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add basic theme; first slides
author Markus Kaiser <markus.kaiser@in.tum.de>
date Sun, 20 Oct 2013 16:30:48 +0200
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children ead11e11a950
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1 \defineUnit{mengen}{%
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2 \begin{frame}
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3 \frametitle{Mengen}
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4 \setbeamercovered{dynamic}
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5
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6 \begin{definition}[Menge]
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7 Eine \structure{Menge} ist eine \alert{ungeordnete} Sammlung \alert{unterscheidbarer} Objekte.\\
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8 Mit \structure{Mengenklammern} werden Objekte zusammengefasst.
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9 \[ A := \left\{ a, b, \ldots, z \right\} \]
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10 Man nennt $a$ ein \structure{Element} von $A$, es gilt $a \in A$.
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11 \end{definition}
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12
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13 \begin{itemize}
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14 \item Reihenfolge ist egal
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15 \item Elemente kommen nicht mehrfach vor
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16 \end{itemize}
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17
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18 \vfill
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19
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20 \begin{example}[]
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21 \begin{itemize}
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22 \item $\left\{ a, b, c, a, c \right\} = \left\{ a, b, c \right\} = \left\{ c, a, b \right\}$
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23 \item $\N := \left\{ 1, 2, 3, \ldots \right\}$
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24 \item $\emptyset := \left\{ \right\}$
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25 \end{itemize}
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26 \end{example}
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27 \end{frame}
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28
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29 \begin{frame}
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30 \frametitle{Schreibweisen}
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31 \setbeamercovered{dynamic}
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32
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33 \begin{definition}[Extensionale Schreibweise]
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34 Die \structure{extensionale Schreibweise} einer Menge zählt ihre Elemente auf.
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35 \[ M := \left\{ x_1, x_2, x_3, \ldots \right\} \]
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36 \end{definition}
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37 \vfill
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38 \begin{example}[]
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39 \begin{itemize}
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40 \item $A := \left\{ 2, 4, 6, \ldots \right\}$
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41 \item $B := \left\{ 1, 2, 3, 4 \right\}$
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42 \item $C := \left\{ 2, 3, 5, 7, 11, \ldots \right\}$
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43 \end{itemize}
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44 \end{example}
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45 \end{frame}
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46
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47 \begin{frame}
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48 \frametitle{Schreibweisen}
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49 \setbeamercovered{dynamic}
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50
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51 \begin{definition}[Intensionale Schreibweise]
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52 Die \structure{intensionale Schreibweise} beschreibt eine Menge durch charakteristische Eigenschaften.
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53 \[ M := \left\{ x \in \Omega \mid P(x) \right\} \]
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54 $M$ enthält alle Elemente im \structure{Universum} $\Omega$ mit der Eigenschaft $P$.
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55 \end{definition}
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56 \vfill
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57 \begin{example}[]
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58 \begin{itemize}
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59 \item $A := \left\{ 2, 4, 6, \ldots \right\} = \left\{ x \in \N \mid x\ \text{gerade} \right\} = \left\{ 2x : x \in \N \right\}$
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60 \item $B := \left\{ 1, 2, 3, 4 \right\} = \left\{ x \in \N \mid x \leq 4 \right\}$
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61 \item $C := \left\{ 2, 3, 5, 7, 11, \ldots \right\} = \left\{ x \in \N \mid x\ \text{prim} \right\}$
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62 \end{itemize}
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63 \end{example}
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64 \end{frame}
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65 }
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66
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67 \defineUnit{mengenoperationen}{%
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68 \begin{frame}
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69 \frametitle{Mengenoperationen}
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70 \setbeamercovered{dynamic}
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71
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72 \begin{block}{Bezeichnungen}
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73 \begin{itemize}
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74 \item Objekte in Mengen
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75 \begin{description}[\qquad\qquad]
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76 \item[$a \in A$] $a$ ist Element von $A$
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77 \item[$b \not\in A$] $b$ ist kein Element von $A$
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78 \end{description}
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79 \item Relationen zwischen Mengen
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80 \begin{description}[\qquad\qquad]
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81 \item[$B \subseteq A$] $B$ ist Teilmenge von $A$, \quad $x \in B \Rightarrow x \in A$
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82 \item[$B \subset A$] $B$ ist echte Teilmenge von $A$
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83 \item[$B = A$] $B \subseteq A$ und $A \subseteq B$
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84 \end{description}
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85 \end{itemize}
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86 \end{block}
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87
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88 \begin{example}[]
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89 \begin{itemize}
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90 \item $1 \in \left\{ 1, 2, 3, 4 \right\}$, aber $9 \not\in \left\{ 1, 2, 3, 4 \right\}$
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91 \item $\left\{ 1, 2 \right\} \subseteq \left\{ 1, 2, 3, 4 \right\}$, aber $\left\{ 1, 5 \right\} \not\subseteq \left\{ 1, 2 \right\}$
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92 \item $\emptyset \subseteq \N \subseteq \N_0 \subseteq \Z \subseteq \Q \subseteq \R \subseteq \C$
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93 \end{itemize}
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94 \end{example}
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95 \end{frame}
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96
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97 \begin{frame}
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98 \frametitle{Mengenoperationen}
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99 \setbeamercovered{dynamic}
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100
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101 \begin{block}{Operationen}
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102 \begin{description}[\qquad\qquad]
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103 \item[$\setnot{A}$] $:= \left\{ x \mid x \not\in A \right\}$\hfill\alert{Komplement}
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104 \item[$A \cup B$] $:= \left\{ x \mid x \in A\ \text{oder}\ x \in B \right\}$\hfill\alert{Vereinigung}
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105 \item[$A \cap B$] $:= \left\{ x \mid x \in A\ \text{und}\ x \in B \right\}$\hfill\alert{Schnitt}
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106 \item[$A \setminus B$] $:= A \cap \setnot{B}$\hfill\alert{Differenz}
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107 \item[$A \setsymdiff B$] $:= \left( A \setminus B \right) \cup \left( B \setminus A \right)$\hfill\alert{Symmetrische Differenz}
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108 \end{description}
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109 \end{block}
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110 \vill
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111 Für mehrere Mengen schreibt man:
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112 \begin{align}
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113 \bigcap_{i=1}^n A_i &:= A_1 \cap A_2 \cap \ldots \cap A_n\\
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114 \bigcup_{i=1}^n A_i &:= A_1 \cup A_2 \cup \ldots \cup A_n
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115 \end{align}
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116 \end{frame}
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117 }
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118
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119 \defineUnit{potenzmenge}{%
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120 content
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121 }
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122
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123 \defineUnit{tupel}{%
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124 content
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125 }