Mercurial > 13ws.ds
annotate notes/tex/basics.tex @ 27:f52078f78e60
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| author | Markus Kaiser <markus.kaiser@in.tum.de> |
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| date | Mon, 25 Nov 2013 23:20:13 +0100 |
| parents | 4436f8006ebd |
| children | e65f4b1a6e32 |
| rev | line source |
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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. |
| 3 | 9 \[ A \defeq \left\{ a, b, \ldots, z \right\} \] |
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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} |
| 3 | 14 \item Reihenfolge ist \alert{egal} |
| 15 \item Elemente kommen \alert{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 $\left\{ a, b, c, a, c \right\} = \left\{ a, b, c \right\} = \left\{ c, a, b \right\}$ |
| 3 | 23 \item $\N \defeq \left\{ 1, 2, 3, \ldots \right\}$ |
| 24 \item $\emptyset \defeq \left\{ \right\}$ | |
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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. |
| 3 | 35 \[ M \defeq \left\{ x_1, x_2, x_3, \ldots \right\} \] |
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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} |
| 3 | 40 \item $A \defeq \left\{ 2, 4, 6, \ldots \right\}$ |
| 41 \item $B \defeq \left\{ 1, 2, 3, 4 \right\}$ = [4] | |
| 42 \item $C \defeq \left\{ 2, 3, 5, 7, 11, \ldots \right\}$ | |
| 43 \item $D \defeq \left\{ \alpha, a, \smiley, 8, \left\{ 1, 2 \right\}, \N \right\}$ | |
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44 \end{itemize} |
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45 \end{example} |
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46 \end{frame} |
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47 |
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48 \begin{frame} |
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49 \frametitle{Schreibweisen} |
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50 \setbeamercovered{dynamic} |
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51 |
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52 \begin{definition}[Intensionale Schreibweise] |
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53 Die \structure{intensionale Schreibweise} beschreibt eine Menge durch charakteristische Eigenschaften. |
| 3 | 54 \[ M \defeq \left\{ x \in \Omega \mid P(x) \right\} \] |
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55 $M$ enthält alle Elemente im \structure{Universum} $\Omega$ mit der Eigenschaft $P$. |
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56 \end{definition} |
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57 \vfill |
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58 \begin{example}[] |
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59 \begin{itemize} |
| 3 | 60 \item $A \defeq \left\{ 2, 4, 6, \ldots \right\} = \left\{ x \in \N \mid x\ \text{gerade} \right\} = \left\{ 2x : x \in \N \right\}$ |
| 61 \item $B \defeq \left\{ 1, 2, 3, 4 \right\} = \left\{ x \in \N \mid x \leq 4 \right\}$ | |
| 62 \item $C \defeq \left\{ 2, 3, 5, 7, 11, \ldots \right\} = \left\{ x \in \N \mid x\ \text{prim} \right\}$ | |
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63 \end{itemize} |
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64 \end{example} |
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65 \end{frame} |
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66 } |
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67 |
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68 \defineUnit{mengenoperationen}{% |
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69 \begin{frame} |
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70 \frametitle{Mengenoperationen} |
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71 \setbeamercovered{dynamic} |
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72 |
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73 \begin{block}{Bezeichnungen} |
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74 \begin{itemize} |
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75 \item Objekte in Mengen |
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76 \begin{description}[\qquad\qquad] |
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77 \item[$a \in A$] $a$ ist Element von $A$ |
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78 \item[$b \not\in A$] $b$ ist kein Element von $A$ |
| 3 | 79 \item[$\abs{A}$] Anzahl der Elemente in $A$, Kardinalität |
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80 \end{description} |
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81 \item Relationen zwischen Mengen |
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82 \begin{description}[\qquad\qquad] |
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83 \item[$B \subseteq A$] $B$ ist Teilmenge von $A$, \quad $x \in B \rightarrow x \in A$ |
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84 \item[$B \subset A$] $B$ ist echte Teilmenge von $A$ |
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85 \item[$B = A$] $B \subseteq A$ und $A \subseteq B$ |
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86 \end{description} |
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87 \end{itemize} |
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88 \end{block} |
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89 |
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90 \begin{example}[] |
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91 \begin{itemize} |
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92 \item $1 \in \left\{ 1, 2, 3, 4 \right\}$, aber $9 \not\in \left\{ 1, 2, 3, 4 \right\}$ |
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93 \item $\left\{ 1, 2 \right\} \subseteq \left\{ 1, 2, 3, 4 \right\}$, aber $\left\{ 1, 5 \right\} \not\subseteq \left\{ 1, 2 \right\}$ |
| 3 | 94 \item $\emptyset \subseteq [5] \subseteq \N \subseteq \N_0 \subseteq \Z \subseteq \Q \subseteq \R \subseteq \C$ |
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95 \end{itemize} |
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96 \end{example} |
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97 \end{frame} |
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98 |
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99 \begin{frame} |
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100 \frametitle{Mengenoperationen} |
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101 \setbeamercovered{dynamic} |
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102 |
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103 \begin{block}{Operationen} |
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104 \begin{description}[\qquad\qquad] |
| 3 | 105 \item[$\setnot{A}$] $\defeq \left\{ x \mid x \not\in A \right\}$\hfill\alert{Komplement} |
| 106 \item[$A \cup B$] $\defeq \left\{ x \mid x \in A\ \text{oder}\ x \in B \right\}$\hfill\alert{Vereinigung} | |
| 107 \item[$A \cap B$] $\defeq \left\{ x \mid x \in A\ \text{und}\ x \in B \right\}$\hfill\alert{Schnitt} | |
| 108 \item[$A \setminus B$] $\defeq A \cap \setnot{B}$\hfill\alert{Differenz} | |
| 109 \item[$A \setsymdiff B$] $\defeq \left( A \setminus B \right) \cup \left( B \setminus A \right)$\hfill\alert{Symmetrische Differenz} | |
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110 \end{description} |
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111 \end{block} |
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112 \vfill |
| 3 | 113 Für mehrere Mengen schreibt man |
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114 \begin{align} |
| 3 | 115 \bigcap_{i=1}^n A_i &\defeq A_1 \cap A_2 \cap \ldots \cap A_n\\ |
| 116 \bigcup_{i=1}^n A_i &\defeq A_1 \cup A_2 \cup \ldots \cup A_n | |
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117 \end{align} |
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118 \end{frame} |
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119 } |
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120 |
| 3 | 121 \defineUnit{venn}{% |
| 122 \begin{frame} | |
| 123 \frametitle{Venn-Diagramme} | |
| 124 \setbeamercovered{dynamic} | |
| 125 | |
| 126 \structure{Venn-Diagramme} visualisieren Mengen $A, B, \ldots$ im Universum $\Omega$. | |
| 127 | |
| 128 { | |
| 129 \def\universe{(-1.5, -1.25) rectangle (2.5, 1.25) node[anchor=north east, black] {$\Omega$}} | |
| 130 \def\first{(0, 0) circle (1)} | |
| 131 \def\second{(1, 0) circle (1)} | |
| 132 \tikzstyle{universe} = [draw, thick, tumblue, fill=tumlightblue!15] | |
| 133 \tikzstyle{inset} = [fill=tumred!35] | |
| 134 \tikzstyle{outline} = [draw, thick, black] | |
| 135 \begin{columns} | |
| 136 \begin{column}{.5\textwidth} | |
| 137 \begin{itemize} | |
| 138 \item $A \cup B$ | |
| 139 \begin{figure} | |
| 140 \begin{tikzpicture} | |
| 141 \draw[universe] \universe; | |
| 142 \fill[inset] \first; | |
| 143 \fill[inset] \second; | |
| 144 \draw[outline] \first node[left=1em] {$A$}; | |
| 145 \draw[outline] \second node[right=1em] {$B$}; | |
| 146 \end{tikzpicture} | |
| 147 \end{figure} | |
| 148 \item $A \setminus B$ | |
| 149 \begin{figure} | |
| 150 \begin{tikzpicture} | |
| 151 \draw[universe] \universe; | |
| 152 \begin{scope} | |
| 153 \clip \first; | |
| 154 \fill[inset, even odd rule] \first \second; | |
| 155 \end{scope} | |
| 156 \draw[outline] \first node[left=1em] {$A$}; | |
| 157 \draw[outline] \second node[right=1em] {$B$}; | |
| 158 \end{tikzpicture} | |
| 159 \end{figure} | |
| 160 \end{itemize} | |
| 161 \end{column} | |
| 162 \begin{column}{.5\textwidth} | |
| 163 \begin{itemize} | |
| 164 \item $A \cap B$ | |
| 165 \begin{figure} | |
| 166 \begin{tikzpicture} | |
| 167 \draw[universe] \universe; | |
| 168 \begin{scope} | |
| 169 \clip \first; | |
| 170 \fill[inset] \second; | |
| 171 \end{scope} | |
| 172 \draw[outline] \first node[left=1em] {$A$}; | |
| 173 \draw[outline] \second node[right=1em] {$B$}; | |
| 174 \end{tikzpicture} | |
| 175 \end{figure} | |
| 176 \item $A \setsymdiff B = (A \setminus B) \cup (B \setminus A)$ | |
| 177 \begin{figure} | |
| 178 \begin{tikzpicture} | |
| 179 \draw[universe] \universe; | |
| 180 \fill[inset, even odd rule] \first \second; | |
| 181 \draw[outline] \first node[left=1em] {$A$}; | |
| 182 \draw[outline] \second node[right=1em] {$B$}; | |
| 183 \end{tikzpicture} | |
| 184 \end{figure} | |
| 185 \end{itemize} | |
| 186 \end{column} | |
| 187 \end{columns} | |
| 188 } | |
| 189 \end{frame} | |
| 190 } | |
| 191 | |
| 192 \defineUnit{mengenrechenregeln}{% | |
| 193 \begin{frame} | |
| 194 \frametitle{Rechnen mit Mengen} | |
| 195 \setbeamercovered{dynamic} | |
| 196 | |
| 197 \begin{theorem}[De Morgansche Gesetze] | |
| 198 Sind $A, B$ Mengen, dann gilt | |
| 199 \begin{alignat}{2} | |
| 200 \setnot{A \cup B} &= \setnot{A} \cap \setnot{B} \qquad\qquad& \setnot{A \cap B} &= \setnot{A} \cup \setnot{B}\\ | |
| 201 \intertext{Für Mengen $A_i$ gilt} | |
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202 \setnot{\bigcup_{i=1}^nA_i} &= \bigcap_{i=1}^n\setnot{A_i} & \setnot{\bigcap_{i=1}^nA_i} &= \bigcup_{i=1}^n\setnot{A_i} |
| 3 | 203 \end{alignat} |
| 204 \end{theorem} | |
| 205 | |
| 206 \vfill | |
| 207 | |
| 208 \begin{itemize} | |
| 209 \item Zusammen mit $\setnot{\setnot{A}} = A$ wichtigste Regel | |
| 210 \item Gilt auch in der Aussagenlogik | |
| 211 \end{itemize} | |
| 212 \end{frame} | |
| 213 } | |
| 214 | |
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215 \defineUnit{potenzmenge}{% |
| 3 | 216 \begin{frame} |
| 217 \frametitle{Potenzmenge} | |
| 218 \setbeamercovered{dynamic} | |
| 219 | |
| 220 \begin{definition}[Potenzmenge] | |
| 221 Die \structure{Potenzmenge} $\powerset{M}$ zu einer Menge $M$ ist die Menge all ihrer Teilmengen. | |
| 222 \[ \powerset{M} \defeq \left\{ X \mid X \subseteq M \right\} \] | |
| 223 \end{definition} | |
| 224 | |
| 225 \begin{itemize} | |
| 226 \item $\powerset{M}$ enthält für endliche Mengen genau $2^{\abs{M}}$ Elemente | |
| 227 \item Man schreibt deshalb auch $2^M$ | |
| 228 \item Es ist $M \in \powerset{M}$ und $\emptyset \in \powerset{M}$ | |
| 229 \end{itemize} | |
| 230 | |
| 231 \vfill | |
| 232 | |
| 233 \begin{example}[] | |
| 234 Für $M = \left\{ a, b, c \right\}$ ist | |
| 6 | 235 \[ \powerset{M} = \left\{ |
| 3 | 236 \emptyset, |
| 237 \left\{ a \right\}, | |
| 238 \left\{ b \right\}, | |
| 239 \left\{ c \right\}, | |
| 240 \left\{ a, b \right\}, | |
| 241 \left\{ a, c \right\}, | |
| 242 \left\{ b, c \right\}, | |
| 243 \left\{ a, b, c \right\} | |
| 244 \right\} \] | |
| 245 mit $\abs{\powerset{M}} = 2^3 = 8$ | |
| 246 \end{example} | |
| 247 \end{frame} | |
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248 } |
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249 |
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250 \defineUnit{tupel}{% |
| 3 | 251 \begin{frame} |
| 252 \frametitle{Tupel} | |
| 253 \setbeamercovered{dynamic} | |
| 254 | |
| 255 \begin{definition}[Tupel] | |
| 256 Ein \structure{$n$-Tupel} ist eine \alert{geordnete} Sammlung $n$ \alert{beliebiger} Objekte.\\ | |
| 257 Mit \structure{Tupelklammern} werden Objekte zusammengefasst. | |
| 258 \[ T \defeq \left( t_1, t_2, \ldots, t_n \right)\] | |
| 259 \end{definition} | |
| 260 | |
| 261 \begin{itemize} | |
| 262 \item Reihenfolge \alert{nicht} egal | |
| 263 \item Elemente \alert{dürfen} mehrmals vorkommen | |
| 264 \end{itemize} | |
| 265 | |
| 266 \vfill | |
| 267 | |
| 268 \begin{example}[] | |
| 269 \begin{itemize} | |
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270 \item $\left( a, b, c \right) \neq \left( c, a, b \right) \neq \left( a, b, c, a, c \right)$ |
| 3 | 271 \item $\left( 1, 2, 3 \right) \neq \left\{ 3, 2, 1 \right\} = \left\{ 1, 2, 3 \right\}$ |
| 272 \item $\left( \left\{ \alpha, \beta \right\}, \emptyset, \N \right)$ | |
| 273 \end{itemize} | |
| 274 \end{example} | |
| 275 \end{frame} | |
| 276 | |
| 277 \begin{frame} | |
| 278 \frametitle{Kreuzprodukt} | |
| 279 \setbeamercovered{dynamic} | |
| 280 | |
| 281 \begin{definition}[Kreuzprodukt] | |
| 282 Sind $A, B$ Mengen, dann ist ihr \structure{kartesisches Produkt} (Kreuzprodukt) | |
| 283 \begin{align} | |
| 284 A \times B &\defeq \left\{ \left( a, b \right) \mid a \in A, b \in B \right\}\\ | |
| 285 \intertext{Für Mengen $A_i$ ist} | |
| 286 A_1 \times \ldots \times A_n &\defeq \left\{ \left(a_1, \ldots, a_n\right) \mid a_1 \in A_1, \ldots, a_n \in A_n \right\} | |
| 287 \end{align} | |
| 288 \end{definition} | |
| 289 | |
| 290 \begin{itemize} | |
| 291 \item Für endliche $A_i$ ist $\abs{A_1 \times \ldots \times A_n} = \abs{A_1} \cdot \ldots \cdot \abs{A_n}$ | |
| 292 \item Man schreibt \structure{$A^n \defeq \underbracket[0.5pt]{A \times \ldots \times A}_{\text{n mal}}$} mit $A^0 = \left\{ \emptyset \right\}$ | |
| 293 \end{itemize} | |
| 294 | |
| 295 \vfill | |
| 296 | |
| 297 \begin{example}[] | |
| 298 \begin{itemize} | |
| 299 \item $\left\{ 1, 2 \right\} \times \left\{ a, b \right\} = \left\{ (1, a), (2, a), (1, b), (2, b) \right\}$ | |
| 300 \item $\left\{ \alpha, \beta \right\}^2 = \left\{ (\alpha, \alpha), (\alpha, \beta), (\beta, \alpha), (\beta, \beta) \right\}$ | |
| 301 \end{itemize} | |
| 302 \end{example} | |
| 303 \end{frame} | |
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304 } |
| 6 | 305 |
| 306 \defineUnit{relationen}{% | |
| 307 \begin{frame} | |
| 308 \frametitle{Relation} | |
| 309 \setbeamercovered{dynamic} | |
| 310 | |
| 311 \begin{definition}[Relation] | |
| 312 Eine binäre \structure{Relation} $R$ verbindet Elemente zweier Mengen $A$ und $B$. | |
| 313 \[ R \subseteq A \times B\] | |
| 314 Ist $(a, b) \in R$, so schreibt man auch \structure{$a\rel{R}b$}. | |
| 315 \end{definition} | |
| 316 | |
| 317 \begin{itemize} | |
| 318 \item Eine Relation über $M \times M$ nennt man homogen | |
| 319 \item Es gibt $\abs{\powerset{A \times B}}$ Relationen über $A, B$ | |
| 320 \end{itemize} | |
| 321 | |
| 322 \begin{example}[] | |
| 323 \begin{itemize} | |
| 324 \item Die \alert{Gleichheitsrelation} über $\N \times \N$ \\ | |
| 325 $\left\{ (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7) \ldots \right\}$ | |
| 326 \medskip | |
| 327 \item Die \alert{Teilbarkeitsrelation} über $\N$ \\ | |
| 328 $\left\{ (1,1), (1,2), (1,3), \ldots, (2,2), (2,4), \ldots, (3,3), (3,6), \ldots \right\}$ | |
| 329 \end{itemize} | |
| 330 \end{example} | |
| 331 \end{frame} | |
| 332 | |
| 333 \begin{frame} | |
| 334 \frametitle{Grafische Darstellung} | |
| 335 \setbeamercovered{dynamic} | |
| 336 | |
| 337 \begin{block}{Grafische Darstellung von Relationen} | |
| 338 Jede Relation $R \subseteq M \times M$ kann als \structure{Graph} dargestellt werden. Die Elemente aus M werden zu \structure{Knoten} und für jedes Tupel $(a, b) \in R$ wird ein \structure{Pfeil} von $a$ nach $b$ eingefügt. | |
| 339 \end{block} | |
| 340 | |
| 341 \begin{example}[] | |
| 342 Sei $R \subseteq [4] \times [4]$ eine Relation über den natürlichen Zahlen. | |
| 343 \[ R \defeq \left\{ (1, 1), (1, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 3) \right\}\] | |
| 344 | |
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345 \centering |
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346 \begin{tikzpicture}[x=5em, y=2.5em] |
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347 \path (0, 0) node[pretty] (1) {$1$} |
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348 +(1, 0) node[pretty] (2) {$2$} |
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349 +(2, 1) node[pretty] (3) {$3$} |
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350 +(2, -1) node[pretty] (4) {$4$}; |
| 6 | 351 |
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352 \path[edge] |
| 6 | 353 (1) edge[loop left] (1) |
| 354 (1) edge (2) | |
| 355 (2) edge (3) | |
| 356 (2) edge (4) | |
| 357 (3) edge[loop above] (3) | |
| 358 (3) edge[bend left] (4) | |
| 359 (4) edge[bend left] (3); | |
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360 \end{tikzpicture} |
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361 \end{example} |
| 6 | 362 \end{frame} |
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363 } |
| 6 | 364 |
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365 \defineUnit{relationeneigenschaften}{% |
| 6 | 366 \begin{frame} |
| 367 \frametitle{Eigenschaften von Relationen} | |
| 368 \setbeamercovered{dynamic} | |
| 369 | |
| 370 \begin{block}{Eigenschaften homogener Relationen} | |
| 371 Sei $R \in M \times M$ eine homogene Relation. Man nennt $R$ | |
| 372 \begin{description}[antisymmetrisch] | |
| 373 \item[reflexiv] $ \forall a\hphantom{, b, c} \in M.\ (a, a) \in R$ | |
| 374 \item[total] $ \forall a, b\hphantom{, c} \in M.\ (a, b) \in R \vee (b, a) \in R$ | |
| 375 \medskip | |
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376 \item[symmetrisch] $ \forall a, b\hphantom{, c} \in M.\ (a, b) \in R \hphantom{{}\wedge (b, a) \in R}\rightarrow (b,a ) \in R$ |
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377 \item[asymmetrisch] $ \forall a, b\hphantom{, c} \in M.\ (a, b) \in R \hphantom{{}\wedge (b, a) \in R}\rightarrow (b,a ) \not\in R$ |
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378 \item[antisymmetrisch] $ \forall a, b\hphantom{, c} \in M.\ (a, b) \in R \wedge (b, a) \in R \rightarrow a \equiv b$ |
| 6 | 379 \medskip |
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380 \item[transitiv] $ \forall a, b, c \in M.\ (a, b) \in R \wedge (b, c) \in R \rightarrow (a, c) \in R$ |
| 6 | 381 \end{description} |
| 382 \end{block} | |
| 383 | |
| 384 \vfill | |
| 385 | |
| 386 \begin{itemize} | |
| 387 \item Jede totale Relation ist reflexiv | |
| 388 \item Jede asymmetrische Relation ist antisymmetrisch | |
| 389 \item \structure{Äquivalenzrelationen} sind reflexiv, symmetrisch und transitiv | |
| 390 \item $R^+$ ist die \structure{transitive Hülle}, $R^*$ die \structure{reflexive transitive Hülle} | |
| 391 \end{itemize} | |
| 392 \end{frame} | |
| 393 } | |
| 394 | |
| 395 \defineUnit{funktionen}{% | |
| 396 \begin{frame} | |
| 397 \frametitle{Funktion} | |
| 398 \setbeamercovered{dynamic} | |
| 399 | |
| 400 \begin{definition}[Funktion] | |
| 401 Eine Relation $f \subseteq A \times B$ ist eine \structure{Funktion von A nach B} wenn es für alle $a \in A$ genau ein Element $b \in B$ mit $a \rel{f} b$ gibt. | |
| 402 \[ \forall a \in A. \abs{\left\{ (a, b) \mid b \in B \right\}} \alert{=} 1 \] | |
| 403 Man schreibt | |
| 404 \begin{align} | |
| 405 f : A &\to B \\ | |
| 406 a &\mapsto f(a) = b | |
| 407 \end{align} | |
| 408 \structure{$A \to B$} bezeichnet die Menge aller Funktionen von $A$ nach $B$. | |
| 409 \end{definition} | |
| 410 | |
| 411 \vfill | |
| 412 \centering | |
| 413 { | |
| 414 \tikzstyle{set} = [draw, thick, tumgreen, fill=tumgreen!15] | |
| 415 \tikzstyle{element} = [thick] | |
| 416 \tikzstyle{arrow} = [thick, tumblue, shorten >=-.4em, shorten <=-.4em] | |
| 417 \begin{tikzpicture}[x=1.5em, y=1.5em] | |
| 418 \draw[set] (0, 0) ellipse (1 and 2); | |
| 419 \draw[set] (5, 0) ellipse (1 and 2); | |
| 420 | |
| 421 \path[element] | |
| 422 (0,0) | |
| 423 +(0, 1.5) node (a1) {$\times$} | |
| 424 +(0.2, 0.85) node (a2) {$\times$} | |
| 425 +(0.1, -0.6) node (a4) {$\times$} | |
| 426 +(-0.1, -1.5) node (a5) {$\times$}; | |
| 427 | |
| 428 \path[element] | |
| 429 (5,0) | |
| 430 +(0, 1.5) node (b1) {$\times$} | |
| 431 +(0.2, 0.85) node (b2) {$\times$} | |
| 432 +(-0.3, 0.0) node (b3) {$\times$} | |
| 433 +(0.1, -0.6) node (b4) {$\times$} | |
| 434 +(-0.1, -1.5) node (b5) {$\times$}; | |
| 435 | |
| 436 \path[arrow] | |
| 437 (a1) edge (b1) | |
| 438 (a2) edge (b5) | |
| 439 (a4) edge (b2) | |
| 440 (a5) edge (b5); | |
| 441 | |
| 442 \path | |
| 443 (0, -2.5) node {$A$} | |
| 444 (5, -2.5) node {$B$} | |
| 445 (2.5, -2.5) node {$f$}; | |
| 446 \end{tikzpicture} | |
| 447 } | |
| 448 \end{frame} | |
| 449 | |
| 450 \begin{frame} | |
| 451 \frametitle{Bild und Urbild} | |
| 452 \setbeamercovered{dynamic} | |
| 453 | |
| 454 \begin{definition}[Bild] | |
| 455 Sei $f : A \to B$ eine Funktion, $X \subseteq A$, $Y \subseteq B$, $b \in B$. Dann ist | |
| 456 \begin{align} | |
| 457 f(X) &\defeq \left\{ f(x) \mid x \in X \right\} \\ | |
| 458 \intertext{das \structure{Bild} der Menge $X$ unter $f$. Außerdem ist} | |
| 459 f^{-1}(b) &\defeq \left\{ a \mid a \in A, f(a) = b \right\} \\ | |
| 460 f^{-1}(Y) &\defeq \bigcup_{y \in Y} \left\{ f^{-1}(y) \right\} | |
| 461 \end{align} | |
| 462 das \structure{Urbild} des Elements $b$ und der Menge $Y$ unter $f$. | |
| 463 \end{definition} | |
| 464 | |
| 465 \vfill | |
| 466 | |
| 467 \begin{itemize} | |
| 468 \item Man nennt $A = f^{-1}(B)$ \structure{Urbild} oder \structure{Definitionsmenge} von $f$ | |
| 469 \item Man nennt $f(A) \subseteq B$ \structure{Bild} oder \structure{Wertemenge} von $f$ | |
| 470 \end{itemize} | |
| 471 \end{frame} | |
| 472 | |
| 473 \begin{frame} | |
| 474 \frametitle{Komposition} | |
| 475 \setbeamercovered{dynamic} | |
| 476 | |
| 477 \begin{definition}[Funktionskomposition] | |
| 478 Seien $f : B \to C$ und $g : A \to B$ Funktionen. Dann ist | |
| 479 \begin{align} | |
| 480 h : A &\to C \\ | |
| 481 a &\mapsto (f \circ g)(a) = f(g(a)) | |
| 482 \end{align} | |
| 483 die \structure{Komposition} der Funktionen $f$ und $g$.\\ | |
| 484 Man ließt $f \circ g$ als \enquote{f \structure{nach} g}. | |
| 485 \end{definition} | |
| 486 | |
| 487 \vfill | |
| 488 | |
| 489 Man definiert die Potenzierung von Funktionen ähnlich der Mengentheorie. | |
| 490 \begin{align} | |
| 491 f^0 &\defeq id\\ | |
| 492 f^n &\defeq \underbracket[0.5pt]{f \circ \ldots \circ f}_{\text{n mal}} | |
| 493 \end{align} | |
| 494 Dabei bezeichnet $id$ die \structure{Identität} mit $id(x) \defeq x$. | |
| 495 \end{frame} | |
| 496 | |
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497 { |
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498 \tikzstyle{set} = [draw, thick, tumgreen, fill=tumgreen!15] |
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499 \tikzstyle{element} = [thick] |
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500 \tikzstyle{head} = [draw, fill=tumblue!15, thick] |
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501 \tikzstyle{arrow} = [thick, tumblue, shorten >=-.4em, shorten <=-.4em] |
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502 \newcommand{\function}[3]{% |
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503 \draw[set] (0, 0) ellipse (1 and 2); |
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504 \draw[set] (4, 0) ellipse (1 and 2); |
| 6 | 505 |
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506 \path[element] |
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507 (0,0) |
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508 +(0, 1.5) node (a1) {$\times$} |
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509 +(0.2, 0.85) node (a2) {$\times$} |
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510 +(0.1, 0.3) node (a3) {$\times$} |
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511 +(-0.1, -0.1) node (a4) {$\times$} |
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512 +(-0.2, -0.7) node (a5) {##2} |
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513 +(-0.1, -1.5) node (a6) {##3}; |
| 6 | 514 |
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515 \path[element] |
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516 (4,0) |
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517 +(0, 1.5) node (b1) {$\times$} |
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518 +(0.2, 0.85) node (b2) {$\times$} |
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519 +(-0.3, 0.0) node (b3) {$\times$} |
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520 +(0.1, -0.6) node (b4) {$\times$} |
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521 +(-0.1, -1.5) node (b5) {$\times$}; |
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522 |
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523 \path |
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524 (0, -2.5) node {$A$} |
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525 (4, -2.5) node {$B$} |
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526 (2, -2.5) node {$f$} |
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527 (2, 3) node[head] {##1}; |
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528 } |
| 6 | 529 |
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530 \begin{frame} |
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531 \frametitle{Eigenschaften von Funktionen} |
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532 \setbeamercovered{dynamic} |
| 6 | 533 |
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534 \begin{block}{Eigenschaften von Funktionen} |
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535 Sei $f: A \to B$ eine Funktion. Man nennt $f$ |
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536 \begin{description}[surjektiv] |
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537 \item[injektiv] $\forall b \in B. \abs{f^{-1}(b)} \leq 1$ \hfill(Kein $b$ wird doppelt getroffen) |
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538 \item[surjektiv] $\forall b \in B. \abs{f^{-1}(b)} \geq 1$\hfill(Jedes $b$ wird getroffen) |
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539 \item[bijektiv] $\forall b \in B. \abs{f^{-1}(b)} = 1$\hfill(Jedes $b$ wird genau einmal getroffen) |
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540 \end{description} |
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541 \end{block} |
| 6 | 542 |
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543 \vfill |
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544 \centering |
| 6 | 545 \begin{tikzpicture}[x=1.5em, y=1.5em] |
| 546 \function{Injektiv}{}{} | |
| 547 | |
| 548 \path[arrow] | |
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549 (a1) edge (b1) |
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550 (a2) edge (b3) |
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551 (a3) edge (b2) |
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552 (a4) edge (b4); |
| 6 | 553 \end{tikzpicture} |
| 554 \hfill | |
| 555 \begin{tikzpicture}[x=1.5em, y=1.5em] | |
| 556 \function{Surjektiv}{$\times$}{$\times$} | |
| 557 | |
| 558 \path[arrow] | |
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559 (a1) edge (b1) |
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560 (a2) edge (b3) |
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561 (a3) edge (b2) |
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562 (a4) edge (b4) |
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563 (a5) edge (b5) |
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564 (a6) edge (b5); |
| 6 | 565 \end{tikzpicture} |
| 566 \hfill | |
| 567 \begin{tikzpicture}[x=1.5em, y=1.5em] | |
| 568 \function{Bijektiv}{}{$\times$} | |
| 569 | |
| 570 \path[arrow] | |
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571 (a1) edge (b1) |
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572 (a2) edge (b3) |
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573 (a3) edge (b2) |
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574 (a4) edge (b4) |
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575 (a6) edge (b5); |
| 6 | 576 \end{tikzpicture} |
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577 \end{frame} |
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578 } |
| 6 | 579 } |