Mercurial > 13ws.ds
annotate notes/tex/basics.tex @ 3:ead11e11a950
finish first slides
| author | Markus Kaiser <markus.kaiser@in.tum.de> |
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| date | Mon, 21 Oct 2013 20:29:08 +0200 |
| parents | ae52d9ffef38 |
| children | fac222767cda |
| 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 \vill |
| 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} | |
| 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} & | |
| 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 | |
| 235 \[ \powerset{M} = \left\{ | |
| 236 \emptyset, | |
| 237 \left\{ a \right\}, | |
| 238 \left\{ b \right\}, | |
| 239 \left\{ c \right\}, | |
| 240 \left\{ a, b \right\}, | |
| 241 \left\{ a, c \right\}, | |
| 242 \left\{ b, c \right\}, | |
| 243 \left\{ a, b, c \right\} | |
| 244 \right\} \] | |
| 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} | |
| 270 \item \left( a, b, c \right) \neq \left( c, a, b \right) \neq $\left( a, b, c, a, c \right)$ | |
| 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 } |