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gitignore, Nummerierung kapitel/sections angepasst
gitignore für temp files, NUmmerierungen angepasst
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.gitattributes
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# Auto detect text files and perform LF normalization
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* text=auto
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# Custom for Visual Studio
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*.cs diff=csharp
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# Standard to msysgit
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*.doc diff=astextplain
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*.docx diff=astextplain
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*.DOCX diff=astextplain
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*.dot diff=astextplain
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*.DOT diff=astextplain
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*.pdf diff=astextplain
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*.PDF diff=astextplain
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*.rtf diff=astextplain
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.gitignore
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## Core latex/pdflatex auxiliary files:
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*.aux
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*.dvi
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*-converted-to.*
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# these rules might exclude image files for figures etc.
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# *.ps
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## Generated if empty string is given at "Please type another file name for output:"
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.pdf
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*.bbl
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## Build tool auxiliary files:
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## Auxiliary and intermediate files from other packages:
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*.mp
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*.t[1-9]
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#(r)(e)ledmac/(r)(e)ledpar
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*.?end
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*.[1-9]
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*-concordance.tex
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# TODO Comment the next line if you want to keep your tikz graphics files
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# minted
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_minted*
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*.pyg
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# morewrites
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*.mw
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*.wrt
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sympy-plots-for-*.tex/
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pythontex-files-*/
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# thmtools
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# TikZ & PGF
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*.dpth
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*.auxlock
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# easy-todo
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*.lod
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# xindy
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# endfloat
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*.ttt
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# Latexian
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TSWLatexianTemp*
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## Editors:
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# WinEdt
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*.bak
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*.sav
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# Texpad
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.texpadtmp
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# Kile
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*.backup
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# KBibTeX
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*~[0-9]*
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# auto folder when using emacs and auctex
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/auto/*
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@ -44,7 +44,7 @@ $\{1;2;3\} = \{3;2;1\} = \{1;2;2;3\}$ \\
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\end{itemize}
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\end{exmpn}
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\paragraph{Bildung und Verknüpfung von Aussagen}
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\section*{Bildung und Verknüpfung von Aussagen}
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\begin{tabular}{|c|c|c|c|c|c|c|}
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\hline
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@ -105,7 +105,7 @@ Aussageform $A(X)$ sei f\"ur jedes $X \in M$ Aussage: neue Aussage mittels Quant
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\end{itemize}
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\end{satz}
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\paragraph{Bildung von Mengen}
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\section*{Bildung von Mengen}
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Seien $M$ und $N$ Mengen
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\begin{compactitem}
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\item Aufzählung der Elemente: $\{1;2;3\}$
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\item leere Menge ist immer Teilmenge jeder Menge $M$
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\item \textbf{Warnung:} $\{\emptyset\} \neq \emptyset$
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\end{compactitem}
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\item Verkn\"upfung von Mengen wie bei Aussagen
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\item Verknüpfung von Mengen wie bei Aussagen
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\end{compactitem}
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\begin{mydef}[Mengensystem]
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Axiome $\to$ Beweise $\to$ Sätze ("neue" wahre Aussagen) \\
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$\to$ ergibt Ansammlung (Menge) wahrer Aussagen
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\paragraph{Formulierung mathematischer Aussagen}
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\paragraph*{Formulierung mathematischer Aussagen}
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\begin{compactitem}
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\item typische Form eines mathematischen Satzes: "Wenn A gilt, dann gilt auch B."
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\item formal: $A \Rightarrow B$ bzw. $A(X) \Rightarrow B(X)$ ist stets wahr (insbesondere falls
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A wahr ist)
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\end{compactitem}
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$\newline$
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Beispiel
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\begin{compactitem}
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\item $X \in \mathbb N$ und ist durch 4 teilbar $\Rightarrow X$ ist durch 2 teilbar
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\Rightarrow \lnot A$
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\end{compactitem}
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\subsection{Relationen und Funktionen}
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\section{Relationen und Funktionen}
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\begin{mydef}[Relation]
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Seien $M$ und $N$ Mengen. Dann ist jede Teilmenge $R$ von
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$M \times N$ eine Relation. \\
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$(x,y) \in R$ hei{\ss}t: $x$ und $y$ stehen in Relation zueinander
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\end{mydef}
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Beispiele
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\begin{exmp}
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$M$ ist die Menge aller Menschen. Die Liebesbeziehung $x$ liebt $y$ sieht als geordnetes Paar
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geschrieben so aus: $(x,y)$. Das hei{\ss}t die Menge der Liebespaare ist das: $L := \{(x,y) \mid
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$\to I: M_2 \to M_2$ mit $I(k)=2k \quad \forall k \in \mathbb N$ \\
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$\to$ Isomorphismus, der die Addition, Ordnung und die Null, aber nicht die Multiplikation erh\"alt
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\subsection{Bemerkungen zum Fundament der Mathematik}
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\subsection*{Bemerkungen zum Fundament der Mathematik}
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Forderungen an eine mathematische Theorie:
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\begin{compactitem}
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\item widerspruchsfrei: Satz und Negation nicht gleichzeitig herleitbar
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Für jedes feste $n \in \mathbb N$ sind beide Definitionen rekursiv und eindeutig definiert. \\
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$\forall n \in \mathbb N$ gilt: $n+1=n+\nu(0)=\nu(n+0) = \nu(n)$
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\begin{framed}
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\textbf{Satz:} Addition und Multiplikation haben folgende Eigenschaften:
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\begin{satz}
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Addition und Multiplikation haben folgende Eigenschaften:
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\begin{compactitem}
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\item es existiert jeweils ein neutrales Element
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\item kommutativ
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\item assoziativ
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\item distributiv
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\end{compactitem}
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\end{framed}
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$\newline$
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\end{satz}
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Es gilt $\forall k,m,n \in \mathbb N$:
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\begin{compactitem}
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\item $m+k=n+k \Rightarrow m=n$ (Kürzungsregel der Addition)
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\item $m \cdot k=n \cdot k \Rightarrow m=n$ (Kürzungsregel der Multiplikation)
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\end{compactitem}
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$\newline$
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Ordnung auf $\mathbb N:$ Relation $R := \{(m,n) \in \mathbb N \times \mathbb N \mid m \le n\}$ \\
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wobei $m \le n \iff n=m+k$ f\"ur ein $k \in \mathbb N$ \\
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ANAG/main.pdf
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ANAG/main.pdf
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\newcommand{\dotsco}{,\dots,} % ,...,
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\newcommand{\expon}{\mathrm{exp}}
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\setlength\parindent{0pt} % noindent whole file!
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\begin{document}
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\title{\textbf{Analysis 1. Semester (WS2017/18)}}
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