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-%%%% START OF PREAMBLE
-% Copyright (c) 2021 Frederik, Franz, Marvin
-% Copyright (c) 2022 Linus, Benny, Marvin
-\usepackage[a4paper, inner=1cm, outer=1cm, top=2cm, bottom=2cm, bindingoffset=0cm]{geometry}
-\usepackage{amsmath,amsthm,amssymb,amsfonts}
-\usepackage{mathrsfs}
-\usepackage{braket}
-\usepackage{enumitem}
-\usepackage{csquotes}
-\usepackage{colortbl}
-\usepackage{environ}
-\usepackage{graphicx,tikz,xcolor,color,float,titlesec}
-\usepackage{pgfplots}
-\usepackage{fancyhdr}
-\usepackage{gauss}
-\usepackage{polynom}
-\usepackage{bm}
-\usepackage[ngerman=ngerman-x-latest]{hyphsubst}
-\usepackage[ngerman]{babel}
-\usetikzlibrary{matrix,shapes,trees}
-\usepgfplotslibrary{fillbetween}
-\pgfplotsset{compat=1.18}
-\setlength\parindent{0pt}
-\definecolor{ochsblau}{RGB}{26,122,219}
-\definecolor{ochsgelb}{RGB}{245,209,168}
-\definecolor{ochsorange}{RGB}{243,92,43}
-\renewcommand{\headrulewidth}{2pt}
-\let\oldheadrule\headrule
-\renewcommand{\headrule}{\color{ochsgelb}\oldheadrule}
-\newcommand\bracketify[1]{\lbrack#1\rbrack}
-\titleformat{\section}{\normalfont\large\bfseries\color{ochsblau}}{Aufgabe \thesection.\ }{0em}{\bracketify}
-%%%% END OF PREAMBLE
-
-%\usepackage{background}
-%\backgroundsetup{
-% position=current page.east,
-% angle=-90,
-% nodeanchor=east,
-% vshift=-5mm,
-% hshift=1cm,
-% opacity=1,
-% scale=3,
-% contents=Entwurf
-%}
-
-\newcommand\namesnstuff{
- \pagestyle{fancy}
- \fancyhf{}
- \fancyhead[L,LO]{\textcolor{ochsblau}{\textbf{Name:}}}
- \fancyhead[C,CO]{\textcolor{ochsblau}{\textbf{MFI2 - SS22}}}
- \fancyhead[R,RO]{\textcolor{ochsblau}{\textbf{Nachklausur}}}
- \fancyfoot[C,CO]{\thepage}
- \setlength{\headheight}{13.6pt}
-
- \vspace{-2.0cm}
- \noindent\includegraphics[width=0.2\textwidth]{../ochs_logo.png}
- \begin{minipage}[b]{0.6\textwidth}
- \centering
- \textcolor{ochsblau}{\textbf{\Large Mathematik 2 für Informatik}}\\
- \vspace{3mm}
- Peter Ochs, Tobias Nordgauer\\
- \vspace{3mm}
- Sommersemester 2022
- \end{minipage}
- \noindent\includegraphics[width=0.2\textwidth]{../uni_logo.png}\\
- \begin{center}\textcolor{ochsblau}{\textbf{Nachklausur Gedächtnisprotokoll}}\end{center}
-}
-
-\newcommand\refiff[1]{\stackrel{\text{#1}}{\iff}}
-\newcommand\refimp[1]{\stackrel{\text{#1}}{\implies}}
-\newcommand\refeq[1]{\stackrel{\text{#1}}{=}}
-\newcommand\refless[1]{\stackrel{\text{#1}}{<}}
-\newcommand\refleads[1]{\stackrel{\text{#1}}{\leadsto}}
-
-\newcommand\NR{\textbf{Nebenrechnung: }}
-\newcommand\proposition{\textbf{Behauptung: }}
-\newcommand\toprove{\textbf{Zu zeigen: }}
-\newcommand\task{\textbf{Aufgabe: }}
-\newcommand\defi{\textbf{Definition: }}
-
-% denglish ftw
-\newcommand\da{\text{ da }}
-\newcommand\with{\text{ mit }}
-\newcommand\und{\text{ und }}
-\newcommand\oder{\text{ oder }}
-\newcommand\for{\text{ für }}
-\newcommand\when{\text{ wenn }}
-\newcommand\sei{\text{ sei }}
-
-\newcommand\Real{\mathrm{Re}} % Realteil
-\newcommand\Imag{\mathrm{Im}} % Imaginärteil
-
-\newcommand\N{\mathbb{N}}
-\newcommand\R{\mathbb{R}}
-\newcommand\Z{\mathbb{Z}}
-\newcommand\C{\mathbb{C}}
-\newcommand\Q{\mathbb{Q}}
-\renewcommand\P{\mathbb{P}}
-\renewcommand\O{\mathcal{O}}
-\newcommand\pot{\mathcal{P}}
-
-\newcommand\rank{\mathrm{rank}}
-\newcommand\lin{\mathrm{Lin}}
-\renewcommand\det{\mathrm{det}}
-\renewcommand\dim{\mathrm{dim}}
-
-\renewcommand\u{\boldsymbol{u}}
-\renewcommand\v{\boldsymbol{v}}
-\newcommand\w{\boldsymbol{w}}
-
-% for vertical line in gmatrix
-\usepackage{etoolbox}
-\makeatletter
-\patchcmd\g@matrix
- {\vbox\bgroup}
- {\vbox\bgroup\normalbaselines}
- {}{}
-\makeatother
-\newcommand{\gvline}{%
- \hspace{-\arraycolsep}%
- \strut\vrule
- \hspace{-\arraycolsep}%
-}
-
-% lol
-\makeatletter
-\renewenvironment{proof}[1][\proofname] {\par\pushQED{\qed}\normalfont\topsep6\p@\@plus6\p@\relax\trivlist\item[\hskip\labelsep\bfseries#1\@addpunct{.}]\ignorespaces}{\popQED\endtrivlist\@endpefalse}
-\makeatother
-\def\qedsymbol{\sc q.e.d.} % hmm?
-
-\newenvironment{induktion}{\renewcommand*{\proofname}{Beweis durch vollständige Induktion}\begin{proof}$ $\newline}{\end{proof}}
-\newcommand\IA[1]{\textbf{Induktionsanfang ($#1$):}}
-\newcommand\IV[1]{\textbf{Induktionsvoraussetzung:} Die Behauptung gelte für ein beliebiges aber festes $#1$.\\}
-\newcommand\IS[1]{\textbf{Induktionsschritt ($#1$):}}
-
-\newenvironment{gegenbeweis}{\renewcommand*{\proofname}{Beweis durch Gegenbeweis}\begin{proof}}{\end{proof}}
-\newenvironment{gegenbeispiel}{\renewcommand*{\proofname}{Beweis durch Gegenbeispiel}\begin{proof}}{\end{proof}}
-
-\NewEnviron{splitty}{\begin{displaymath}\begin{split}\BODY\end{split}\end{displaymath}}
-\newcolumntype{C}{>{$}c<{$}} % math-mode column
-
-\DeclareMathOperator{\ggT}{ggT}
-
-\newlist{abc}{enumerate}{10}
-\setlist[abc]{label=(\alph*)}
-
-\newlist{num}{enumerate}{10}
-\setlist[num]{label=\arabic*.}
-
-\newlist{rom}{enumerate}{10}
-\setlist[rom]{label=(\roman*)}
-