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#LyX 1.6.4 created this file. For more info see http://www.lyx.org/
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Hinweis:
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Dies ist eine
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inoffizielle
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, vorläufige Zusammenfassung des Vorlesungsstoffs.
Aktuelle Mitschriften, eine Fehlermeldeseite sowie Updates gibt es im MedienInf
o-Wiki unseres Semesters:
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Cheatsheet
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Analysis für Informatiker und Statistiker
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WS 2009/2010
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Dozent: Dr.
Walter Spann
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Satz: Das Analysis TeX Team
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Beteiligte:
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LaTeX-Satz: Marc Winking, Sebastian Straub, Frederic Kettelhoit, Sascha
Böckmann
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Grafiken: Elisabeth Friedrich, Harald Brunner
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Cheatsheet: Thomas Gorny
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Stand: §0 bis einschließlich §2
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Grundlagen
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Aussagenlogik
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Aussagenlogik
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Mengenoperationen
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von
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ist gleichbedeutend mit
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übliche Ergänzung -
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\end_inset
Ex falso quodlibet
\begin_inset Quotes grd
\end_inset
,
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $f$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $f$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $w$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
also
\begin_inset Quotes gld
\end_inset
aus Falschem folgt Beliebiges
\begin_inset Quotes grd
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Implikation"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Vorrangregel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Vorrangregel
\end_layout
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\underset{\mbox{abnehmender Vorrang}}{\underrightarrow{\begin{array}{cccc}
& & & \Leftrightarrow\\
\neg & \wedge & \vee & \Rightarrow\end{array}}}\]
\end_inset
\end_layout
\begin_layout Section
Gesetze der Aussagenlogik
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Gesetzte der Aussagenlogik
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features>
<column alignment="right" valignment="top" width="0">
<column alignment="right" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<row interlinespace="0.5em">
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{l}
\mbox{(a)}\\
\mbox{(b)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\left.\begin{array}{rcl}
\mathcal{A}\vee\mathcal{B} & \Leftrightarrow & \mathcal{B}\vee\mathcal{A}\\
\mathcal{A}\wedge\mathcal{B} & \Leftrightarrow & \mathcal{B}\wedge\mathcal{A}\end{array}\right\} $
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Kommutativgesetze
\begin_inset Index
status open
\begin_layout Plain Layout
Kommutativgesetze, Aussagenlogik
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row interlinespace="0.5em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{c}
\mbox{(c)}\\
\mbox{(d)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\left.\begin{array}{rcl}
\left(\mathcal{A}\vee\mathcal{B}\right)\vee\mathcal{C} & \Leftrightarrow & \mathcal{A}\vee\left(\mathcal{B}\vee\mathcal{C}\right)\\
\left(\mathcal{A}\wedge\mathcal{B}\right)\wedge\mathcal{C} & \Leftrightarrow & \mathcal{A}\wedge\left(\mathcal{B}\wedge\mathcal{C}\right)\end{array}\right\} $
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Assoziativgesetze
\begin_inset Index
status open
\begin_layout Plain Layout
Assoziativgesetze, Aussagenlogik
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row interlinespace="0.5em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{c}
\mbox{(e)}\\
\mbox{(f)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\left.\begin{array}{rcl}
\left(\mathcal{A}\vee\mathcal{B}\right)\vee\mathcal{C} & \Leftrightarrow & \left(\mathcal{A}\wedge\mathcal{C}\right)\vee\left(\mathcal{B}\wedge\mathcal{C}\right)\\
\left(\mathcal{A}\wedge\mathcal{B}\right)\wedge\mathcal{C} & \Leftrightarrow & \left(\mathcal{A}\vee\mathcal{C}\right)\wedge\left(\mathcal{B}\vee\mathcal{C}\right)\end{array}\right\} $
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Distributivgesetze
\begin_inset Index
status open
\begin_layout Plain Layout
Distributivgesetze, Aussagenlogik
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row interlinespace="0.5em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{c}
\mbox{(g)}\\
\mbox{(h)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\left.\begin{array}{rcl}
\neg\left(\mathcal{A}\vee\mathcal{B}\right) & \Leftrightarrow & \neg\mathcal{A}\wedge\neg\mathcal{B}\\
\neg(\mathcal{A}\wedge\mathcal{B}) & \Leftrightarrow & \neg\mathcal{A}\vee\neg\mathcal{B}\end{array}\right\} $
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\noun on
DeMorgansche
\noun default
Regeln
\begin_inset Index
status open
\begin_layout Plain Layout
DeMorgansche Regeln
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row interlinespace="0.5em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\begin{array}{c}
\mbox{(i)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\begin{array}{rcl}
\neg\neg\mathcal{A} & \Leftrightarrow & \mathcal{A}\end{array}$
\end_inset
\begin_inset space \quad{}
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Doppelte Verneinung
\begin_inset Index
status open
\begin_layout Plain Layout
Doppelte Verneinung
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Gesetze der Aussagenlogik"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Schlussweisen
\begin_inset Index
status open
\begin_layout Plain Layout
Schlussweisen, allgemeingültig
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\begin{array}{rrcl}
\mbox{(a)} & \left(\mathcal{A}\Rightarrow\mathcal{B}\right)\wedge\left(\mathcal{B}\Rightarrow\mathcal{A}\right) & \Rightarrow & \left(\mathcal{A}\Leftrightarrow\mathcal{B}\right)\\
\mbox{(b)} & \left(\mathcal{A}\Leftrightarrow\mathcal{B}\right)\wedge\left(\mathcal{B}\Leftrightarrow\mathcal{C}\right) & \Rightarrow & \left(\mathcal{A}\Leftrightarrow\mathcal{C}\right)\\
\mbox{(c)} & \left(\mathcal{A}\Rightarrow\mathcal{B}\right)\wedge\left(\mathcal{B}\Rightarrow\mathcal{C}\right) & \Rightarrow & \left(\mathcal{A}\Rightarrow\mathcal{C}\right)\end{array}\]
\end_inset
\end_layout
\begin_layout Section
Allgemeingültige Aussagen
\begin_inset Index
status open
\begin_layout Plain Layout
Allgemeingültige Aussage
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="5">
<features>
<column alignment="right" valignment="top" width="0">
<column alignment="right" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<row>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(a)
\end_layout
\end_inset
</cell>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\mathcal{A}\Rightarrow\mathcal{B}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\Leftrightarrow$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\neg\mathcal{A}\vee\mathcal{B}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(b)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $(\mathcal{A}\Rightarrow\mathcal{B})$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\Leftrightarrow$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\neg\mathcal{B}\Rightarrow\neg\mathcal{A}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Kontrapositionsgesetz
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Kontrapositionsgesetz
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Schlussweisen Kontrapositionsgesetz"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Formula $\mathcal{A}\vee\mathcal{A}\Leftrightarrow\mathcal{A}$
\end_inset
,
\begin_inset Formula $\mathcal{A}\wedge\mathcal{A}\Leftrightarrow\mathcal{A}$
\end_inset
,
\begin_inset Formula $\mathcal{A}\vee f\Leftrightarrow\mathcal{A}$
\end_inset
,
\begin_inset Formula $\mathcal{A}\wedge f\Leftrightarrow f$
\end_inset
\end_layout
\begin_layout Subsection*
Quantoren
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Quantoren
\end_layout
\end_inset
\begin_inset Formula \[
\neg\left(\forall x\in M:\,\mathcal{A}\left(x\right)\right)\Leftrightarrow\exists x\in M:\,\neg\mathcal{A}\left(x\right)\text{\qquad(\#)}\]
\end_inset
\begin_inset Formula \[
\neg\left(\exists x\in M:\,\mathcal{A}\left(x\right)\right)\Leftrightarrow\forall x\in M:\,\neg\mathcal{A}\left(x\right)\qquad\text{(\#\#)}\]
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Formula $\forall x\in M\,\forall y\in N:\,\mathcal{A}\left(x,\, y\right)\Leftrightarrow\forall y\in N\,\forall x\in M:\,\mathcal{A}\left(x,\, y\right)$
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Formula $\forall x\in M:\,\mathcal{A}\left(x\right):\Leftrightarrow\forall x:\, x\in M\Rightarrow\mathcal{A}\left(x\right)$
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Formula $\exists x\in M:\,\mathcal{A}\left(x\right):\Leftrightarrow\exists x:\, x\in M\wedge\mathcal{A}\left(x\right)$
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Formula $\neg\forall x:\,\mathcal{A}\left(x\right)\Leftrightarrow\exists x:\,\neg\mathcal{A}\left(x\right)$
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Formula $\neg\exists x:\,\mathcal{A}\left(x\right)\Leftrightarrow\forall x:\,\neg\mathcal{A}\left(x\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\begin{split}\forall x:\,(A\left(x\right)\wedge\mathcal{B})\Leftrightarrow\left(\forall x:\,\mathcal{A}(x)\right)\wedge\mathcal{B}\\
\forall x:\,(A\left(x\right)\vee\mathcal{B})\Leftrightarrow\left(\forall x:\,\mathcal{A}(x)\right)\vee\mathcal{B}\\
\exists x:\,(A\left(x\right)\wedge\mathcal{B})\Leftrightarrow\left(\exists x:\,\mathcal{A}(x)\right)\wedge\mathcal{B}\\
\exists x:\,(A\left(x\right)\vee\mathcal{B})\Leftrightarrow\left(\exists x:\,\mathcal{A}(x)\right)\vee\mathcal{B}\end{split}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\begin{split}\forall x:(A\left(x\right)\wedge\mathcal{B}\left(x\right)) & \Leftrightarrow\left(\forall x:\,\mathcal{A}\left(x\right)\right)\wedge\left(\forall x:\,\mathcal{B}\left(x\right)\right)\\
\exists x:(A\left(x\right)\vee\mathcal{B}\left(x\right)) & \Leftrightarrow\left(\exists x:\,\mathcal{A}\left(x\right)\right)\vee\left(\exists x:\,\mathcal{B}\left(x\right)\right)\end{split}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
Nicht allgemeingültig
\emph default
sind
\begin_inset Formula \[
\begin{split}\forall x:\,(A(x)\vee\mathcal{B}(x) & \Leftrightarrow\left(\forall x:\,\mathcal{A}(x)\right)\vee\left(\forall x:\,\mathcal{B}(x)\right)\\
\exists x:\,(A(x)\wedge\mathcal{B}(x) & \Leftrightarrow\left(\exists x:\,\mathcal{A}(x)\right)\wedge\left(\exists x:\,\mathcal{B}(x)\right)\end{split}
\]
\end_inset
\end_layout
\begin_layout Section
Mengenoperationen
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="5">
<features>
<column alignment="left" valignment="top" width="0">
<column alignment="right" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(a)
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $A=B$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $:\Leftrightarrow$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\forall x:\, x\in A\Leftrightarrow x\in B$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Mengengleichheit formal
\end_layout
\end_inset
Mengengleichheit formal
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(b)
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $A\subset B$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $:\Leftrightarrow$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\forall x:\, x\in A\Rightarrow x\in B$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $A$
\end_inset
Teilmenge
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Teilmenge
\end_layout
\end_inset
von
\begin_inset Formula $B$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(c)
\end_layout
\end_inset
</cell>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\mathcal{P}(A)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $:=$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\left\{ M:\, M\subset A\right\} $
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Potenzmenge
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Potenzmenge
\end_layout
\end_inset
von
\begin_inset Formula $A$
\end_inset
: Menge aller Teilmengen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Menge aller Teilmengen
\end_layout
\end_inset
von
\begin_inset Formula $A$
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Folgerung (Mengenoperationen)
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Mengenoperationen
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\begin{array}{rrcl}
\mbox{(a)} & A\subset B\wedge B\subset A & \Leftrightarrow & A=B\\
\mbox{(b)} & A\subset B\wedge B\subset C & \Leftrightarrow & A\subset C\end{array}\]
\end_inset
\end_layout
\begin_layout Section
Regeln (Mengenoperationen)
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features>
<column alignment="right" valignment="top" width="0">
<column alignment="right" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<row interlinespace="0.5em">
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{l}
\mbox{(a)}\\
\mbox{(b)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\left.\begin{array}{rcl}
A\cup B & = & B\cup A\\
A\cap B & = & B\cap A\end{array}\right\} $
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Kommutativgesetz
\begin_inset Index
status open
\begin_layout Plain Layout
Kommutativgesetz, Mengen
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row interlinespace="0.5em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{c}
\mbox{(c)}\\
\mbox{(d)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\left.\begin{array}{rcl}
(A\cup B)\cup C & = & A\cup(B\cup C)\\
(A\cap B)\cap C & = & A\cap(B\cap C)\end{array}\right\} $
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Assoziativgesetz
\begin_inset Index
status open
\begin_layout Plain Layout
Assoziativgesetz, Mengen
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row interlinespace="0.5em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\begin{array}{c}
\mbox{(e)}\\
\mbox{(f)}\end{array}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\left.\begin{array}{rcl}
(A\cup B)\cap C & = & (A\cap C)\cup(B\cap C)\\
(A\cap B)\cup C & = & (A\cup C)\cap(B\cup C)\end{array}\right\} $
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Distributivgesetz
\begin_inset Index
status open
\begin_layout Plain Layout
Distributivgesetz, Mengen
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Gesetze der Mengenoperationen"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Formula $A\cup A=A$
\end_inset
,
\begin_inset Formula $A\cap A=A$
\end_inset
,
\begin_inset Formula $A\cup\emptyset=A$
\end_inset
,
\begin_inset Formula $A\cap\emptyset=\emptyset$
\end_inset
\end_layout
\begin_layout Section
Satz (Mengenoperationen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Mengenoperationen
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\begin{array}{rrcl}
\mbox{(a)} & M\setminus\left(A\cup B\right) & = & \left(M\setminus A\right)\cap\left(M\setminus B\right)\\
\mbox{(b)} & M\setminus\left(A\cap B\right) & = & \left(M\setminus A\right)\cup\left(M\setminus B\right)\\
\mbox{(c)} & M\setminus\left(M\setminus A\right) & = & M\cap A\end{array}\]
\end_inset
\end_layout
\begin_layout Section
Folgerung (Complement)
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Complement
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $A,\, B,\, M$
\end_inset
Mengen,
\begin_inset Formula $A\subset M$
\end_inset
,
\begin_inset Formula $B\subset M$
\end_inset
.
\begin_inset Formula $\complement_{M}A:=M\setminus A,\,\complement_{M}B:=M\setminus B$
\end_inset
\begin_inset Formula \[
\begin{array}{rrcl}
\mbox{(a)} & \complement_{M}\left(A\cup B\right) & = & \complement_{M}A\cap\complement_{M}B\\
\mbox{(b)} & \complement_{M}\left(A\cap B\right) & = & \complement_{M}A\cup\complement_{M}B\\
\mbox{(c)} & \complement_{M}\complement_{M}A & = & A\end{array}\]
\end_inset
\end_layout
\begin_layout Section
Definition (Indexmengen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Indexmengen
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $I$
\end_inset
eine nicht leere (Index-)Menge und
\begin_inset Formula $A_{i}$
\end_inset
für jede
\begin_inset Formula $i\in I$
\end_inset
eine Menge.
Dann
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="3">
<features>
<column alignment="left" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<row interlinespace="1em">
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(a)
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\underset{\mbox{Menge aller \ensuremath{x}, die in mind. einem \ensuremath{A_{i}}enthalten sind}}{\underset{i\in I}{\bigcup}A_{i}:=\underbrace{\{x:\exists i\in I:x\in A_{i}\}}}$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Vereinigung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Indexmengen, Vereinigung
\end_layout
\end_inset
der Mengen
\begin_inset Formula $A_{i}(i\in I)$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
(b)
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\underset{\mbox{Menge aller \ensuremath{x}, die in jedem \ensuremath{A_{i}}enthalten sind.}}{\underset{i\in I}{\bigcap}A_{i}:=\underbrace{\{x:\forall i\in I:x\in A_{i}\}}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Durchschnitt
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Indexmengen, Durchschnitt
\end_layout
\end_inset
der Mengen
\begin_inset Formula $A_{i}(i\in I)$
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Quote
Bemerkungen:
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset Formula $I$
\end_inset
kann unendlich sein.
\end_layout
\begin_layout Enumerate
Falls
\begin_inset Formula $I=\left\{ 1,\dotsc,n\right\} $
\end_inset
, dann gilt
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="3">
<features>
<column alignment="left" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<column alignment="left" valignment="top" width="0">
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\underset{i\in\{1,\dotsc,n\}}{\bigcup}A_{i}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $=A_{1}\cup A_{2}\cup\dots\cup A_{n}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
und
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\underset{i\in\{1,\dots,n\}}{\bigcap}A_{i}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $=A_{1}\cap A_{2}\cap\dots\cap A_{n}$
\end_inset
.
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Section
Regeln (Durchschnitt u.
Vereinigung von Indexmengen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Durchschnitt u.
Vereinigung von Indexmengen
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $M$
\end_inset
eine Menge,
\begin_inset Formula $I$
\end_inset
und
\begin_inset Formula $A_{i}$
\end_inset
wie in Definition 0.9.
Dann gilt
\begin_inset Formula \[
\begin{array}{rrcl}
\mbox{(a)} & \bigl(\underset{i\in I}{\bigcup}A_{i}\bigr)\cup M & = & \underset{i\in I}{\bigcup}\left(A_{i}\cup M\right)\\
\mbox{(b)} & \bigl(\underset{i\in I}{\bigcap}A_{i}\bigr)\cap M & = & \underset{i\in I}{\bigcap}\left(A_{i}\cap M\right)\\
\mbox{(c)} & \bigl(\underset{i\in I}{\bigcup}A_{i}\bigr)\cap M & = & \underset{i\in I}{\bigcup}\left(A_{i}\cap M\right)\\
\mbox{(d)} & \bigl(\underset{i\in I}{\bigcap}A_{i}\bigr)\cup M & = & \underset{i\in I}{\bigcap}\left(A_{i}\cup M\right)\\
\mbox{(e)} & M\setminus\underset{i\in I}{\bigcup}A_{i} & = & \underset{i\in I}{\bigcap}\left(M\setminus A_{i}\right)\\
\mbox{(f)} & M\setminus\underset{i\in I}{\bigcap}A_{i} & = & \underset{i\in I}{\bigcup}\left(M\setminus A_{i}\right)\end{array}\]
\end_inset
\end_layout
\begin_layout Section
Definition (
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Kartesisches Produkt in Mengen
\end_layout
\end_inset
Kartesisches Produkt von Mengen)
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $A_{1},\,\dotsc,A_{k}$
\end_inset
endlich viele Mengen
\begin_inset Formula $(k\geq2)$
\end_inset
.
Dann heißt
\begin_inset Formula \[
A_{1}\times A_{2}\times\dots\times A_{k}:=\{(a_{1},a_{2},\dotsc,a_{k}):\,\forall i\in\{1,\dotsc,k\}:\, a_{i}\in A_{i}\}\]
\end_inset
das Kartesisches Produkt der Mengen
\begin_inset Formula $A_{1},\,\dotsc,A_{k}$
\end_inset
.
Dabei sind zwei
\begin_inset Quotes gld
\end_inset
k-Tupel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Tupel
\end_layout
\end_inset
\begin_inset Quotes grd
\end_inset
(
\begin_inset Formula $a_{1},\,\dotsc,a_{k})$
\end_inset
und
\begin_inset Formula $(a_{1}',\,\dotsc,a_{k}')$
\end_inset
genau dann gleich, wenn
\begin_inset Formula $a_{1}=a_{1}'\wedge a_{2}=a_{2}'\wedge\dotsc\wedge a_{k}=a_{k}'$
\end_inset
.
\end_layout
\begin_layout Standard
Falls
\begin_inset Formula $A_{1}=A_{2}=\dots=A_{k}$
\end_inset
, schreibt man
\begin_inset Formula $A^{k}:=A\times\dots\times A$
\end_inset
und setzt
\begin_inset Formula $A^{1}:=A$
\end_inset
.
\end_layout
\begin_layout Section
Abzählungen einiger endlicher Mengen
\end_layout
\begin_layout Subsection*
Fakultät
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Fakultät
\end_layout
\end_inset
, Binomialkoeffizient
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Binomialkoeffizient
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
n!:=n\times(n-1)\times\dots\times1\quad(n\in\mathbb{N},0!:=1)\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\binom{n}{k}:=\frac{n!}{k!(n-k)!}\quad(n\in\mathbb{N}_{0},k\in\mathbb{N},0!:=1).\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Folgerung (Fakultät, Binomialkoeffizient)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\begin_inset Formula \[
\binom{n}{k}=\frac{n\cdot(n-1)\cdot\dots\cdot\overbrace{(n-k+1)}}{k!}\quad(n\in\mathbb{N},k\in\mathbb{N})\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula \[
\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}\quad(n\in\mathbb{N},k\in\mathbb{N},k\leq n)\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Satz (Mengen)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $A$
\end_inset
eine Menge aus
\begin_inset Formula $n$
\end_inset
Elementen,
\begin_inset Formula $n\in\mathbb{N}$
\end_inset
.
(Schreibweise
\begin_inset Formula $\underbrace{\vert A\vert}_{\text{Anzahl der Elemente}}=n)$
\end_inset
.
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $k\in\mathbb{N}$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\begin_inset Formula \[
\vert A^{k}\vert=\vert{\underbrace{(x_{1},\dots,x_{k})}_{\text{k-Tupel aus Elementen von }A}:x_{n},\dots,x_{k}\in A}\vert=n^{k}\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula \[
\begin{split}\vert{\underbrace{(x_{1},\dots,x_{k})}_{\substack{\text{k-Tupel ohne Wiederholung}\\
\text{aus Elementen von A}}
}:x_{1},\dots,x_{k}\in A,\underbrace{\forall i\neq j:x_{i}\neq x_{j}}_{x_{1},x_{2}\text{ paarweise verschieden}}}\vert\\
=n\cdot(n-1)\cdot\dots\cdot(n-k+1)\text{ für }1\leq k\leq n\end{split}
\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\begin_inset Formula \[
\begin{split}\vert\{\underbrace{\{x_{1},\dots,x_{k}\}}_{\text{k-elementige Teilmenge von A}}:x_{1},\dots,x_{k}\in A,\:\forall i\neq j:x_{i}\neq x_{j}\}\vert\\
=\binom{n}{k}\text{ für }0\leq k\leq n\end{split}
\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\begin_inset Formula \[
\vert\underbrace{\mathbb{P}(A)}_{\text{Potenzmenge = Menge aller Teilmengen von A}}\vert=2^{n}\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $x\neq y,0\leq k\leq n$
\end_inset
.
\begin_inset Formula \[
\vert\{(z_{1},\dotsc,z_{n})\in\{x,y\}^{n}:\vert\{i:\, z_{i}=x\}\vert=k\}\vert=\binom{n}{k}\]
\end_inset
\emph on
\begin_inset Formula $n$
\end_inset
-Tupel, bei denen
\begin_inset Formula $x$
\end_inset
k-fach vorkommt und
\begin_inset Formula $y$
\end_inset
n-k-fach vorkommt.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Relationen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Relationen
\end_layout
\end_inset
und Funktionen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Funktionen
\end_layout
\end_inset
\end_layout
\begin_layout Subsection*
Definition
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $X,\, Y$
\end_inset
Mengen,
\begin_inset Formula $R\subset X\times Y$
\end_inset
.
Dann heißt das Tripel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Tripel
\end_layout
\end_inset
\begin_inset Formula $(X,\, Y,\, R)$
\end_inset
Relation zwischen
\begin_inset Formula $X$
\end_inset
und
\begin_inset Formula $Y$
\end_inset
.
Falls
\begin_inset Formula $X=Y$
\end_inset
spricht man von einer
\emph on
binären
\series bold
\emph default
\series default
Relation
\begin_inset Index
status collapsed
\begin_layout Plain Layout
binäre Relation
\end_layout
\end_inset
oder einer Relation auf
\begin_inset Formula $X$
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Relation auf X
\end_layout
\end_inset
.
In diesem Fall benutzt man die Schreibweise
\begin_inset Formula \[
x\sim y:\Leftrightarrow(x,y)\in\mathcal{R}\]
\end_inset
und schreibt:
\begin_inset Formula $(x,\sim)$
\end_inset
an Stelle von
\begin_inset Formula $(X,\, Y,\, R)$
\end_inset
\end_layout
\begin_layout Section
Definition (Umkehrrelation)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\mathcal{R}=(X,\, Y,\, R)$
\end_inset
eine Relation.
Die Relation
\begin_inset Formula $(Y,\, X,\overset{-1}{\,\mathcal{R}})$
\end_inset
mit
\begin_inset Formula $\overset{-1}{\mathcal{R}}=\lbrace(y,x)\in Y\times X:\,(x,y)\in\mathcal{R}\rbrace$
\end_inset
heißt Umkehrrelation von
\begin_inset Formula $\mathcal{R}$
\end_inset
(bzw.
\begin_inset Formula $\overset{-1}{\mathcal{R}}$
\end_inset
).
\end_layout
\begin_layout Standard
Bemerkung:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Umkehrrelationen existieren immer (im Unterschied zu den noch zu betrachtenden
Umkehrfunktionen).
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Es gilt
\begin_inset Formula $\overset{-1}{\mathcal{R}}^{-1}=\mathcal{R}$
\end_inset
wegen
\begin_inset Formula $(x,y)\in\mathcal{R}\Leftrightarrow(y,x)\in\overset{-1}{\mathcal{R}}\Leftrightarrow(x,y)\in\left(\overset{-1}{\mathcal{R}}^{-1}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Abbildung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Abbildung
\end_layout
\end_inset
, Funktion
\end_layout
\begin_layout Standard
Eine Relation
\begin_inset Formula $f=(X,\, Y,\,\mathcal{R})$
\end_inset
heißt Abbildung der Funktion, wenn gilt
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\forall x\in X\underbrace{\exists_{1}}_{\text{"'genau ein"'}}y\in Y(x,y\in\mathcal{R})\]
\end_inset
\end_layout
\begin_layout Standard
d.h.
jedem
\begin_inset Formula $x\in X$
\end_inset
wird genau ein
\begin_inset Formula $y\in Y$
\end_inset
zugeordnet.
Wir nennen
\begin_inset Formula $y$
\end_inset
Funktionswert
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Funktionswert
\end_layout
\end_inset
von
\begin_inset Formula $f$
\end_inset
an der Stelle
\begin_inset Formula $x$
\end_inset
und schreiben
\begin_inset Formula $y=f(x)$
\end_inset
.
\end_layout
\begin_layout Standard
An Stelle von
\begin_inset Formula $f=(X,\, Y,\,\mathcal{R})$
\end_inset
schreiben wir
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
und nennen
\begin_inset Formula $X$
\end_inset
Definitionsbereich
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Definitionsbereich
\end_layout
\end_inset
oder Quelle
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Quelle
\end_layout
\end_inset
von
\begin_inset Formula $f$
\end_inset
und
\begin_inset Formula $Y$
\end_inset
Wertebereich
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Wertebereich
\end_layout
\end_inset
oder Ziel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Ziel
\end_layout
\end_inset
von
\begin_inset Formula $f$
\end_inset
und
\begin_inset Formula $R$
\end_inset
Graph
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Graph
\end_layout
\end_inset
von
\begin_inset Formula $f$
\end_inset
(Schreibweise Graph
\begin_inset Formula $f$
\end_inset
).
\end_layout
\begin_layout Section
Bezeichnungen
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $X$
\end_inset
eine Menge,
\begin_inset Formula $id_{x}:\, X\rightarrow X,\, x\rightarrow x$
\end_inset
heißt Identität
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Identität
\end_layout
\end_inset
auf
\begin_inset Formula $X$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $X$
\end_inset
eine Menge
\begin_inset Formula $A\subset X$
\end_inset
.
\begin_inset Formula $L_{A}:A\rightarrow X,x\rightarrow x$
\end_inset
heißt Einbettung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Einbettung
\end_layout
\end_inset
oder Inklusion
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Inklusion
\end_layout
\end_inset
von
\begin_inset Formula $A$
\end_inset
in
\begin_inset Formula $X$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $f:\, x\rightarrow y$
\end_inset
eine Funktion,
\begin_inset Formula $A\subset X.f|_{A},\, A\rightarrow Y,\, x\rightarrow f(x)$
\end_inset
heißt Einschränkung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Einschränkung
\end_layout
\end_inset
oder Restriktion
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Restriktion
\end_layout
\end_inset
von
\begin_inset Formula $f$
\end_inset
auf
\begin_inset Formula $A$
\end_inset
\end_layout
\begin_layout Standard
Bemerkung:
\begin_inset Formula $L_{A}=id_{x}|A$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Bild
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Bild
\end_layout
\end_inset
, Urbild
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Urbild
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
eine Abbildung (
\begin_inset Formula $\rightarrow$
\end_inset
Funktion),
\begin_inset Formula $A\subset X$
\end_inset
,
\begin_inset Formula $B\subset Y$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f(A):=\lbrace f(x):x\in A\rbrace=\lbrace y\in Y:\exists x\in A:y=f(x)\rbrace$
\end_inset
heißt
\begin_inset Quotes gld
\end_inset
Bild von
\begin_inset Formula $A$
\end_inset
unter
\begin_inset Formula $f$
\end_inset
\begin_inset Quotes grd
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\overset{-1}{f}(B):=\{x\in X:f(x)\in B\}=\{x\in X:\exists y\in B:y=f(x)\}$
\end_inset
heißt
\begin_inset Quotes gld
\end_inset
Urbild von
\begin_inset Formula $B$
\end_inset
unter
\begin_inset Formula $f$
\end_inset
\begin_inset Quotes grd
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Verkettung von Abbildungen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Verkettung von Abbildungen
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $f:X\rightarrow Y_{1}$
\end_inset
und
\begin_inset Formula $g:Y_{2}\rightarrow Z$
\end_inset
Abbildungen mit
\begin_inset Formula $f(x)\subset Y_{2}$
\end_inset
.
\end_layout
\begin_layout Standard
Die Abbildung
\begin_inset Formula \[
X\rightarrow Z,x\rightarrow g(f(x))\]
\end_inset
heißt Verkettung von
\begin_inset Formula $g$
\end_inset
mit
\begin_inset Formula $f$
\end_inset
und wird mit
\begin_inset Formula $g\circ f$
\end_inset
bezeichnet.
\end_layout
\begin_layout Section
Satz - Assoziativität der Verkettung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Assoziativität der Verkettung
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $f:X\rightarrow Y_{1}$
\end_inset
,
\begin_inset Formula $g:Y_{2}\rightarrow Z_{1}$
\end_inset
,
\begin_inset Formula $h:Z_{2}\rightarrow W$
\end_inset
Abbildungen mit
\begin_inset Formula $f(x)\subset Y_{2}$
\end_inset
und
\begin_inset Formula $g(Y_{2})\subset Z_{2}$
\end_inset
.
Dann gilt:
\begin_inset Formula \[
(h\circ g)\circ f=h\circ(g\circ f)\]
\end_inset
\end_layout
\begin_layout Subsubsection*
Bemerkung
\end_layout
\begin_layout Standard
Auch bei identischem Definitions- und Wertebereich gilt im allgemeinen:
\begin_inset Formula \[
g\circ f\neq f\circ g\]
\end_inset
\end_layout
\begin_layout Section
Definition
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
eine Abbildung.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f$
\end_inset
heißt
\emph on
injektiv
\emph default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
injektiv
\end_layout
\end_inset
, wenn
\begin_inset Formula \[
\forall x_{1},\, x_{2}\in X:\, x_{1}\neq x_{2}\Rightarrow f(x_{1})\neq f(x_{2})\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f$
\end_inset
heißt
\emph on
surjektiv
\emph default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
surjektiv
\end_layout
\end_inset
, wenn
\begin_inset Formula \[
f(X)=Y\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f$
\end_inset
heißt
\emph on
bijektiv
\emph default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
bijektiv
\end_layout
\end_inset
, wenn
\begin_inset Formula $f$
\end_inset
zugleich
\emph on
injektiv
\emph default
und
\emph on
surjektiv
\emph default
ist.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Bemerkung zu (a): Oft ist es einfacher, die Kontraposition zu beweisen:
\begin_inset Formula \[
\forall x_{1},\, x_{2}\in X:\, f(x_{1})=f(x_{2})\Rightarrow x_{1}\in x_{2}\]
\end_inset
\end_layout
\begin_layout Section
Lemma
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
und
\begin_inset Formula $g:Y\rightarrow Z$
\end_inset
Abbildungen.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $g\circ f$
\end_inset
injektiv
\begin_inset Formula $\Rightarrow f$
\end_inset
injektiv
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $g\circ f$
\end_inset
surjektiv
\begin_inset Formula $\Rightarrow g$
\end_inset
surjektiv
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Umkehrfunktion
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Umkehrfunktion
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Eine Abbildung
\begin_inset Formula $g:Y\rightarrow X$
\end_inset
heißt Umkehrfunktion oder Umkehrdarstellung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Umkehrdarstellung
\end_layout
\end_inset
zu
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
, wenn gilt:
\begin_inset Formula \[
g\circ f=idx\text{, d.h. }g\left(f(x)\right)=x(x\in X)\]
\end_inset
und
\begin_inset Formula \[
f\circ g=idy\text{, d.h. }f\left(g(x)\right)=y(y\in Y)\]
\end_inset
Falls
\begin_inset Formula $f$
\end_inset
eine Umkehrabbildung besitzt, ist diese eindeutig bestimmt und bijektiv
(Bezeichnung
\begin_inset Formula $f^{-1}$
\end_inset
).
In diesem Fall ist
\begin_inset Formula $f$
\end_inset
ebenfalls bijektiv.
\end_layout
\begin_layout Section
Satz
\end_layout
\begin_layout Standard
Es gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
bijektiv
\begin_inset Formula $\Leftrightarrow f$
\end_inset
besitzt eine Umkehrabbildung
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
bijektiv,
\begin_inset Formula $B\subset Y\Rightarrow\overset{-1}{f}(B)=f^{-1}(B)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
bijektiv
\begin_inset Formula $\Rightarrow(f^{-1})^{-1}=f$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f:X\rightarrow Y$
\end_inset
bijektiv,
\begin_inset Formula $g:Y\rightarrow Z$
\end_inset
bijektiv
\begin_inset Formula $\Rightarrow g\circ f$
\end_inset
bijektiv und
\begin_inset Formula $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Chapter
Reelle Zahlen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Reelle Zahlen
\end_layout
\end_inset
\end_layout
\begin_layout Section*
Körperaxiome
\end_layout
\begin_layout Standard
Auf der Menge
\begin_inset Formula $\mathbb{R}$
\end_inset
sind zwei Verknüpfungen gegeben
\begin_inset Formula \[
+:\mathbb{R}\times\mathbb{R}\to\mathbb{R},\ (x,y)\leftrightarrow x+y\]
\end_inset
\begin_inset Formula \[
\cdot:\mathbb{R}\times\mathbb{R}\to\mathbb{R},\quad(x,y)\leftrightarrow x\cdot y\qquad\text{(bzw. \ensuremath{xy}})\]
\end_inset
die folgende Regeln (Axiome) erfüllen:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(K1)]
\end_layout
\end_inset
Assoziativgesetze
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Reelle Zahlen, Assoziativgesetze
\end_layout
\end_inset
\begin_inset Formula \[
\forall_{x,y,z\in\mathbb{R}}:\quad\begin{array}[t]{rcl}
(x+y)+z & = & x+(y+z),\\
(x\cdot y)\cdot z & = & x\cdot(y\cdot z)\end{array}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(K2)]
\end_layout
\end_inset
Kommutativgesetze
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Reelle Zahlen, Kommutativgesetze
\end_layout
\end_inset
\begin_inset Formula \[
\forall_{x,y\in\mathbb{R}}:\quad\begin{array}[t]{rlc}
x+y & = & y+x,\\
x\cdot y & = & y\cdot x\end{array}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(K3)]
\end_layout
\end_inset
Existenz und Eindeutigkeit der neutralen Elemente
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Reelle Zahlen, Existenz und Eindeutigkeit der neutralen Elemente
\end_layout
\end_inset
\begin_inset Formula $0$
\end_inset
und
\begin_inset Formula $1$
\end_inset
,
\begin_inset Formula $0\neq1$
\end_inset
mit
\begin_inset Formula \begin{eqnarray*}
\begin{array}{ccc}
\forall x\in\mathbb{R}: & x+0 & =x\\
\forall x\in\mathbb{R}: & x\cdot1 & =x\end{array}\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(K4)]
\end_layout
\end_inset
Existenz und Eindeutigkeit der inversen Elemente
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Reelle Zahlen, Existenz und Eindeutigkeit der inversen Elemente
\end_layout
\end_inset
\begin_inset Formula \[
\begin{array}{lrcl}
\forall_{x\in\mathbb{R}}\ \exists_{1}\,_{y\in\mathbb{R}}: & x+y & = & 0\text{\quad(Bez. \ensuremath{-x}})\\
\forall_{x\in\mathbb{R}\setminus\lbrace0\rbrace}\ \underbrace{\exists_{1}}_{\text{exist. genau 1}}\,_{y\in\mathbb{R}}: & x\cdot y & = & 1\quad\text{(Bez. }\mbox{\ensuremath{x^{-1})}}\end{array}\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(K5)]
\end_layout
\end_inset
Distributivgesetz
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Reelle Zahlen, Distributivgesetz
\end_layout
\end_inset
\begin_inset Formula \[
\forall_{x,y,z\in\mathbb{R}}:\quad x\cdot(y+z)=x\cdot y+x\cdot z\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition (Subtraktion und Division)
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $x,y\in\mathbb{R}$
\end_inset
\begin_inset Formula \[
x-y:=x+(-y)\]
\end_inset
\begin_inset Formula \[
\frac{x}{y}:=x\cdot y^{-1}\qquad\text{, (falls }y\neq0)\]
\end_inset
Insbesondere:
\begin_inset Formula $\frac{1}{y}=y^{-1}]$
\end_inset
\end_layout
\begin_layout Section
Rechenregeln
\end_layout
\begin_layout Standard
Im Folgenden seien
\begin_inset Formula $x,\, y,\, u,\, v\in\mathbb{R}$
\end_inset
.
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(a)]
\end_layout
\end_inset
\begin_inset Formula $-(-x)=x,\ \frac{1}{\frac{1}{x}}=x\quad(x\neq0)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula $-(x+y)=-x-y,$
\end_inset
\begin_inset Formula $\frac{1}{xy}=\frac{1}{x}\cdot\frac{1}{y}\qquad(x\neq0,$
\end_inset
\begin_inset Formula $y\neq0)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\begin_inset Formula $x\cdot y=0\Longleftrightarrow x=0\vee y=0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\begin_inset Formula $(-1)\cdot x=-x$
\end_inset
,
\begin_inset Formula $(-x)\cdot(-y)=x\cdot y$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\begin_inset Formula \begin{eqnarray*}
\frac{u}{x}+\frac{v}{y} & = & \frac{uy+vx}{xy}\quad(x\neq0,\ y\neq0)\\
\frac{u}{x}-\frac{v}{y} & = & \frac{uy-vx}{xy}\quad(x\neq0,\ y\neq0)\\
\frac{u}{x}\cdot\frac{v}{y} & = & \frac{uv}{xy}\quad(x\neq0,y\neq0)\\
\frac{\frac{u}{x}}{\frac{v}{y}} & = & \frac{uy}{vx}\quad(x\neq0,y\neq0,v\neq0)\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Summen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Summenschreibweise
\end_layout
\end_inset
-, Produkt
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Produktschreibweise
\end_layout
\end_inset
- und Potenzschreibweise
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Potenzschreibweise
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
{\displaystyle \sum_{i=1}^{n}x_{i}:=x_{1}+x_{2}+\cdots x_{n}\ (n\in\mathbb{N}),\ \sum_{i=1}^{0}x_{i}:=0\qquad\text{(leere Summe)}}\]
\end_inset
\begin_inset Formula \[
\prod_{i=1}^{n}x_{i}:=x_{1}\cdot x_{2}\cdot\ldots\cdot x_{n}\ (n\in\mathbb{N}),\ \prod_{i=1}^{0}x_{i}:=1\qquad\text{(leeres Produkt)}\]
\end_inset
\begin_inset Formula \[
x^{n}:=\underbrace{x\cdot\ldots\cdots x}=\prod_{i=1}^{n}x\ (n\in\mathbb{N}),\]
\end_inset
\begin_inset Formula $x^{0}:=1$
\end_inset
(auch
\begin_inset Formula $0^{0}:=1$
\end_inset
);
\begin_inset Formula $x^{-n}=(\frac{1}{x})^{n}\qquad(n\in\mathbb{N},x\neq0)$
\end_inset
\end_layout
\begin_layout Subsubsection*
Verallgemeinerung
\begin_inset Formula \[
\sum_{i=m}^{n}x_{i}:=\begin{cases}
x_{m}+x_{m+1}+\ldots x_{n} & (m,\, n\in\mathbb{Z},m\leq n)\\
0 & (m,\, n\in\mathbb{Z},m>n)\end{cases}\]
\end_inset
\begin_inset Formula \[
\prod_{i=m}^{n}x_{i}:=\begin{cases}
x_{m}\cdot x_{m+1}\cdot\ldots\cdot x_{n} & (m,n\in\mathbb{Z},m\leq n)\\
1 & (m,n\in\mathbb{Z},m>n)\end{cases}\]
\end_inset
Folgerungen
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(1)]
\end_layout
\end_inset
\begin_inset Formula \[
\sum_{i=\ell}^{n}x_{i}=\biggl(\sum_{i=\ell}^{m}x_{i}\biggr)+\biggl(\sum_{i=m+1}^{n}x_{i}\biggr)\quad(l,\, m,\, n\in\mathbb{Z},l\leq m\leq n)\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(2)]
\end_layout
\end_inset
\begin_inset Formula \[
\prod_{i=\ell}^{n}x_{i}=\biggl(\prod_{i=\ell}^{m}x_{i}\biggr)\cdot\biggl(\prod_{i=m+1}^{n}x_{i}\biggr)\quad(l\,,m\,,n\in\mathbb{Z},l\leq m\leq n)\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(3)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
x^{m}\cdot x^{n} & = & x^{m+n}\qquad(m,\, n\in\mathbb{N_{\mbox{0}}\mbox{,\,\ensuremath{x,\, y\in\mathbb{R})}}}\\
(x^{m})^{n} & = & x^{m\cdot n}\\
(x\cdot y)^{n} & = & x^{n}\cdot y^{n}\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(4)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Für jede bijektive Abbildung
\begin_inset Formula $\varphi:\lbrace1,\,\ldots,\, n\rbrace\to\lbrace1,\,\ldots,\, n\rbrace$
\end_inset
gilt:
\begin_inset Formula \[
\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}x_{\varphi(i)}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(5)]
\end_layout
\end_inset
\end_layout
\begin_layout Paragraph*
\begin_inset Formula \[
\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}x_{n+1-i}\qquad[x_{1}+\cdots+x_{n}=x_{n}+x_{n-1}+\cdots+x_{1}]\]
\end_inset
\end_layout
\begin_layout Subparagraph*
\begin_inset Formula $\varphi:\lbrace m^{*},\, m^{*}+1,\,\ldots,\, n^{*}\rbrace\to\lbrace m,\, m+1,\,\ldots,\, n\rbrace\text{bijektiv}$
\end_inset
:
\begin_inset Formula \[
\sum_{i=m}^{n}x_{i}=\sum_{i=m^{*}}^{n^{*}}x_{\varphi(i)}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(6)]
\end_layout
\end_inset
\begin_inset Formula \[
\sum_{i=1}^{n}x_{i}=\sum_{i=0}^{n-1}x_{i+1}\qquad\text{Indexverschiebung}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(7)]
\end_layout
\end_inset
\begin_inset Formula \[
\sum_{i=1}^{n}(x_{i}+y_{i})=\biggl(\sum_{i=1}^{n}x_{i}\biggr)+\biggl(\sum_{i=1}^{n}y_{i}\biggr)\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(8)]
\end_layout
\end_inset
\begin_inset Formula \[
\sum_{i=1}^{m}\biggl(\sum_{j=1}^{n}a_{ij}\biggr)=\sum_{j=1}^{n}\biggl(\sum_{i=1}^{m}a_{ij}\biggr)\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(9)]
\end_layout
\end_inset
\begin_inset Formula \[
\lambda\cdot\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}(\lambda\cdot x_{i})=\sum_{i=1}^{n}\lambda\cdot x_{i}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(10)]
\end_layout
\end_inset
\begin_inset Formula \[
\biggl(\sum_{i=1}^{m}x_{i}\biggr)\biggl(\sum_{j=1}^{n}y_{j}\biggr)=\sum_{i=1}^{m}\biggl(\sum_{j=1}^{n}x_{i}y_{j}\biggr)\overset{(8)}{=}\sum_{j=1}^{n}\biggl(\sum_{i=1}^{m}x_{i}y_{j}\biggr)\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subsection*
Bemerkungen:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[1.]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\sum_{k=0}^{n}(a_{k}-a_{_{k+1}})=a_{o}-a_{n+1}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\sum_{k=0}^{n}(a_{k+1}-a_{k}) & = & a_{n+1}-a_{0}\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[2.]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\sum_{k=0}^{n}k & = & \frac{n\left(n+1\right)}{2}\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Binomischer Satz
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Binomischer Satz
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $x,y\in\mathbb{R}$
\end_inset
,
\begin_inset Formula $n\in\mathbb{N}_{0}$
\end_inset
.
Dann gilt
\begin_inset Formula \[
(x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}\]
\end_inset
\end_layout
\begin_layout Section
Geometrische Summenformel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Geometrische Summenformel
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $x,\, y\in\mathbb{R}$
\end_inset
,
\begin_inset Formula $n\in\mathbb{N}_{0}$
\end_inset
, dann gilt:
\begin_inset Formula \begin{eqnarray*}
x^{n+1}-y^{n+1} & = & (x-y)\cdot\sum_{k=0}^{n}x^{n-k}y^{k}\\
& = & (x-y)\cdot\sum_{k=0}^{n}x^{k}y^{n-k}\end{eqnarray*}
\end_inset
\shape italic
\emph on
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\shape italic
\emph on
Sei
\family roman
\emph default
\begin_inset Formula $q\in\mathbb{R}$
\end_inset
,
\begin_inset Formula $q\neq1$
\end_inset
,
\begin_inset Formula $n\in\mathbb{N}_{0}$
\end_inset
.
\family default
\emph on
Dann gilt
\family roman
\emph default
\begin_inset Formula \[
\sum_{k=0}^{n}q^{k}=\frac{1-q^{n+1}}{1-q}=\frac{q^{n+1}-1}{q-1}\]
\end_inset
\family default
\shape default
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition (Polynom vom Grad n
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Polynom vom Grad n
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $n\in\mathbb{N}_{0}$
\end_inset
,
\begin_inset Formula $a_{0},\,\ldots,\, a_{n}\in\mathbb{R}$
\end_inset
.
Die Funktion
\begin_inset Formula $p:\mathbb{R}\to\mathbb{R}$
\end_inset
,
\begin_inset Formula $p(x)=\overset{n}{\underset{k=0}{\sum}}a_{k}x^{k}$
\end_inset
wird als Polynom bezeichnet.
Ist
\begin_inset Formula $a_{n}\neq0$
\end_inset
, so heißt
\begin_inset Formula $n$
\end_inset
Grad des Polynoms
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Grad des Polynoms
\end_layout
\end_inset
(Bez.:
\begin_inset Formula $\text{grad }p$
\end_inset
) und
\begin_inset Formula $a_{n}$
\end_inset
Höchstkoeffizient
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Höchstkoeffizient
\end_layout
\end_inset
.
\begin_inset Formula \[
\xi\in\mathbb{R}\text{ heisst Nullstelle von }p\text{, wenn }p(\xi)=0\]
\end_inset
\end_layout
\begin_layout Standard
Bemerkung: Das Nullpolynom
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Nullpolynom
\end_layout
\end_inset
, definiert durch
\begin_inset Formula $p(x):=0$
\end_inset
\begin_inset Formula $(x\in\mathbb{R})$
\end_inset
, zählt zu den Polynomen und erhält den Grad -1.
\end_layout
\begin_layout Standard
[
\begin_inset Formula $P_{n}:=\{p:p$
\end_inset
Polynom vom Grad
\begin_inset Formula $\leq n\}$
\end_inset
soll auch das Nullpolynom enthalten.]
\end_layout
\begin_layout Section
Satz
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $p$
\end_inset
ein Polynom von Grad
\begin_inset Formula $n\geq1$
\end_inset
und
\begin_inset Formula $\zeta$
\end_inset
eine Nullstelle von
\begin_inset Formula $p$
\end_inset
.
Dann gibt es ein Polynom
\begin_inset Formula $q$
\end_inset
mit Grad
\begin_inset Formula $q=n-1$
\end_inset
, so dass
\begin_inset Formula \begin{eqnarray*}
p(x) & = & (x-\zeta)\cdot q(x)\quad(x\in\mathbb{R})\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Jedes Polynom von Grad
\begin_inset Formula $n$
\end_inset
lässt sich schreiben als
\begin_inset Formula \begin{eqnarray*}
p(x) & = & (x-\xi)^{m_{1}}\ldots(x-\xi_{\ell})^{m_{\ell}}\cdot q(x)\\
& = & \biggl(\prod_{i=1}^{\ell}(x-\xi_{i})^{m_{i}}\biggr)\cdot q(x)\quad(x\in\mathbb{R})\end{eqnarray*}
\end_inset
mit
\begin_inset Formula $\ell\in\mathbb{N}_{0}$
\end_inset
,
\begin_inset Formula $m_{1},\,\ldots,\, m_{\ell}\in\mathbb{N}$
\end_inset
,
\begin_inset Formula $\xi_{1},\,\ldots,\,\xi_{\ell}\in\mathbb{N}$
\end_inset
und
\begin_inset Formula $q(x)\neq0\ (x\in\mathbb{R})$
\end_inset
.
\end_layout
\begin_layout Standard
Außerdem gilt grad
\begin_inset Formula $p=$
\end_inset
\begin_inset Formula $m_{1}+\ldots m_{\ell}+\text{ grad }q$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Jedes Polynom von Grad
\begin_inset Formula $n\geq0$
\end_inset
besitzt höchstens
\begin_inset Formula $n$
\end_inset
verschiedene Nullstellen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Nullstelle
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Falls zwei Polynome
\begin_inset Formula $\underset{k=0}{\overset{n}{\sum}}a_{k}\cdot x^{k}$
\end_inset
und
\begin_inset Formula $\underset{k=0}{\overset{n}{\sum}}b_{k}\cdot x^{k}$
\end_inset
an mindestens
\begin_inset Formula $n+1$
\end_inset
Stellen übereinstimmen, dann gilt
\begin_inset Formula $a_{k}=b_{k}$
\end_inset
für alle
\begin_inset Formula $k=0,\,\ldots,\, n$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section*
Anordnungsaxiome
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Anordnungsaxiome
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\begin_layout Plain Layout
\backslash
item[(A1)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Für jedes
\begin_inset Formula $x\in\mathbb{R}$
\end_inset
ist genau eine der folgenden Aussagen
\begin_inset Formula \[
x>0,\quad x=0,\quad-x>0\]
\end_inset
wahr.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(A2)]
\end_layout
\end_inset
\begin_inset Formula \begin{eqnarray*}
\forall_{x,y\in\mathbb{R}}:\, x & > & 0\wedge y>0\Rightarrow x+y>0\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(A3)]
\end_layout
\end_inset
\begin_inset Formula \[
\forall_{x,y\in\mathbb{R}}:\, x>0\wedge y>0\Rightarrow x\cdot y>0\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $x,\, y\in\mathbb{R}$
\end_inset
.
\begin_inset Formula \begin{eqnarray*}
x>y & :\Longleftrightarrow & x-y>0\\
x\ge y & :\Longleftrightarrow & x>y\vee x=y\\
x<y & :\Longleftrightarrow & y>x\\
x\le y & :\Longleftrightarrow & x<y\vee x=y\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Section
Rechenregeln
\begin_inset Index
status collapsed
\begin_layout Plain Layout
reelle Zahlen, Rechenregeln
\end_layout
\end_inset
(Auszug)
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $x,\, y,\, z,\, a,\, b\in\mathbb{R}$
\end_inset
.
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\begin_inset Formula \[
x>y\wedge y>z\Rightarrow x>z\quad\text{(Transitivität)}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula \[
\begin{array}[t]{lcr}
x>y & \Leftrightarrow & x+a>y+a\\
x>y\wedge a>b & \Rightarrow & x+a>y+b\end{array}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\begin_inset Formula \begin{eqnarray*}
x>y\wedge a>0 & \Rightarrow & a\cdot x>a\cdot y\\
x>y\wedge a<0 & \Rightarrow & a\cdot x<a\cdot y\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\begin_inset Formula \[
x\neq0\Rightarrow x^{2}>0\quad\text{(insbesondere \ensuremath{1>0})}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\begin_inset Formula \[
\underbrace{x>y\wedge y>0}_{x>y>0}\Rightarrow\frac{1}{y}>\frac{1}{x}>0\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition (Betrag
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Betrag
\end_layout
\end_inset
von
\begin_inset Formula $x$
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $x\in\mathbb{R}$
\end_inset
.
Dann
\begin_inset Formula $\vert x\vert:=\begin{cases}
x & \text{falls \ensuremath{x\geq0}}\\
-x & \text{falls \ensuremath{x<0}}\end{cases}$
\end_inset
\end_layout
\begin_layout Section
Folgerungen
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $x\in\mathbb{R}$
\end_inset
,
\begin_inset Formula $M\geq0$
\end_inset
.
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(a)]
\end_layout
\end_inset
\begin_inset Formula \[
|x|\geq0,\quad|-x|=|x\vert>,\quad|x|=0\Leftrightarrow x=0\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula \begin{eqnarray*}
|x|\leq M & \Leftrightarrow & -M\leq x\leq M\\
|x|<M & \Leftrightarrow & -M<x<M\\
-|x| & \leq & x\leq|x|\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\begin_inset Formula \[
|xy|=|x|\cdot|y|\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\begin_inset Formula \[
\biggl|\frac{x}{y}\biggr|=\frac{|x|}{|y|}\quad(\text{falls }y\neq0)\]
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Satz (Dreiecksungleichung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Dreiecksungleichung
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $x,\, y\in\mathbb{R}$
\end_inset
.
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(a)]
\end_layout
\end_inset
\begin_inset Formula $|x+y|\leq|x|+|y|$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula $\bigl||x|-|y|\bigr|\leq|x-y|$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section*
Natürliche Zahlen und vollständige Induktion
\begin_inset Index
status collapsed
\begin_layout Plain Layout
vollständige Induktion
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition (min M
\begin_inset Index
status collapsed
\begin_layout Plain Layout
min M
\end_layout
\end_inset
, max M
\begin_inset Index
status collapsed
\begin_layout Plain Layout
max M
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $M$
\end_inset
eine nichtleere Teilmenge von
\begin_inset Formula $\mathbb{R}.$
\end_inset
Dann heißt
\begin_inset Formula $m\in M$
\end_inset
kleinstes bzw.
größtes Element von
\begin_inset Formula $M$
\end_inset
, wenn gilt:
\begin_inset Formula \begin{eqnarray*}
\forall a\in M & : & m\leq a\\
\forall a\in M & : & a\leq m\end{eqnarray*}
\end_inset
Bezeichner: min M bzw.
max M.
\end_layout
\begin_layout Section
\noindent
Lemma
\end_layout
\begin_layout Standard
\noindent
Sei
\begin_inset Formula $M\subset\mathbb{N}$
\end_inset
,
\begin_inset Formula $M\neq\emptyset$
\end_inset
.
Dann existiert min
\begin_inset Formula $M$
\end_inset
.
\end_layout
\begin_layout Section
\noindent
Prinzip der vollständigen Induktion
\begin_inset Index
status open
\begin_layout Plain Layout
vollständige Induktion
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
Sei
\begin_inset Formula $\mathcal{A}(n)$
\end_inset
für jedes
\begin_inset Formula $n\in\mathbb{N}$
\end_inset
eine Aussage.
\end_layout
\begin_layout Standard
\noindent
Sofern:
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Formula $\mathcal{A}(1)$
\end_inset
wahr ist
\emph on
(Induktionsanfang
\begin_inset Index
status collapsed
\begin_layout Plain Layout
\emph on
Induktionsanfang
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
\noindent
und für jedes
\begin_inset Formula $n\in\mathbb{N}$
\end_inset
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Formula $\mathcal{A}(n)\Rightarrow\mathcal{A}(n+1)$
\end_inset
wahr ist
\emph on
(Induktionsschluss
\begin_inset Index
status collapsed
\begin_layout Plain Layout
\emph on
Induktionsschluss
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
\noindent
dann gilt für jedes
\begin_inset Formula $n\in\mathbb{N}$
\end_inset
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Formula $\mathcal{A}(n)$
\end_inset
wahr
\end_layout
\begin_layout Section
\noindent
\noun on
Bernoullische
\noun default
Ungleichungen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Bernoullische Ungleichungen
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
Sei
\begin_inset Formula $x\in\mathbb{R},\, x\geq-1,\, n\in\mathbb{N}_{0}$
\end_inset
.
Dann gilt
\begin_inset Formula $\left(1+x\right)^{n}\geq1+n\cdot x$
\end_inset
\end_layout
\begin_layout Section
\noindent
Potenzsummen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Potenzsummen
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
Sei
\begin_inset Formula $k\in\mathbb{N}_{0}$
\end_inset
.
Dann ist
\begin_inset Formula $S_{k}:\,\mathbb{N}\rightarrow\mathbb{N},\, S_{k}\left(n\right)=\overset{n}{\underset{i=0}{\sum}}i^{k}$
\end_inset
ein Polynom
\begin_inset Formula $\left(\mbox{in }n\right)$
\end_inset
vom Grad
\begin_inset Formula $k+1$
\end_inset
und es gilt:
\begin_inset Formula \begin{eqnarray*}
\mbox{(*)\hspace{1.5cm}}S_{k}\left(n\right) & = & \frac{1}{k+1}(\left(n+1\right)^{k+1}-1-\overset{k-1}{\underset{j=0}{\sum}}\left(\begin{array}{c}
k+1\\
j\end{array}\right)S_{j}\left(n\right))\hspace{0.5cm}\left(n\in\mathbb{N}\right)\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Subsubsection*
\noindent
Folgebegriff, reflexive Definition
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Folgebegriff, reflexive Definition
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
Unter einer Folge von reellen Zahlen versteht man eine Abbildung
\begin_inset Formula $\mathbb{N}\rightarrow\mathbb{R},\, n\mapsto a_{n}$
\end_inset
.
(Jedem
\begin_inset Formula $n\in\mathbb{N}$
\end_inset
wird also ein
\begin_inset Formula $a_{n}\in\mathbb{R}$
\end_inset
zugeordnet)
\end_layout
\begin_layout Standard
\noindent
Schreibweise:
\begin_inset Formula $\left(a_{n}\right)\, n\in\mathbb{N}$
\end_inset
oder
\begin_inset Formula $\left(a_{1},\, a_{2},\, a_{3},\,...\right)$
\end_inset
\end_layout
\begin_layout Standard
\noindent
Ersetzt man
\begin_inset Formula $\mathbb{R}$
\end_inset
durch eine beliebige Menge
\begin_inset Formula $X$
\end_inset
, so erhält man eine Folge von Elementen in
\begin_inset Formula $X$
\end_inset
(Folge in
\begin_inset Formula $X$
\end_inset
).
\end_layout
\begin_layout Standard
Bsp:
\begin_inset Formula \[
\begin{array}[t]{ccl}
a_{0} & = & 1,\, a_{n+1}=-\frac{1}{n+2}\underset{k=0}{\overset{n}{\sum}}\left(\begin{array}{c}
n+2\\
k\end{array}\right)a_{k}\,\left(n\in\mathbb{N}_{0}\right)\\
& & \left(a_{n}\right)_{n\in\mathbb{N}}\mbox{ Bernoulli-Zahlen (Bezeichnung: \ensuremath{B_{n}})}\end{array}\]
\end_inset
\begin_inset Formula \[
\begin{array}{c|ccccccccccccc}
n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline B_{n} & 1 & -\frac{1}{2} & \frac{1}{6} & 0 & -\frac{1}{30} & 0 & \frac{1}{42} & 0 & -\frac{1}{30} & 0 & \frac{5}{66} & 0 & -\frac{691}{2730}\end{array}\]
\end_inset
keine explizite Formel bekannt!
\end_layout
\begin_layout Standard
\noindent
Mit den
\noun on
Bernoulli
\noun default
-Zahlen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Bernoulli-Zahlen
\end_layout
\end_inset
erhält man Formeln für die Potenzsummen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Potenzsummen
\end_layout
\end_inset
\begin_inset Formula \[
S_{k}\left(n\right)=\frac{1}{k+1}\overset{k}{\underset{j=0}{\sum}}B_{j}\left(\begin{array}{c}
k+1\\
j\end{array}\right)\left(n+1\right)^{k+1-j}\]
\end_inset
\end_layout
\begin_layout Subsection*
(A4)
\noun on
Archimedisches
\noun default
Axiom
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Archimedisches Axiom
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\forall x\in\mathbb{R}\,\exists n\in\mathbb{N}:\, n>x$
\end_inset
\end_layout
\begin_layout Standard
[Hieraus folgt, dass es
\emph on
kein
\emph default
Element
\begin_inset Formula $\infty$
\end_inset
(
\begin_inset Quotes pld
\end_inset
unendlich
\begin_inset Index
status collapsed
\begin_layout Plain Layout
unendlich
\end_layout
\end_inset
\begin_inset Quotes prd
\end_inset
) in
\begin_inset Formula $\mathbb{R}$
\end_inset
gibt]
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
zu jedem
\begin_inset Formula $\ensuremath{x\in\mathbb{R}}$
\end_inset
gibt es genau ein
\begin_inset Formula $\ensuremath{z\in\mathbb{Z}}$
\end_inset
, so dass
\begin_inset Formula $z\leq x<z+1:\,\forall x\in\mathbb{R}\,\exists_{1}z\in\mathbb{Z}:\, z\leq x<z+1$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\forall_{\varepsilon}>0\,\exists n\in\mathbb{N}:\,\frac{1}{n}<\varepsilon$
\end_inset
(Satz des
\noun on
Eudoxos
\noun default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Satz des Eudoxos
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\alpha>0$
\end_inset
.
Dann
\begin_inset Formula $\forall\varepsilon>0\,\exists n\in\mathbb{N}:\,\frac{\alpha}{2^{n}}<\varepsilon$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subsection*
Bezeichnung für Intervalle:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Beschränkte Intervalle
\end_layout
\end_inset
Beschränkte Intervalle:
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[a,\, b\right]:=\left\{ x\in\mathbb{R}:\, a\leq x\leq b\right\} \,\left(a,\, b\in\mathbb{R},\, a\leq b\right)$
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
abgeschlossenes Intervall
\end_layout
\end_inset
abgeschlossenes Intervall
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Intervall, abgeschlossenes
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[a,\, b\right[:=\left\{ x\in\mathbb{R}:\, a\leq x<b\right\} \,\left(a,\, b\in\mathbb{R},\, a<b\right)$
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
halboffenes Intervall
\end_layout
\end_inset
halboffenes Intervall
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Intervall, halboffenes
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left]a,\, b\right]:=\left\{ x\in\mathbb{R}:\, a<x\leq b\right\} $
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left]a,\, b\right[:=\left\{ x\in\mathbb{R}:\, a<x<b\right\} \,\left(a,\, b\in\mathbb{R},\, a<b\right)$
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
offenes Intervall
\end_layout
\end_inset
offenes Intervall
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Intervall, offenes
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Wir ordnen allen diesen Intervallen die Länge
\begin_inset Formula $\left|J\right|=b-a$
\end_inset
zu.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Uneingeschränkte Intervalle
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Uneingeschränkte Intervalle
\end_layout
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[a,\,\infty\right[:=\left\{ x\in\mathbb{R}:\, a\leq x\right\} \,\left(a\in\mathbb{R}\right)$
\end_inset
abgeschlossenes Intervall
\end_layout
\begin_layout Standard
\begin_inset Formula $\left]-\infty,\, a\right]:=\left\{ x\in\mathbb{R}:\, x\leq a\right\} \,\left(a\in\mathbb{R}\right)$
\end_inset
abgeschlossenes Intervall
\end_layout
\begin_layout Standard
\begin_inset Formula $\left]a,\,\infty\right[:=\left\{ x\in\mathbb{R}:\, a<x\right\} \,\left(a\in\mathbb{R}\right)$
\end_inset
offenes Intervall
\end_layout
\begin_layout Standard
\begin_inset Formula $\left]-\infty,\, a\right[:=\left\{ x\in\mathbb{R}:\, x<a\right\} \,\left(a\in\mathbb{R}\right)$
\end_inset
offenes Intervall
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition (Intervallschachtelung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Intervallschachtelung
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(J_{n}\right)\, n\in\mathbb{N}$
\end_inset
eine Folge abgeschlossener, beschränkter Intervalle.
Diese bildet eine Intervallschachtelung, wenn
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(i)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\forall n\in\mathbb{N}:\, J_{n+1}\subset J_{n}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(ii)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\forall\varepsilon>0\,\exists n\in\mathbb{N}:\,\left|J_{n}\right|<\varepsilon$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection*
Vollständigkeitsaxiom
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Vollständigkeitsaxiom
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(V)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Für jede Intervallschachtelung gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\underset{n\in\mathbb{N}}{\bigcap}J_{n}\neq\emptyset\]
\end_inset
d.h.
\begin_inset Formula $\exists x\in\mathbb{R}\,\forall n\in\mathbb{N}:\, x\in J_{n}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x$
\end_inset
ist eindeutig bestimmt, d.h.
\begin_inset Formula $\exists_{1}x\in\mathbb{R}\,\forall n\in\mathbb{N}:\, x\in J_{n}$
\end_inset
\end_layout
\begin_layout Section
\begin_inset Formula $B$
\end_inset
-adische Entwicklung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
B-adische Entwicklung
\end_layout
\end_inset
(für reelle Zahlen
\begin_inset Formula $\in\left[0,\,1\right[$
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $B\in\mathbb{N},\, B\geq2$
\end_inset
.
Wir nennen
\begin_inset Formula $Z\in\left\{ 0,\,...,\, B-1\right\} $
\end_inset
eine Ziffer zur Basis
\begin_inset Formula $B$
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Ziffer zur Basis B
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(Z_{k}\right)_{k\in\mathbb{N}}$
\end_inset
eine Folge von Ziffern zur Basis
\begin_inset Formula $B$
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Folge von Ziffern zur Basis B
\end_layout
\end_inset
.
Dann ist
\begin_inset Formula $\left(J_{n}\right)_{n\in\mathbb{N}}$
\end_inset
mit
\begin_inset Formula \[
J_{n}=\left[\underset{k=1}{\overset{n}{\sum}}Z_{k}\cdot B^{-k},\,\underset{k=1}{\overset{n}{\sum}}Z_{k}\cdot B^{-k}+B^{-n}\right]\subset\left[0,\,1\right]\]
\end_inset
eine Intervallschachtelung.
\end_layout
\begin_layout Standard
Die Folge
\begin_inset Formula \[
\left(\underset{k=1}{\overset{n}{\sum}}Z_{k}\cdot B^{-k}\right)_{n\in\mathbb{N}}\]
\end_inset
heißt
\begin_inset Formula $B$
\end_inset
-adische Entwicklung der eindeutig bestimmten reellen Zahl
\begin_inset Formula $x\in\underset{n\in\mathbb{N}}{\bigcap}J_{n}$
\end_inset
.
\end_layout
\begin_layout Standard
(Hierfür schreiben wir später
\begin_inset Formula $x=\underset{k=1}{\overset{\infty}{\sum}}Z_{k}\cdot B^{-k}$
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $0\leq x<1$
\end_inset
.
Dann besitzt
\begin_inset Formula $x$
\end_inset
eine
\begin_inset Formula $B$
\end_inset
-adische Entwicklung
\begin_inset Index
status collapsed
\begin_layout Plain Layout
B-adische Entwicklung
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
\noindent
Bemerkung:
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
Im Fall
\begin_inset Formula $x\geq1$
\end_inset
erhält man
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $x=\overset{\infty}{\underset{k=-k_{0}}{\sum}}Z_{k}B^{-k}$
\end_inset
für ein
\begin_inset Formula $k_{0}\in\mathbb{N}_{0}$
\end_inset
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
Das ergibt sich aus der Ungleichung
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $B^{k_{0}}\leq x<B^{k_{0}+1}$
\end_inset
und der Anwendung von 1.20 auf
\begin_inset Formula $\frac{x}{B^{k_{0}+1}}$
\end_inset
.
\end_layout
\begin_layout Section
\noindent
Definition (Lipschitzstetigkeit
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Lipschitzstetigkeit
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
Sei
\begin_inset Formula $I\in\mathbb{R}$
\end_inset
ein Intervall.
\begin_inset Formula $f:I\rightarrow\mathbb{R}$
\end_inset
heißt lipschitzstetig
\begin_inset Index
status collapsed
\begin_layout Plain Layout
lipschitzstetig
\end_layout
\end_inset
, wenn
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $\exists L>0\,\forall x_{1},x_{2}\in I:\,\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right|\leq L\left|x_{1}-x_{2}\right|$
\end_inset
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Index
status collapsed
\begin_layout Plain Layout
L (Lipschitzkonstante)
\end_layout
\end_inset
L wird als Lipschitzkonstante
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Lipschitzkonstante
\end_layout
\end_inset
bezeichnet.
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $\left|\frac{f\left(x_{1}\right)-f\left(x_{2}\right)}{x_{1}-x_{2}}\right|\leq L\,\left(x_{1},x_{2}\in I,\, x_{1}\neq x_{2}\right)$
\end_inset
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
Betrag der Sehnensteigungen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Sehnensteigungen
\end_layout
\end_inset
ist durch L beschränkt.
L unabhängig von
\begin_inset Formula $x_{1},\, x_{2}$
\end_inset
!
\end_layout
\begin_layout Subparagraph*
\noindent
Bemerkung:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $f$
\end_inset
lipschitzstetig,
\begin_inset Formula $\Rightarrow$
\end_inset
\begin_inset Formula $f$
\end_inset
stetig
\begin_inset Index
status collapsed
\begin_layout Plain Layout
stetig
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $I$
\end_inset
beschränktes Intervall,
\begin_inset Formula $f:I\rightarrow\mathbb{R}$
\end_inset
lipschitzstetig
\begin_inset Formula $\Rightarrow$
\end_inset
\begin_inset Formula $f$
\end_inset
beschränkt, d.h.
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\begin_inset Formula $\exists M>0\,\forall x\in I.\,\left|f\left(x\right)\right|\leq M$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
\noindent
Zwischenwertsatz
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Zwischenwertsatz
\end_layout
\end_inset
(Vorläufige Fassung)
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\noindent
Sei
\begin_inset Formula $I=\left[a,b\right]$
\end_inset
ein beschränktes Intervall,
\begin_inset Formula $f:I\rightarrow\mathbb{R}$
\end_inset
sei lipschitzstetig,
\begin_inset Formula $f\left(a\right)<0$
\end_inset
,
\begin_inset Formula $f\left(b\right)>0$
\end_inset
(oder umgekehrt
\begin_inset Formula $f\left(a\right)>0$
\end_inset
und
\begin_inset Formula $f\left(b\right)<0$
\end_inset
).
Dann gibt es [mindestens] ein
\begin_inset Formula $x\in I$
\end_inset
mit
\begin_inset Formula $f\left(x\right)=0$
\end_inset
.
\end_layout
\begin_layout Section
\noun on
Corollar
\noun default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Corollar
\end_layout
\end_inset
(Vorläufige Fassung)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $I=\left[a,b\right]$
\end_inset
ein abgeschlosses beschränktes Intervall,
\begin_inset Formula $f:\, I\rightarrow\mathbb{R}$
\end_inset
sei lipschitzstetig und
\begin_inset Formula $y$
\end_inset
eine reelle Zahl zwischen
\begin_inset Formula $f\left(a\right)$
\end_inset
und
\begin_inset Formula $f\left(b\right)$
\end_inset
.
Dann gibt es (mindestens) ein
\begin_inset Formula $x\in I$
\end_inset
mit
\begin_inset Formula $f\left(x\right)=y$
\end_inset
.
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
Wegen der Eigenschaft, dass
\begin_inset Formula $f$
\end_inset
jeden Wert zwischen
\begin_inset Formula $f\left(a\right)$
\end_inset
und
\begin_inset Formula $f\left(b\right)$
\end_inset
annimmt, bezeichnet man 1.22 bzw.
1.23 als Zwischenwertsatz.
\end_layout
\begin_layout Section
(k-te Wurzel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
k-te Wurzel
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $k\in\mathbb{N},\, k\geq2$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Die Funktion
\begin_inset Formula $[0,\,\infty[\rightarrow[0,\,\infty[\,,x\mapsto x^{k}$
\end_inset
besitzt genau eine Umkehrfunktion.
\begin_inset Newline newline
\end_inset
(Bezeichnung:
\begin_inset Formula $x\mapsto\sqrt[k]{x}$
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(\sqrt[k]{a}\right)^{k}=a=\sqrt[k]{a^{k}}\,\left(a\geq0\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\sqrt[k]{a\cdot b}=\sqrt[k]{a}\cdot\sqrt[k]{b}\,\left(a,b\geq0\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $0\leq a<b\Rightarrow\sqrt[k]{a}<\sqrt[k]{b}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Enumerate
Für
\begin_inset Formula $k$
\end_inset
ungerade lässt sich zeigen, dass die Funktion
\begin_inset Formula $\mathbb{R}\rightarrow\mathbb{R},\, x\mapsto x^{k}$
\end_inset
bijektiv ist.
Man kann also in diesem Fall die
\begin_inset Formula $k-te$
\end_inset
Wurzel auf ganz
\begin_inset Formula $\mathbb{R}$
\end_inset
definieren.
\end_layout
\begin_layout Enumerate
Die Funktion
\begin_inset Formula $x\mapsto\sqrt[k]{x}\,\left(x\geq0\right)$
\end_inset
,
\begin_inset Formula $k\in\mathbb{N},\, k\geq2$
\end_inset
fest, ist lipschitzstetig auf
\begin_inset Formula $[a,\,\infty[$
\end_inset
für jedes feste
\begin_inset Formula $a>0$
\end_inset
, aber
\emph on
nicht
\emph default
lipschitzstetig auf
\begin_inset Formula $[0,\,\infty[$
\end_inset
.
\end_layout
\begin_layout Chapter
Folgen und Grenzwerte
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Folgen und Grenzwerte
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definitionen (Grenzwert
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Grenzwert
\end_layout
\end_inset
, Limes
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Limes
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen und
\begin_inset Formula $a\in\mathbb{R}.$
\end_inset
Dann heißt
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
konvergent
\begin_inset Index
status collapsed
\begin_layout Plain Layout
konvergent
\end_layout
\end_inset
gegen
\begin_inset Formula $a\,:\Longleftrightarrow\forall\varepsilon>0\;\exists n_{0}\in\mathbb{N}\;\forall n\geq n_{0}:\,|a_{n}-a|<\varepsilon$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a$
\end_inset
wird als Grenzwert oder Limes der Folge
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
bezeichnet.
\end_layout
\begin_layout Standard
Schreibweise:
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}a_{n}=a$
\end_inset
oder
\begin_inset Formula $a_{n}\rightarrow a\,\left(n\rightarrow\infty\right)$
\end_inset
\end_layout
\begin_layout Standard
Ist
\begin_inset Formula $a=0$
\end_inset
, so heißt
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
Nullfolge
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Nullfolge
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
Eine Folge wird als
\emph on
konvergent
\emph default
bezeichnet, wenn sie gegen eine reelle Zahl konvergiert.
Andernfalls heißt sie
\emph on
divergent
\emph default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
divergent
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
\paragraph_spacing onehalf
\begin_inset Graphics
filename grafiken/2009-12-01_04.pdf
lyxscale 40
height 2.5cm
\end_inset
\begin_inset Formula $\left|a_{n}-a\right|<\varepsilon\Leftrightarrow a_{n}\in U_{\varepsilon}\left(a\right)$
\end_inset
\end_layout
\begin_layout Standard
Mit Hilfe der offenen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
epsilon-Umgebung
\end_layout
\end_inset
\begin_inset Formula $\varepsilon$
\end_inset
- Umgebung von
\begin_inset Formula $a\,\left(\varepsilon>0\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $U_{\varepsilon}\left(a\right)=\left]a-\varepsilon,\, a+\varepsilon\right[=\left\{ x\in\mathbb{R}:\,\left|x-a\right|<\varepsilon\right\} $
\end_inset
\end_layout
\begin_layout Standard
lässt sich auch sagen:
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
konvergiert gegen
\begin_inset Formula $a\Leftrightarrow$
\end_inset
\end_layout
\begin_layout Standard
In jeder
\begin_inset Formula $\varepsilon$
\end_inset
- Umgebung liegen fast alle Folgeglieder
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Folgeglieder
\end_layout
\end_inset
an.
\end_layout
\begin_layout Standard
\begin_inset Quotes pld
\end_inset
Fast alle
\begin_inset Quotes prd
\end_inset
bedeutet hier
\begin_inset Quotes pld
\end_inset
alle mit Ausnahme höchstens endlich vieler
\begin_inset Quotes prd
\end_inset
.
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Jede konvergente Folge reeller Zahlen ist beschränkt, d.h.
\end_layout
\begin_layout Standard
\begin_inset Formula $\exists M>0\;\forall n\in\mathbb{N}:\;\left|a_{n}\right|\leq M$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Falls
\begin_inset Formula $a_{n}\rightarrow a\neq0\,(n\rightarrow\infty)$
\end_inset
, dann sind fast alle
\begin_inset Formula $a_{n}\neq0$
\end_inset
und es gibt
\begin_inset Formula $n_{0}\in\mathbb{N}$
\end_inset
, so dass
\begin_inset Formula $\left(\frac{1}{a_{n}}\right)_{n\geq n_{0}}$
\end_inset
beschränkt ist, d.h.
\begin_inset Formula $\exists M>0\;\exists n_{0}\in\mathbb{N}\;\forall n\geq n_{0}:\,\frac{1}{\left|a_{n}\right|}\leq M$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Falls
\begin_inset Formula $a_{n}\geq0\,\left(n\in\mathbb{N}\right)$
\end_inset
und
\begin_inset Formula $a_{n}\rightarrow a$
\end_inset
, dann gilt
\begin_inset Formula $a\geq0$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
Aus
\begin_inset Formula $a_{n}>0\,\left(n\in\mathbb{N}\right)$
\end_inset
und
\begin_inset Formula $a_{n}\rightarrow a$
\end_inset
folgt im allg.
\emph on
nicht
\emph default
\begin_inset Formula $a>0$
\end_inset
.
\end_layout
\begin_layout Section
Satz
\end_layout
\begin_layout Standard
Jede Folge reeller Zahlen besitzt höchstens einen Grenzwert.
\end_layout
\begin_layout Section
Lemma
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
und
\begin_inset Formula $\left(b_{n}\right)_{n\in\mathbb{N}}$
\end_inset
Folgen reeller Zahlen,
\begin_inset Formula $\lambda\in\mathbb{R}$
\end_inset
.
\end_layout
\begin_layout Standard
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow0\Leftrightarrow\left|a_{n}\right|\rightarrow0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow0\Rightarrow\lambda\cdot a_{n}\rightarrow0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow0\wedge b_{n}\rightarrow0\text{\Rightarrow}a_{n}+b_{n}\rightarrow0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $b_{n}\geq0\,\left(n\in\mathbb{N}\right)$
\end_inset
,
\begin_inset Formula $\left|a_{n}\right|\leq b_{n}\,\left(n\in\mathbb{N}\right)$
\end_inset
,
\begin_inset Formula $b_{n}\rightarrow0\Rightarrow a_{n}\rightarrow0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Rechenregeln I
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Folgen und Grenzwerte, Rechenregeln I
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
,
\begin_inset Formula $\left(b_{n}\right)_{n\in\mathbb{N}}$
\end_inset
,
\begin_inset Formula $\left(c_{n}\right)_{n\in\mathbb{N}}$
\end_inset
Folgen reeller Zahlen.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\Leftrightarrow a_{n}-a\rightarrow0\Leftrightarrow\left|a_{n}-a\right|\rightarrow0\Rightarrow\left|a_{n}\right|\rightarrow\left|a\right|$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\underset{n\rightarrow\infty}{\lim}a_{n}=a & \Leftrightarrow & \underset{n\rightarrow\infty}{\lim}\left(a_{n}-a\right)=0\\
\Leftrightarrow\underset{n\rightarrow\infty}{\lim}\left|a_{n}-a\right|=0 & \Leftrightarrow & \underset{n\rightarrow\infty}{\lim}\left|a_{n}\right|=\left|\underset{n\rightarrow\infty}{\lim}a_{n}\right|\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\,\wedge\, b_{n}\rightarrow b\Rightarrow a_{n}+b_{n}\rightarrow a+b\,\wedge\, a_{n}-b_{n}\rightarrow a-b$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{ existiert }\wedge\underset{n\rightarrow\infty}{\lim}b_{n}\mbox{ existiert }\Rightarrow\underset{n\rightarrow\infty}{\lim}\left(a_{n}+b_{n}\right)=\underset{n\rightarrow\infty}{\lim}a_{n}+\underset{n\rightarrow\infty}{\lim}b_{n}\wedge\underset{n\rightarrow\infty}{\lim}\left(a_{n}-b_{n}\right)=\underset{n\rightarrow\infty}{\lim}a_{n}-\underset{n\rightarrow\infty}{\lim}b_{n}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\,\wedge\, b_{n}\rightarrow b\Rightarrow a_{n}\cdot b_{n}\rightarrow a\cdot b$
\end_inset
(insbesondere:
\begin_inset Formula $a_{n}\rightarrow a,\,\lambda\in\mathbb{R}\Rightarrow\lambda\cdot a_{n}\rightarrow\lambda\cdot a$
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{ existiert }\wedge\underset{n\rightarrow\infty}{\lim}b_{n}\mbox{ existiert }\Rightarrow\underset{n\rightarrow\infty}{\lim}\left(a_{n}\cdot b_{n}\right)=\underset{n\rightarrow\infty}{\lim}a_{n}\cdot\underset{n\rightarrow\infty}{\lim}b_{n}\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a$
\end_inset
,
\begin_inset Formula $b_{n}\rightarrow b\neq0\Rightarrow\frac{a_{n}}{b_{n}}\rightarrow\frac{a}{b}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{ existiert }\wedge\underset{n\rightarrow\infty}{\lim}b_{n}\mbox{ existiert und ist ungleich }0\Rightarrow\underset{n\rightarrow\infty}{\lim}\frac{a_{n}}{b_{n}}=\frac{\underset{n\rightarrow\infty}{\lim}a_{n}}{\underset{n\rightarrow\infty}{\lim}b_{n}}\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\,\wedge\, b_{n}\rightarrow b\,\wedge\, a_{n}\geq b_{n}\,\left(n\in\mathbb{N}\right)\Rightarrow a\geq b$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{ existiert }\wedge\underset{n\rightarrow\infty}{\lim}b_{n}\mbox{ existiert }\wedge a_{n}\geq b_{n}\,\left(n\in\mathbb{N}\right)\Rightarrow\underset{n\rightarrow\infty}{\lim}a_{n}\geq\underset{n\rightarrow\infty}{\lim}b_{n}\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(f)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\,\wedge\, b_{n}\rightarrow a\,\wedge\, a_{n}\leq c_{n}\leq b_{n}\,\left(n\in\mathbb{N}\right)\Rightarrow c_{n}\rightarrow a$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $[\,\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{ existiert }\wedge\underset{n\rightarrow\infty}{\lim}b_{n}\mbox{ existiert }\wedge\underset{n\rightarrow\infty}{\lim}a_{n}=$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $=\underset{n\rightarrow\infty}{\lim}b_{n}\wedge a_{n}\leq c_{n}\leq b_{n}\,\left(n\in\mathbb{N}\right)\Rightarrow\underset{n\rightarrow\infty}{\lim}c_{n}=\underset{n\rightarrow\infty}{\lim}a_{n}=\underset{n\rightarrow\infty}{\lim}b_{n}\,]$
\end_inset
\end_layout
\begin_layout Standard
\emph on
(Sandwich-Lemma)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Rechenregeln II
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Folgen und Grenzwerte, Rechenregeln II
\end_layout
\end_inset
(vorläufige Fassung)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $I$
\end_inset
ein Intervall,
\begin_inset Formula $a\in I$
\end_inset
,
\begin_inset Formula $a_{n}\in I\,\left(n\in\mathbb{N}\right)$
\end_inset
und
\begin_inset Formula $f:I\rightarrow\mathbb{R}$
\end_inset
lipschitzstetig.
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\,\Rightarrow\, f\left(a_{n}\right)\rightarrow f\left(a\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{existiert, }\underset{n\rightarrow\infty}{\lim}a_{n}\in I\Rightarrow\underset{n\rightarrow\infty}{\lim}f\left(a_{n}\right)=f\left(\underset{n\rightarrow\infty}{\lim}a_{n}\right)\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen,
\begin_inset Formula $a\geq0,\: k\in\mathbb{N},\: k\geq2$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}\rightarrow a\,\Rightarrow\sqrt[k]{a_{n}}\rightarrow\sqrt[k]{a}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[\underset{n\rightarrow\infty}{\lim}a_{n}\mbox{existiert \wedge\ }a\geq0\;(n\in\mathbb{N})\Rightarrow\underset{n\rightarrow\infty}{\lim}\sqrt[k]{a_{n}}=\sqrt[k]{\underset{n\rightarrow\infty}{\lim}a_{n}}\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Enumerate
Mit dem später zu behandelnden Begriff der Stetigkeit
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Stetigkeit
\end_layout
\end_inset
lässt sich (a) folgendermaßen formulieren:
\begin_inset Newline newline
\end_inset
\emph on
Jede lipschitzstetige Funktion ist stetig.
\end_layout
\begin_layout Enumerate
(b) enthält die die Aussage, dass die k-te Wurzel auf
\begin_inset Formula $[0,\,\infty[$
\end_inset
stetig ist.
\end_layout
\begin_layout Enumerate
In (b) kann für
\begin_inset Formula $k$
\end_inset
ungerade die Voraussetzung
\begin_inset Formula $a_{n}\geq0$
\end_inset
entfallen.
\end_layout
\begin_layout Section
Einige Grenzwerte
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Grenzwerte
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $k\in\mathbb{N}.$
\end_inset
Dann gilt:
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}\frac{1}{n^{k}}=0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $k\in\mathbb{N}$
\end_inset
,
\begin_inset Formula $k\geq2$
\end_inset
.
Dann gilt:
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}\frac{1}{\sqrt[k]{n}}=0$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a\in\mathbb{R}$
\end_inset
,
\begin_inset Formula $a>0$
\end_inset
.
Dann gilt:
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}\sqrt[n]{a}=1$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Es gilt:
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}\sqrt[n]{n}=1$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $q\in\mathbb{R},\,\left|q\right|<1,\, k\in\mathbb{N}_{0}$
\end_inset
.
Dann gilt:
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}n^{k}\cdot q^{n}=0$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition (Häufungspunkt
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Häufungspunkt
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen.
\begin_inset Formula $a$
\end_inset
heißt Häufigkeitspunkt der Folge, wenn gilt:
\begin_inset Formula \[
\forall\varepsilon>0\,\forall n_{0}\in\mathbb{N}\,\exists n\geq n_{0}:\,\left|a_{n}-a\right|<\varepsilon\]
\end_inset
Mit Hilfe der
\begin_inset Formula $\varepsilon$
\end_inset
-Umgebung von
\begin_inset Formula $a$
\end_inset
lässt sich äquivalent schreiben:
\end_layout
\begin_layout Standard
\begin_inset Formula $a$
\end_inset
Häufigkeitspunkt in
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
, genau dann, wenn in jeder
\begin_inset Formula $\varepsilon$
\end_inset
-Umgebung
\begin_inset Formula $U_{\varepsilon}\left(a\right)$
\end_inset
liegen unendlich viele Folgenglieder
\begin_inset Formula $a_{n}$
\end_inset
.
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine gegen
\begin_inset Formula $a$
\end_inset
konvergente Folge.
Dann ist
\begin_inset Formula $a$
\end_inset
der einzige Häufigkeitspunkt von
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
.
\end_layout
\begin_layout Section
Lemma
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(I_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Intervallschachtelung,
\begin_inset Formula $\underset{n\in\mathbb{N}}{\bigcap}I_{n}=\left\{ a\right\} $
\end_inset
,
\begin_inset Formula $a_{n}\in I_{n}\,\left(n\in\mathbb{N}\right)$
\end_inset
.
Dann gilt
\begin_inset Formula \[
\underset{n\rightarrow\infty}{\lim}\left|I_{n}\right|=0\mbox{ und }\underset{n\rightarrow\infty}{\lim}a_{n}=a\]
\end_inset
gilt insbesondere für
\begin_inset Formula $I_{n}=\left[A_{n},\, B_{n}\right]$
\end_inset
\end_layout
\begin_layout Standard
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula $\underset{n\rightarrow\infty}{\lim}A_{n}=a=\underset{n\rightarrow\infty}{\lim}B_{n}$
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
Aus Lemma 2.10 folgt, dass für die
\begin_inset Index
status collapsed
\begin_layout Plain Layout
B-adische Entwicklung
\end_layout
\end_inset
\begin_inset Formula $B$
\end_inset
-adische Entwicklung
\begin_inset Formula $\left(\overset{n}{\underset{k=1}{\sum}}a_{k}b_{k}\right)_{n\in\mathbb{N}}$
\end_inset
von
\begin_inset Formula $x\in\left[0,\,1\right[$
\end_inset
gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula $x=\underset{n\rightarrow\infty}{\lim}\overset{n}{\underset{k=1}{\sum}}a_{k}B^{-k}=:\overset{\infty}{\underset{k=1}{\sum}}a_{k}B^{-k}$
\end_inset
\end_layout
\begin_layout Section
Satz von Bolzano-Weierstraß
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Satz von Bolzano-Weierstraß
\end_layout
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Bolzano-Weierstraß
\end_layout
\end_inset
(1.
Fassung)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine
\emph on
beschränkte
\emph default
Folge reeller Zahlen, d.h.
\begin_inset Formula $\exists M>0\,\forall n\in\mathbb{N}:\,\left|a_{n}\right|\leq M$
\end_inset
\end_layout
\begin_layout Standard
Dann besitzt
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
mindestens einen Häufungspunkt.
\end_layout
\begin_layout Section
Satz von Bolzano-Weierstraß
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Bolzano-Weierstraß
\end_layout
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Satz von Bolzano-Weierstraß
\end_layout
\end_inset
(2.
Fassung)
\end_layout
\begin_layout Standard
Jede beschränkte Folge
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
reeller Zahlen besitzt einen kleinsten und einen größten Häufungspunkt.
\begin_inset Index
status collapsed
\begin_layout Plain Layout
limes inferior
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Index
status collapsed
\begin_layout Plain Layout
limes superior
\end_layout
\end_inset
\end_layout
\begin_layout Standard
(Bezeichnung:
\begin_inset Formula $\underset{\mbox{Limes inferior}}{\underset{n\rightarrow\infty}{\liminf a_{n}}},\,\underset{\mbox{Limes superior}}{\underset{n\rightarrow\infty}{\limsup a_{n}}}$
\end_inset
)
\end_layout
\begin_layout Standard
Für alle
\begin_inset Formula $\varepsilon>0$
\end_inset
gilt: Fast alle (mit Ausnahme höchstens endlich vieler) Folgeglieder liegen
in
\begin_inset Formula $\left]\underset{n\rightarrow\infty}{\liminf}a_{n}-\varepsilon,\,\underset{n\rightarrow\infty}{\limsup}a_{n}+\varepsilon\right[$
\end_inset
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine beschränkte Folge mit einem einzigen Häufungspunkt
\begin_inset Formula $a$
\end_inset
.
Dann konvergiert
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
gegen
\begin_inset Formula $a$
\end_inset
.
\end_layout
\begin_layout Section
Definition (monotone Folgen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
monotone Folgen
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen.
Sie heißt
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
(streng) monoton wachsend
\begin_inset Index
status collapsed
\begin_layout Plain Layout
monoton wachsend
\end_layout
\end_inset
\begin_inset Formula $:\Leftrightarrow\forall n\in\mathbb{N}:\, a_{n+1}\overset{\left(>\right)}{\geq}a_{n}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
(streng) monoton fallend
\begin_inset Index
status collapsed
\begin_layout Plain Layout
monoton fallend
\end_layout
\end_inset
\begin_inset Formula $:\Leftrightarrow\forall n\in\mathbb{N}:\, a_{n+1}\overset{\left(<\right)}{\leq}a_{n}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
(streng) monoton, wenn sie (streng) monoton wachsend oder (streng) monoton
fallend ist.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Satz (Konvergenz beschränkter monotoner Folgen
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Konvergenz beschränkter monotoner Folgen
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Jede beschränkte monotone Folge konvergiert.
\end_layout
\begin_layout Standard
Bemerkung:
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
monoton wachsend und nach oben beschränkt
\begin_inset Formula $\Rightarrow a_{n}\leq\underset{k\rightarrow\infty}{\lim}a_{k}\,\left(n\in\mathbb{N}\right)$
\end_inset
\end_layout
\begin_layout Standard
Analog für monoton fallende nach unten beschränkte Folgen.
\end_layout
\begin_layout Section
Definition (Teilfolge
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Teilfolge
\end_layout
\end_inset
)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen und
\begin_inset Formula $\left(n_{k}\right)_{k\in\mathbb{N}}$
\end_inset
eine streng monoton wachsende Folge natürlicher Zahlen (d.h.
\begin_inset Formula $n_{1}<n_{2}<n_{3}<...$
\end_inset
)
\end_layout
\begin_layout Standard
Dann heißt
\begin_inset Formula $\left(a_{n_{k}}\right)_{k\in\mathbb{N}}=\left(a_{n_{1}},a_{n_{2}},a_{n_{3}},\,...\right)$
\end_inset
Teilfolge von
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
.
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
konvergiert gegen
\begin_inset Formula $a\Leftrightarrow$
\end_inset
Jede Teilfolge von
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
konvergiert gegen
\begin_inset Formula $a$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a$
\end_inset
ist Häufungspunkt von
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}\Leftrightarrow$
\end_inset
Es gibt eine Teilfolge
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
, die gegen
\begin_inset Formula $a$
\end_inset
konvergiert.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Satz von Bolzano-Weierstraß
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Satz von Bolzano-Weierstraß
\end_layout
\end_inset
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Bolzano-Weierstraß
\end_layout
\end_inset
(3.
Fassung)
\end_layout
\begin_layout Standard
Jede beschränkte Folge enthält (mindestens) eine konvergente Teilfolge.
\end_layout
\begin_layout Section
Ungleichung vom arithmetischen und geometrischen Mittel
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Ungleichung vom arithmetischen und geometrischen Mittel
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Seien
\begin_inset Formula $a_{1},\, a_{2},\,...,\, a_{n}\geq0\,\left(n\geq2\,\mbox{fest}\right)$
\end_inset
.
Dann gilt
\begin_inset Formula $\sqrt[n]{a_{1}\cdot a_{2}\cdot...\cdot a_{n}}\leq\frac{a_{1}+...+a_{n}}{n}$
\end_inset
, wobei Gleichheit genau dann eintritt, wenn
\begin_inset Formula $a_{1}=a_{2}=...=a_{n}$
\end_inset
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
Ein Produkt aus
\begin_inset Formula $n$
\end_inset
positiven reellen Zahlen mit konstanter Summe ist am größten, wenn alle
Faktoren gleich sind.
\end_layout
\begin_layout Section
Definition und Satz
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Es gilt
\begin_inset Formula \begin{eqnarray*}
\underbrace{\mathrm{e}}_{\text{Eulersche Zahl}}:=\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n} & = & \lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{1}{k!}\qquad\left[0!=1\right]\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Die Funktion
\begin_inset Formula $\text{exp}:\mathbb{R}\rightarrow\mathbb{R},\text{ exp}\left(x\right)=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}$
\end_inset
ist wohldefiniert und wird als Exponentialfunktion
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Exponentialfunktion
\end_layout
\end_inset
bezeichnet.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\forall x_{1},x_{2}\in\mathbb{R}:\,\exp\left(x_{1}+x_{2}\right)=\exp\left(x_{1}\right)\cdot\exp\left(x_{2}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Bemerkung:
\begin_inset Formula $\mathrm{e}=2.71828...$
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\underbrace{\exp(x)}_{\lim_{n\rightarrow\infty}(1+\frac{x}{n})^{n}}=\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{x^{k}}{k!} & =: & \sum_{k=0}^{\infty}\frac{x^{k}}{k!}\text{ (Exponentialreihe)}\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Section
Folgerung
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\exp\left(0\right)=1,\,\exp\left(1\right)=\mathrm{e},\,\exp\left(-\mathrm{e}\right)=\frac{1}{\exp(x)}\quad\left(x\in\mathbb{R}\right),\,\exp(x)>0\quad\left(x\in\mathbb{R}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\begin_inset Formula \[
\begin{array}{rcl}
\exp\left(m\right) & = & \mathrm{e}^{m}\,\left(m\in\mathbb{Z}\right)\\
\exp\left(\frac{1}{n}\right) & = & \sqrt[n]{\mathrm{e}}\quad\left(n\in\mathbb{N},\, n\geq2\right)\\
\exp\left(m\cdot x\right) & = & \left(\exp\left(x\right)\right)^{m}\quad\left(m\in\mathbb{Z},\, x\in\mathbb{R}\right)\\
\exp\left(\frac{x}{n}\right) & = & \sqrt[n]{\exp(x)}\quad\left(n\in\mathbb{N},\, n\geq2,\, x\in\mathbb{R}\right)\end{array}\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{1}>x_{2}\Rightarrow\exp\left(x_{1}\right)>\exp\left(x_{2}\right)\quad\left(x_{1},\, x_{2}\in\mathbb{R}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\exp\left(x\right)$
\end_inset
ist lipschitzstetig auf
\begin_inset Formula $\left]-\infty,\, a\right]$
\end_inset
\begin_inset Formula $\left(a\in\mathbb{R}\mbox{ fest}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}\rightarrow x\Rightarrow\exp\left(x_{n}\right)\rightarrow\exp\left(x\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[\underset{n\rightarrow\infty}{\lim}x_{n}\mbox{exist.}\Rightarrow\underset{n\rightarrow\infty}{\lim}\exp\left(x_{n}\right)=\exp\underset{n\rightarrow\infty}{\lim}x_{n}\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(f)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\exp\left(\mathbb{R}\right)=\underset{\mathbb{R}_{+}}{\underbrace{\left]0,\,\infty\right[}}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
Wegen 2.22b und 2.21c schreiben wir
\begin_inset Formula $\exp\left(x\right)=\mathrm{e}^{x}$
\end_inset
, insbesondere
\begin_inset Formula $\mathrm{e}^{0}=1,\,\mathrm{e}^{1}=\mathrm{e},\,\mathrm{e}^{-x}=\frac{1}{e^{x}},\,\mathrm{e}^{x_{1}+x_{2}}=\mathrm{e}^{x_{1}}\cdot\mathrm{e}^{x_{2}}$
\end_inset
und
\begin_inset Formula $\mathrm{e}^{m\cdot x}=\left(\mathrm{e}^{x}\right)^{m}\,\left(m\in\mathbb{Z}\right)$
\end_inset
\end_layout
\begin_layout Standard
[!
\begin_inset Formula $\left(\mathrm{e}^{x}\right)^{m}$
\end_inset
ist bisher nur für
\begin_inset Formula $m\in\mathbb{Z}$
\end_inset
definiert !]
\end_layout
\begin_layout Section
Satz und Definition
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Die Funktion
\begin_inset Formula $\mathbb{R}\rightarrow\mathbb{R}_{+},\, x\mapsto\exp\left(x\right)$
\end_inset
besitzt genau eine Umkehrfunktion
\end_layout
\begin_layout Standard
(Bezeichnung:
\begin_inset Formula $\ln:\mathbb{R}_{+}\rightarrow\mathbb{R}$
\end_inset
, natürlicher Logarithmus)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\mathrm{e}^{\ln x}=x\quad(x>0),\ln\mathrm{e}^{x}=x\quad\left(x\in\mathbb{R}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\ln\left(x_{1}\cdot x_{2}\right)=\ln x_{1}+\ln x_{2},\,\ln\frac{x_{1}}{x_{2}}=\ln x_{1}-\ln x_{2}\quad\left(x_{1},\, x_{2}>0\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\ln x^{m}=m\cdot\ln x\quad\left(m\in\mathbb{Z},\, x>0\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $0<x_{1}<x_{2}\Rightarrow\ln x_{1}<\ln x_{2}\quad\left(x_{1},\, x_{2}>0\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\ln$
\end_inset
ist lipschitzstetig auf
\begin_inset Formula $[a,\,\infty[\quad\left(a>0\mbox{ fest}\right)$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}>0\quad\left(n\in\mathbb{N}\right),\, x>0,\, x_{n}\rightarrow x\Rightarrow\ln x_{n}\rightarrow\ln x$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Vorbemerkung:
\end_layout
\begin_layout Standard
Es gilt für
\begin_inset Formula $a>0$
\end_inset
\begin_inset Formula \[
\begin{array}{rcl}
a^{m} & = & \left(\mathrm{e}^{\ln a}\right)^{m}=\mathrm{e}^{m\ln a}\,\left(m\in\mathbb{Z}\right)\\
\sqrt[n]{a} & = & \underset{\sqrt[n]{\exp\left(\ln a\right)}=\exp\left(\ln a\right)}{\underbrace{\sqrt[n]{\mathrm{e}^{\ln a}}}}=\mathrm{e}^{\frac{1}{n}\ln a}\\
\underset{a^{\frac{m}{n}}:=}{\underbrace{\sqrt[n]{a^{m}}}} & = & \mathrm{e}^{\frac{1}{n}\ln a^{m}}=\mathrm{e}^{\frac{m}{n}\ln a}\end{array}\]
\end_inset
Wir definieren deshalb für
\begin_inset Formula $a>0$
\end_inset
\begin_inset Formula \[
a^{x}:=\mathrm{e}^{x\ln a}\,\left(x\in\mathbb{R}\right)\]
\end_inset
(Exponentialfunktion zur Basis
\begin_inset Formula $a$
\end_inset
)
\end_layout
\begin_layout Section
Satz
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>0,\, b>0$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\begin{array}{rcl}
a^{x_{1}+x_{2}} & = & a^{x_{1}}\cdot a^{x_{2}}\,\left(x_{1},\, x_{2}\in\mathbb{R}\right)\\
\left(a\cdot b\right)^{x} & = & a^{x}\cdot b^{x},\,\left(\frac{a}{b}\right)^{x}=\frac{a^{x}}{b^{x}}\,\left(x\in\mathbb{R}\right)\\
\left(a^{x_{1}}\right)^{x_{2}} & = & a^{x_{1}\cdot x_{2}}\,\left(x_{1},\, x_{2}\in\mathbb{R}\right)\end{array}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1$
\end_inset
.
Dann
\begin_inset Formula $x_{1}<x_{2}\Rightarrow a^{x_{1}}<a^{x_{2}}\,\left(x_{1},\, x_{2}\in\mathbb{R}\right)$
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $0<a<1$
\end_inset
.
Dann
\begin_inset Formula $x_{1}<x_{2}\Rightarrow a^{x_{1}}>a^{x_{2}}\,\left(x_{1},\, x_{2}\in\mathbb{R}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{1}\rightarrow x\Rightarrow a^{x_{n}}\rightarrow a^{x}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei ab jetzt
\begin_inset Formula $a>0$
\end_inset
und
\begin_inset Formula $a\neq1.$
\end_inset
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Die Funktion
\begin_inset Formula $\mathbb{R}_{+},\, x\rightarrow a^{x}$
\end_inset
besitzt genau eine Umkehrfunktion (Bez.
\begin_inset Formula $\log_{a}x$
\end_inset
Logarithmus zur Basis
\begin_inset Formula $a$
\end_inset
).
\end_layout
\begin_layout Standard
Es gilt
\begin_inset Formula $\log_{a}(x)=\frac{\ln x}{\ln a}\quad(x\in\mathbb{R})$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\begin_inset Formula \begin{eqnarray*}
\log_{a}a^{x} & = & x\quad\left(x\in\mathbb{R}\right),\, a^{\log_{a}x}=x\quad\left(x>0\right)\\
\log\left(x_{1}\cdot x_{2}\right) & = & \log_{a}x_{1}+\log_{a}x_{2}\quad\left(x_{1,}x_{2}>0\right)\\
\log_{a}\left(\frac{x_{1}}{x_{2}}\right) & = & \log_{a}x_{1}-\log_{a}x_{2}\quad\left(x_{1},x_{2}>0\right)\\
\log_{a}\left(x_{1}^{x_{2}}\right) & = & x_{2}\log_{a}x_{1}\quad\left(x_{1}>0,x_{2}\in\mathbb{R}\right)\end{eqnarray*}
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(f)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{1}>0\quad\left(n\in\mathbb{N}\right),\, x>0,\, x_{n}\rightarrow x\Rightarrow\log_{a}x_{n}\rightarrow\log_{a}x$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(g)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei jetzt
\begin_inset Formula $a>0,\, b\in\mathbb{R}.$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}>0\quad(n\in\mathbb{N}),\, a_{n}\rightarrow a,\, b_{n}\rightarrow b\Rightarrow a_{n}^{b_{n}}\rightarrow a^{b}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left[a_{n}>0\,\left(n\in\mathbb{N}\right),\,\underset{n\rightarrow\infty}{\lim}a_{n}>0,\,\underset{n\rightarrow\infty}{\lim}b_{n}\mbox{ exist. }\Rightarrow\underset{n\rightarrow\infty}{\lim}a_{n}^{b_{n}}=\left(\underset{n\rightarrow\infty}{\lim}a_{n}\right)^{\underset{n\rightarrow\infty}{\lim}b_{n}}\right]$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[1.]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Neben
\begin_inset Formula $\log_{\mathrm{e}}=\ln$
\end_inset
werden häufig noch
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begingroup
\backslash
leftskip1em
\backslash
rightskip2em
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\log_{2}=:\mbox{ld}$
\end_inset
(Logarithmus dualis)
\end_layout
\begin_layout Standard
\begin_inset Formula $\log_{10}=:\lg$
\end_inset
(Zehnerlogarithmus)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
endgroup
\end_layout
\end_inset
\end_layout
\begin_layout Standard
verwendet.
Sie sind nach (d) jeweils konstante Vielfache von
\begin_inset Formula $\ln$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[2.]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Ohne die Voraussetzung
\begin_inset Formula $a>0$
\end_inset
kann (g) falsch sein.
(ÜA)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Definition
\end_layout
\begin_layout Standard
Eine Folge
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
heißt
\emph on
bestimmt divergent
\emph default
gegen
\begin_inset Formula $+\infty$
\end_inset
bzw.
\begin_inset Formula $\left(-\infty\right)$
\end_inset
, wenn
\end_layout
\begin_layout Standard
\begin_inset Formula $\begin{array}{rl}
\forall M>0\,\exists n_{0}\in\mathbb{N}\,\forall n\geq n_{0}: & a_{n}\geq M\\
\mbox{bzw.} & \left(a_{n}\leq-M\right)\end{array}$
\end_inset
\end_layout
\begin_layout Standard
Schreibweise:
\begin_inset Formula \begin{eqnarray*}
a_{n} & \rightarrow & \infty\quad(\text{bzw. }a_{n}\rightarrow-\infty)\\
\lim_{n\rightarrow\infty}a_{n} & = & \infty\quad(\text{bzw. }\lim_{n\rightarrow\infty}a_{n}=-\infty)\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
a_{n}>0\quad\left(n\in\mathbb{N}\right),\, a_{n}\rightarrow0 & \Rightarrow & \frac{1}{a_{n}}\rightarrow\infty\\
a_{n}<0\quad\left(n\in\mathbb{N}\right),\, a_{n}\rightarrow0 & \Rightarrow & \frac{1}{a_{n}}\rightarrow-\infty\\
a_{n}\rightarrow\infty & \Rightarrow & \frac{1}{a_{n}}\rightarrow0\\
a_{n}\rightarrow-\infty & \Rightarrow & \frac{1}{a_{n}}\rightarrow0\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Section
Satz
\size large
(Wachstumsverhalten der Exponentialfunktion und des Logarithmus)
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{itemize}
\backslash
item[(a)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1,\,\beta\geq0$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}>0\,\left(n\in\mathbb{N}\right),\, x_{n}\rightarrow\infty\Rightarrow\frac{a^{x_{n}}}{x^{\beta}}\rightarrow\infty$
\end_inset
(bzw.
\begin_inset Formula $\frac{x^{\beta}}{a^{x_{n}}}\rightarrow0$
\end_inset
)
\end_layout
\begin_layout Standard
[Später schreiben wir hierfür:
\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}\frac{a^{x}}{x^{\beta}}=\infty$
\end_inset
]
\end_layout
\begin_layout Standard
\begin_inset Quotes pld
\end_inset
\begin_inset Formula $a^{x}$
\end_inset
wächst für
\begin_inset Formula $x\rightarrow\infty$
\end_inset
schneller als jede Potenz
\begin_inset Quotes prd
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(b)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1,\, k\in\mathbb{N}_{0}$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}\rightarrow-\infty\Rightarrow x_{n}^{k}\cdot a^{x_{n}}\rightarrow0$
\end_inset
\end_layout
\begin_layout Standard
[später:
\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}x_{n}^{k}\cdot a^{x_{n}}=0$
\end_inset
]
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(c)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}\rightarrow\infty\Rightarrow\log_{a}x\rightarrow\infty$
\end_inset
\end_layout
\begin_layout Standard
[Später:
\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}\log_{a}x=\infty$
\end_inset
]
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(d)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}>0\,\left(n\in\mathbb{N}\right),\, x_{n}\rightarrow0\Rightarrow\log_{a}\left(x_{n}\right)\rightarrow-\infty$
\end_inset
\end_layout
\begin_layout Standard
[Später:
\begin_inset Formula $\underset{x\rightarrow0^{+}}{\lim}\log_{a}\left(x\right)=-\infty$
\end_inset
]
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(e)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1,\,\beta>0$
\end_inset
.
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}>0\,\left(n\in\mathbb{N}\right),\, x_{n}\rightarrow\infty\Rightarrow\frac{\log_{a}\left(x_{n}\right)}{x_{n}^{\beta}}\rightarrow0$
\end_inset
\end_layout
\begin_layout Standard
[Später:
\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}\frac{\log_{a}\left(x\right)}{x^{\beta}}=0$
\end_inset
]
\end_layout
\begin_layout Standard
\begin_inset Quotes pld
\end_inset
\begin_inset Formula $\log_{a}\left(x\right)$
\end_inset
wächst für
\begin_inset Formula $x\rightarrow\infty$
\end_inset
langsamer als jede Potenz
\begin_inset Quotes prd
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
item[(f)]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $a>1,\,\beta>0.$
\end_inset
Dann gilt
\end_layout
\begin_layout Standard
\begin_inset Formula $x_{n}>0\quad(n\in\mathbb{N}),x_{n}\rightarrow0\Rightarrow x_{n}^{\beta}\log_{a}(x_{n})=0$
\end_inset
\end_layout
\begin_layout Standard
[später:
\begin_inset Formula $\lim_{x\rightarrow0^{+}}x^{\beta}\log_{a}(x)=0$
\end_inset
]
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{itemize}
\end_layout
\end_inset
\end_layout
\begin_layout Subparagraph*
Bemerkung:
\end_layout
\begin_layout Standard
Auf die Formulierung entsprechender Aussagen für
\begin_inset Formula $0<a<1$
\end_inset
wird verzichtet, weil sie sich leicht über die Beziehungen
\begin_inset Formula $a^{x}=\left(\frac{1}{a}\right)^{-x}$
\end_inset
und
\begin_inset Formula $\log_{a}\left(x\right)=-\log_{\frac{1}{a}}\left(x\right)$
\end_inset
gewinnen lassen.
\end_layout
\begin_layout Section
Definition (Cauchyfolge)
\end_layout
\begin_layout Standard
Eine Folge
\begin_inset Formula $(a_{n})_{n\in\mathbb{N}}$
\end_inset
reeller Zahlen heißt
\emph on
Cauchyfolge
\emph default
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Cauchyfolge
\end_layout
\end_inset
, wenn
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\forall\epsilon>0\quad\exists N\in\mathbb{N}\quad\forall n,\, m\geq N:\left|a_{n}-a_{m}\right| & < & \epsilon\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Section
Satz (Cauchykriterium
\begin_inset Index
status collapsed
\begin_layout Plain Layout
Cauchykriterium
\end_layout
\end_inset
für Folgen)
\end_layout
\begin_layout Standard
Sei
\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
\end_inset
eine Folge reeller Zahlen.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\left(a_{n}\right)_{n\in\mathbb{N}}\text{ konvergent } & \Leftrightarrow & \left(a_{n}\right)_{n\in\mathbb{N}}\text{ Cauchyfolge}\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset index_print
LatexCommand printindex
\end_inset
\end_layout
\end_body
\end_document
