`OhH79mXVGXn(XlQ>J^%qIRTND6u`v5$->76?d0reVqgR9wtl509fQLT z+x?<0W7|PlDY1yN`mT`6`H1dpCyQ|5;$@*sSn~a)gdmk@YJKC1Qz(exR~IcNth6{1 zgM4=MkdGerv$$_?M;DHmG9o__^D?pV*Aj>-6QpA^2i-;lwCl1GiEn@OZiSnwD`g(< z4gjK5E>1O-=f(29v)lS?!oxe0V}!?l`FEW&@fU$ZAs;LJ&-i_F#F<_Ckwdb(gyS=} zi+WCGd9-j6K?Lf-bI;aK=cltC`mYEkAsh$`pz(mvY1)q6pojA6TnRc4KN^Hm4D0!# 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Ich füge hier nur nach Lust und Laune Dinge ein, so dass dies in keinster Weise als vollständig oder stets korrekt angesehen werden darf. Die Quelldateien sind öffentlich unter \url{https://git.webschneider.org/uni/sammlung} einsehbar. -Jeder ist dazu aufgerufen, sich an der Entwicklung zu beiteiligen! +Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \tableofcontents \bigskip @@ -44,6 +44,27 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beiteiligen! M \cup N = \{x : x \in M \lor x \in N\} \end{align} +\section{Relationen} + Eine Relation ist zunächst eine Teilmenge beliebiger Mengen $M, N$. + Man schreibt für ein beliebiges Paar aus $(x,y) \in M \times N$ entweder + $xRy$ oder seltener $(x,y) \in R$. + + \subsection{Eigenschaften} + Zur Beschreibung einer Relation gibt es folgende Eigenschaften, dazu + betrachten wir eine zweistellige Relation $R$ auf $M$: + \begin{table}[h] + \centering\label{relation:eigenschaften} + \begin{tabular}{rl} + \textbf{reflexiv} & $\forall x\in M $ gilt $xRx$, das Element steht also zu sich selbst in Relation.\\ + \textbf{symmetrisch} & $\forall x,y \in M$ aus $xRy$ auch $yRx$ folgt. Die Reihenfolge ist also egal.\\ + \textbf{transitiv} & $\forall x,y,z \in M$ und $xRy$ und $yRz$ folgt $xRz$. + \end{tabular} + \end{table} + + \subsection{Äquivalenzrelation}\label{relationen:aequivalenz} + Eine Relation heißt Äquivalenzrelation, wenn alle Eigenschaften aus~\ref{relation:eigenschaften} + zutreffen. + \section{Vektorräume}\label{vektorraum} \subsection{Vektorraum}\label{vektorraum:def} Eine nichtleere Menge $V$ heißt Vektorraum über einem Körper $\K$, wenn die folgenden Eigenschaften zutreffen:\\ @@ -75,10 +96,10 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beiteiligen! Seien $v_1, \dots, v_m$ Elemente eines Vektorraums $V$ über einem Skalarenkörper $\K$, und es sei $\alpha := (\alpha_1, \dots, \alpha_m)$ ein m-Tupel aus $\K^m$ für ein $m\in \N \backslash \{0\}$. \subsubsection{Linearkombination} - Eine Linearkombination ist eine Vektroaddition, bei der jeder Vektor einer Menge $V$ zunächst mit einem Skalar $\alpha$ multipliziert wird. + Eine Linearkombination ist eine Vektoraddition, bei der jeder Vektor einer Menge $V$ zunächst mit einem Skalar $\alpha$ multipliziert wird. \[\sum_{j=1}^{m} \alpha_j \cdot v_j \] \subsubsection{Affinkombination} - Wenn außerdem gilt, dass die Summe aller Koeffizienten 1 ergibt, also + Wenn außerdem gilt, dass die Summe aller Koeffizienten $1$ ergibt, also \[\sum_{j=1}^{m} \alpha_j = 1\] dann spricht man von einer \emph{Affinkombination}. \subsubsection{Konvexkombination}