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zTqSp_*nGuFB~ZUlT7@A+a>2UjO72jvvey=~x)d`i=#ieT zMoXK;Qq4udJM<3caa7`@78t6eaiY<2e9_fNlT#x1HXXutlU;Gjka64V#!8SpQxGcU2})NUrE!TFd1eIYKKjn#~NEQ z_okc7y{fB}4b4|O#F{{tjP BH2eSn literal 0 HcmV?d00001 diff --git a/MafIA1/mafia.tex b/MafIA1/mafia.tex new file mode 100644 index 0000000..1f1d598 --- /dev/null +++ b/MafIA1/mafia.tex @@ -0,0 +1,74 @@ +\documentclass[a4paper,11pt]{scrartcl} +\input{../env/packages} +\input{../env/commands} +\input{../env/meta} + +\begin{document} + +\title{Zusammenfassung MafIA 1} +\maketitle +\tableofcontents +\bigskip + +\section{Mengenlehre} + \subsection{Menge} + Eine Zusammenfassung von unterschiedlichen Elementen zu einem Ganzen + + \subsection{Teilmenge} + \begin{align} + N \subset M \Leftrightarrow M \subset N + \end{align} + + \subsection{Potzenmenge} + Sei \(M\) eine Menge.\\ + \(P(M)\) (auch \(2^M\)):= Menge von allen Teilmengen von M. + Sei \(l\) die Anzahl der Elemente von \(M\), so ist die + Anzahl der Möglichkeiten ist \(2^l\) + + \subsection{Schnittmenge} + \begin{align} + M \cap N := \{x : x \in M \land x \in N\} + \end{align} + Die Schnittmenge besteht also aus den gemeinsamen Elementen der beiden Mengen. + Falls \(M \cap N = \emptyset\) sind, sind M und N \emph{disjunkt} + + \subsection{Vereinigung} + \begin{align} + M \cup N = \{x : x \in M \lor x \in N\} + \end{align} + +\section{Vektorräume}\label{vektorraum} + \subsection{Vektorraum}\label{vektorraum:def} + Eine nichtleere Menge $V$ heißt Vektorraum über einem Körper $\K$, wenn die folgenden Eigenschaften zutreffen:\\ + \emph{Addition} + \begin{alignat}{2} + (u + v) + w &= u + (v+w) \;&& \forall u,v,w \in V\\ + u + v &= v + u \; && \forall u,v \in V\\ + u + 0 &= u \; &&\forall u, v \in V \\ + v + (-v) &= 0 \; && \forall v \in V + \end{alignat} + \emph{Skalarmultiplikation}: + \begin{alignat}{2} + (\alpha \cdot \beta) \cdot v &= \alpha \cdot (\beta \cdot v )\\ + \alpha \cdot (u+v) &= \alpha \cdot u + \alpha \cdot v \\ + (\alpha \cdot \beta) \cdot v & = \alpha \cdot(\beta \cdot v)\\ + 1 \cdot v &= v + \end{alignat} + + \subsection{Untervektorraum}\label{vektorraum:unterraum} + Sei $V$ ein Vektorraum über dem Körper $\K$. + Dann ist $U \subset V$ ein Untervektorraum, wenn gilt: + \begin{enumerate} + \item $U$ ist nicht leer, also muss mindestens $\colvec{0}{0} \in U$ gelten. + \item Die Addition muss abgeschlossen sein. + \item Die Skalarmultiplikation muss abgeschlossen sein. + \end{enumerate} + %\subsection{Kombinationen}\label{vektorraum:kombination} + %Voraussetzungen für die nächsten Definitionen: + %Es 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} + +\vspace{\fill} +\doclicenseThis{} +\end{document} diff --git a/env/commands.tex b/env/commands.tex new file mode 100644 index 0000000..15d1431 --- /dev/null +++ b/env/commands.tex @@ -0,0 +1,22 @@ +% custom commands go here % + +\newcommand{\N}{\mathbb{N}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\K}{\mathbb{K}} +\newcommand{\Q}{\mathbb{Q}} + +\newcommand{\Rq}{\R^2} +\newcommand{\Qq}{\Q^2} + +\newcommand{\lrarr}{\Leftrightarrow} +\newcommand{\linf}{\lim\limits_{n \rightarrow{} \infty}} + +%running fraction with slash - requires math mode. +\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}} + +% Vector +\newcommand*\colvec[3][]{ + \begin{pmatrix}\ifx\relax#1\relax\else#1\\\fi#2\\#3\end{pmatrix} +} +\newcommand{\xyvec}{\colvec{x}{y}} diff --git a/env/meta.tex b/env/meta.tex new file mode 100644 index 0000000..4c364bb --- /dev/null +++ b/env/meta.tex @@ -0,0 +1,9 @@ +%opening +\author{Marcel Schneider \\ \href{mailto:matheschneider@webschneider.org}{matheschneider@webschneider.org}} + +% License +\usepackage[ + type={CC}, + modifier={by-nc-sa}, + version={3.0}, +]{doclicense} diff --git a/env/packages.tex b/env/packages.tex new file mode 100644 index 0000000..812e0ee --- /dev/null +++ b/env/packages.tex @@ -0,0 +1,9 @@ +\usepackage[utf8]{inputenc} +\usepackage[ngerman]{babel} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amsthm,amssymb,amsfonts} +\usepackage{mathtools} +\usepackage{pdfpages} +\usepackage{hyperref} +\usepackage{stmaryrd} +\usepackage{enumerate}