From 564379dc6b8a00d13343b3880c2e793c86b8230f Mon Sep 17 00:00:00 2001 From: JirR02 Date: Tue, 3 Jun 2025 10:46:38 +0200 Subject: [PATCH] Analysis Chapter 2 Added Analysis Chapter 2 --- fs25/analysis_I_II/.DS_Store | Bin 6148 -> 0 bytes fs25/analysis_I_II/grundlagen/funktionen.tex | 16 +- fs25/analysis_I_II/grundlagen/logik.tex | 10 +- fs25/analysis_I_II/grundlagen/mengenlehre.tex | 6 +- fs25/analysis_I_II/grundlagen/quantoren.tex | 4 +- .../notizen_analysis_ruh_jirayu.bbl | 2 +- .../notizen_analysis_ruh_jirayu.pdf | Bin 310745 -> 373405 bytes fs25/analysis_I_II/preamble.tex | 57 +++---- fs25/analysis_I_II/sources.bib | 9 +- fs25/analysis_I_II/vektorfeld/satz_green.aux | 12 +- .../zahlen_und_vektoren/komplexe_zahlen.tex | 5 + .../natuerlichen_ganzen_rationalen_zahlen.tex | 30 ++++ .../zahlen_und_vektoren/reellen_zahlen.tex | 142 ++++++++++++++++++ .../zahlen_und_vektoren/supremum_infimum.tex | 58 +++++++ .../zahlen_und_vektoren.tex | 7 + 15 files changed, 297 insertions(+), 61 deletions(-) delete mode 100644 fs25/analysis_I_II/.DS_Store create mode 100644 fs25/analysis_I_II/zahlen_und_vektoren/komplexe_zahlen.tex create mode 100644 fs25/analysis_I_II/zahlen_und_vektoren/natuerlichen_ganzen_rationalen_zahlen.tex create mode 100644 fs25/analysis_I_II/zahlen_und_vektoren/reellen_zahlen.tex create mode 100644 fs25/analysis_I_II/zahlen_und_vektoren/supremum_infimum.tex diff --git a/fs25/analysis_I_II/.DS_Store b/fs25/analysis_I_II/.DS_Store deleted file mode 100644 index f6c670c800d84ab00d4b3dd3b386c680e0023f36..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 zcmeHK%}T>S5T0$TZ74zx3OxqA7EF;!#7l_v1+3^nr6#1+U`$Grl0zxvtS{t~_&m<+ zZpBi)NKq*>F#FBU&g?Q@!u|k&@TOr0zyyE>Dq+RI<_n>5(iN#_522trvT+cn0rp;l z@k%s1{v!kQ?W!<>2=0U8^Y@FREVu19QK{Cpw(E7H-mKsDAJU?q`}s8U{K+--F48m( z7PcQ;M#ExY?w+Pe?nlXRA`_x;h#@yuQ4*$wmrj!~lev90U>FT!V76wnW4qh7I-Pmj znmLX%JNA5TH1_rnPR<99@l%pMD+&evZBlkLF5m@?DvTk7(O*|{WIKy$y3Qe)KdNR? z86z{m3@`)hWI*qtjpjNh<++#vW?&;2p#8x@C3H2G3gy;;4P75;yhccZHoYYXEsd_m zQX!6@2$hPcQiXnE2$hb0OUJnyONA;Ogj^Zt(JKr6LJ@Lx^jk6&3S1SBSK|>zJ7)wWS3snmG YEiw>Yjio~LpzuXN(!dQf@T&~G18mHDE&u=k diff --git a/fs25/analysis_I_II/grundlagen/funktionen.tex b/fs25/analysis_I_II/grundlagen/funktionen.tex index d6648a0..d9f26f7 100644 --- a/fs25/analysis_I_II/grundlagen/funktionen.tex +++ b/fs25/analysis_I_II/grundlagen/funktionen.tex @@ -15,10 +15,10 @@ Intuitiv ist eine Funktion (oder Abbildung) von $X$ nach $Y$ eine Vorschrift, di } \dfn{Definitionsbereich, Zielbereich, Graph, Wert in einem Punkt}{ - Für die Funktion $f$ haben wir folgende Definitonen um die Eigenschaften einer Funktion zu definieren. + Für die Funktion $f$ haben wir folgende Definitionen um die Eigenschaften einer Funktion zu definieren. \begin{itemize} - \item $\text{dom } f := \text{dom}(f) := \text{Definitionsbereich von } f := X$ Der Definitonsbereich sind die Werte von $x$, welche für diese Funktion erlaubt sind. + \item $\text{dom } f := \text{dom}(f) := \text{Definitionsbereich von } f := X$ Der Definitionsbereich sind die Werte von $x$, welche für diese Funktion erlaubt sind. \item $\text{codom } f := \text{codom}(f) := \text{Zielbereich von } f := Y$ Der Zielbereich sind die Werte von $y$, welche für diese Funktion erlaubt sind. \item Graph von $f := G$ \item Wert von $f$ an der Stelle $x \in X := f(x) := y$ @@ -81,13 +81,13 @@ Wir werden nun weitere Eigenschaften von Funktionen kennenlernen: Die Injektivit \begin{itemize} \item Die Identität $\text{id}_x$ ist bijektiv \item Die Funktion $f : [ 0, \infty ) \rightarrow \mathbb{R}, f(x) := x$, ist injektiv und nicht surjektiv. - \item Die Funktion $f :\mathbb{R} \rightarrow [ 0, \infty ) , f(x) := x ^2$, sit nicht injektiv, aber surjektiv. + \item Die Funktion $f :\mathbb{R} \rightarrow [ 0, \infty ) , f(x) := x ^2$, ist nicht injektiv, aber surjektiv. \item Die Funktion $f : \mathbb{R} \rightarrow \mathbb{R}, f(x) := x ^2$, ist weder injektiv noch surjektiv. \end{itemize} } \nt{ - Die Identität ist eine Funktion welches sich selber wieder ausgibt. + Die Identität ist eine Funktion, welches sich selber wieder ausgibt. \[ f(x) = x @@ -96,7 +96,7 @@ Wir werden nun weitere Eigenschaften von Funktionen kennenlernen: Die Injektivit } \dfn{Umkehrfunktion / Inverse}{ - Die Umkehrfunktion oder Inverse einer Funktion ist eine Funktion, welches das Gegenteil der Ursprünglichen Funktion $f$ macht. + Die Umkehrfunktion oder Inverse einer Funktion ist eine Funktion, welches das Gegenteil der ursprünglichen Funktion $f$ macht. \[ f ^{(-1)} : Y \rightarrow X, f ^{(-1)}(y) := x @@ -120,9 +120,9 @@ Wir werden nun weitere Eigenschaften von Funktionen kennenlernen: Die Injektivit g \circ f : X \rightarrow Z, g \circ f(x) := g(f(x)) .\] - Dies bdeuetet nichts weiter, dass der codom($f$) in die Funktion $g$ eingesetzt wird, und ELemente von $Z$ dabei herauskommen. + Dies bedeutet nichts weiter, dass der codom($f$) in die Funktion $g$ eingesetzt wird, und Elemente von $Z$ dabei herauskommen. \\ - Wichtig zu erwähnen ist, dass die codom($f$) = dom($g$) ist weil sonst die Verknüpfung nicht funktionieren würde. + Wichtig zu erwähnen ist, dass die codom($f$) = dom($g$) ist, weil sonst die Verknüpfung nicht funktionieren würde. } \exa{Verknüpfung, Reihenfolge davon \cite{Ziltner2024}}{ @@ -163,6 +163,6 @@ Wir werden nun weitere Eigenschaften von Funktionen kennenlernen: Die Injektivit g \circ f : \mathbb{R} ^2 \rightarrow \mathbb{R}, g \circ f(x,y) = \text{exp}(x + y) = e ^{x + y} .\] - Die umgekehrte Verknüpfung $f \circ g$ ist nicht wohldefiniert (= sinnvoll), da der Yielbereich von $g$, also $\mathbb{R}$, nivcht gleich dem Definitionsbereich von $f$, also $\mathbb{R} ^2$, ist. + Die umgekehrte Verknüpfung $f \circ g$ ist nicht wohldefiniert (= sinnvoll), da der Zielbereich von $g$, also $\mathbb{R}$, nicht gleich dem Definitionsbereich von $f$, also $\mathbb{R} ^2$, ist. \end{itemize} } diff --git a/fs25/analysis_I_II/grundlagen/logik.tex b/fs25/analysis_I_II/grundlagen/logik.tex index 376681c..39e5f04 100644 --- a/fs25/analysis_I_II/grundlagen/logik.tex +++ b/fs25/analysis_I_II/grundlagen/logik.tex @@ -2,13 +2,13 @@ \subsection{Grundlagen} -In der Logik werden (mathematische) Aussagen untersucht. Eine Aussage ist eine Äusserung, die entweder wahr oder falsch ist. \cite{Ziltner2024} (Wahr oder Falsch). +In der Logik werden (mathematische) Aussagen untersucht. Eine Aussage ist eine Äusserung, die entweder wahr oder falsch ist. \cite{Ziltner2024} (wahr oder falsch). \\ \\ In der mathematischen Logik gelten die folgenden Sätze. \begin{itemize} - \item \textbf{Satz vom ausgeschlossenen Wiederspruch:} Eine Aussage ist nicht sowohl war als auch falsch. + \item \textbf{Satz vom ausgeschlossenen Widerspruch:} Eine Aussage ist nicht sowohl war als auch falsch. \item \textbf{Satz vom ausgeschlossenen Dritten:} Jede Aussage ist wahr oder falsch. \end{itemize} \cite{Ziltner2024} @@ -42,7 +42,7 @@ Für Verknüpfungen verwenden wir folgende Notationen. \end{tabular} \end{center} -Die Wahrheitstabelle der vorher erwähnten Verknüpfungen sind wie folgt. +Die Wahrheitstabelle der vorher erwähnten Verknüpfungen ist wie folgt. \begin{center} \begin{tabular}{| c | c | c | c | c | c | c |} @@ -58,7 +58,7 @@ Die Wahrheitstabelle der vorher erwähnten Verknüpfungen sind wie folgt. Aus der Tabelle kann man die Zusammenhänge der Verknüpfungen erkennen. \nt{ - Wir unterscheiden zwischen dem inklusiven Oder und dem exklusiven Oder. Beim inklusiven Oder können beide Aussagen warh sein während beim exklusiven oder nur einer der beiden Aussagen wahr sein kann. + Wir unterscheiden zwischen dem inklusiven Oder und dem exklusiven Oder. Beim inklusiven Oder können beide Aussagen wahr sein während beim exklusiven oder nur einer der beiden Aussagen wahr sein kann. } \nt{ @@ -105,7 +105,7 @@ In der Mathematik sind Axiome von grosser Bedeutung. Sie sind das Fundament der Um Aussagen zu Beweisen, verwenden wir in der Logik den Modus Ponens. \dfn{Modus Ponens}{ - Ein Beweis einer Aussage $A$ ist eine sukzessive Herleitung von $A$ aus dem Axiomen, in der logische Schlussregeln angewendet werden. Eine solche Regel ist der Modus Ponens. + Ein Beweis einer Aussage $A$ ist eine sukzessive Herleitung von $A$ aus den Axiomen, in der logische Schlussregeln angewendet werden. Eine solche Regel ist der Modus Ponens. \begin{tabular}{ c } $A$ \\ diff --git a/fs25/analysis_I_II/grundlagen/mengenlehre.tex b/fs25/analysis_I_II/grundlagen/mengenlehre.tex index 1090c02..110425f 100644 --- a/fs25/analysis_I_II/grundlagen/mengenlehre.tex +++ b/fs25/analysis_I_II/grundlagen/mengenlehre.tex @@ -71,7 +71,7 @@ Wenn die Menge $A$ auch in der Menge $B$ liegt, so ist $A$ eine Teilmenge von $B } \thm{Tupel}{ - Wenn wir eine Liste von Elementen, bestehen aus $x_1, ..., x_{n}$ haben, so nennen wir diese Liste ein Tupel. Die Anzahl von Elementen $n$ sowie die Anordunng der Elementen spielt eine Rolle. + Wenn wir eine Liste von Elementen, bestehen aus $x_1, ..., x_{n}$ haben, so nennen wir diese Liste ein Tupel. Die Anzahl von Elementen $n$ sowie die Anordnung der Elementen spielt eine Rolle. \begin{itemize} \item Für $n = 2$ nennen wir den Tupel ein Paar. @@ -79,7 +79,7 @@ Wenn die Menge $A$ auch in der Menge $B$ liegt, so ist $A$ eine Teilmenge von $B \item Für alle anderen $n$ bezeichnet man die Liste als n-Tupel. \end{itemize} - Wie schon vorher erwähnt spielt die Anordung eine grosse Rolle. ($(x_1, x_2) \neq (x_2, x_1)$) + Wie schon vorher erwähnt spielt die Anordnung eine grosse Rolle. ($(x_1, x_2) \neq (x_2, x_1)$) } \thm{Karthesisches Produkt}{ @@ -144,7 +144,7 @@ Die euklidische Norm $|| \cdot ||$ ist die Distanz von einem Punkt, z.B. $\nu$ z \end{figure} \nt{ - Das die Sphäre eine Dimension verliert ($n-1$) ist auf dem ersten Blick verwirrend, macht aber Sinn. Bei einer Sphäre wird nur der Mantel betrachtet. Dadurch wird der Freiheitsgrad verringert was dazu führt, dass eine Dimension verloren geht i.e. der Mantel einer Kugel ist eine Fläche oder der Rand einer Kreisscheibe ist eine Linie. + Dass die Sphäre eine Dimension verliert ($n-1$) ist auf den ersten Blick verwirrend, macht aber Sinn. Bei einer Sphäre wird nur der Mantel betrachtet. Dadurch wird der Freiheitsgrad verringert was dazu führt, dass eine Dimension verloren geht i.e. der Mantel einer Kugel ist eine Fläche oder der Rand einer Kreisscheibe ist eine Linie. } \thm{Fall $r = 0$ und $r = \infty$}{ diff --git a/fs25/analysis_I_II/grundlagen/quantoren.tex b/fs25/analysis_I_II/grundlagen/quantoren.tex index 4306b0d..a044a22 100644 --- a/fs25/analysis_I_II/grundlagen/quantoren.tex +++ b/fs25/analysis_I_II/grundlagen/quantoren.tex @@ -23,10 +23,10 @@ Quantoren sind logische Operatoren, die angeben, wie viele Objekte $x$ eine Bedi } \nt{ - Die Reihenfolge der Quantoren spielt eine Rolle. Dies können wir an den vorheringen Beispielen erkennen. + Die Reihenfolge der Quantoren spielt eine Rolle. Dies können wir an den vorherigen Beispielen erkennen. } -\thm{Verneinung einer quantifizierten Assageform}{ +\thm{Verneinung einer quantifizierten Aussageform}{ Die Verneinung von den Quantoren $\forall$ und $\exists$ ist wie folgt definiert. \[ diff --git a/fs25/analysis_I_II/notizen_analysis_ruh_jirayu.bbl b/fs25/analysis_I_II/notizen_analysis_ruh_jirayu.bbl index d6c5f59..f547742 100644 --- a/fs25/analysis_I_II/notizen_analysis_ruh_jirayu.bbl +++ b/fs25/analysis_I_II/notizen_analysis_ruh_jirayu.bbl @@ -5,7 +5,7 @@ \newblock {Manim - Mathematical Animation Framework}. \bibitem[Ziltner, 2024]{Ziltner2024} -Ziltner, Prof.~Dr., F. (2024). +Ziltner, F. (2024). \newblock {Notizen zur Vorlesung Analysis 1 für ITET und RW, Herbstsemester 2024}. \end{thebibliography} diff --git a/fs25/analysis_I_II/notizen_analysis_ruh_jirayu.pdf b/fs25/analysis_I_II/notizen_analysis_ruh_jirayu.pdf index e9045f67c5ce636bf58fcf7adb42488534a0018e..697282ebb4da032cdf5f8f16288bdd9b2db8eca5 100644 GIT binary patch delta 219716 zcmZs?V{oopu(li9wr$(CXFOxuPTttIZQIU_oy^#_?X%YU_E)vfsZ%w6^wR@Bx@z1{ zUEMv_K2UOwFzOLuK$w}*9DRY&K)IPXl7tDV0oogOxSUA7D;l@TAdPG{vd{Fw`$W>9 ze;ejNTc3ralqd%@DsbLdZem!7#Zk@8py8Wv$1ozS;<~(Ue72aM_%{? zN5b>ljwnO=z?6%r%%leN3#mGDpL>asJfpgG;|%PY#Sq8Xhx!UpNpWCtioy+tj91{_ z0i=mPb$)%h@3CO8uhifn3ShJq?BdQ!Ch`wuDR#)J5VV|#e1pl*oQqY4Hqf`Y5u@e?7?@Q_f1k1xj6p#_FL}E5ZXHN?^X+rCw>g1?R_jD+K19s;SBYgV3rrk z*|}Igh^JZhJhc`OXyC{emf#}$)Io_a0Kk@P#ocwiHw>q`@tGgZ4+2QjVi(fq12b|5qp zKxzQDDMYthHfRNAu>ab4W9gEX1Q=(1A@SWF*ko_^LhT5y!x0<`&Ok~$rmbC(WSDZm z`mIlL+#a{7m;ylsJ72mwCWGVX`x2i*5cEnC9Q!*_@`$V9$d!lQP}`6`{H4};uPDGOgP>Gu zs0Y`c+Fzqt#TWK@1_3V53~)UXJ{p*`pFI2!8u?PTY8pkzSrV*vPQN^|AGfj652j}9 zSxBa#BWe<#OPj0Yj%}SRo!_4iwW|jL6wq}R}J^gb}<6uzymWXGO_VEBayyq?xvRU%HwGLM_dq>-ba`iJvQPt1p>2WFv7q9U(Hf~X7O;n#9w-K)M1-2 zu2C{hhrwx6#8}Fv2*~0!C#WkZv5uq^ij~_Ta~wnsZB9iN18`YhSo}D+x$c6Xt{sLMx641^{Yu(n8&}DK|`JGZcHTGVt8&`6YlN1u4POCBK2&Ljl^YucB1;HbW z@bp0>A^25U{TgWb&ed+SvarByt2h@N{z z!65Hg3bLcQo0g~mpU&6t$$HvV1<7BkRW5%x=H@2?D0wCj$uje`-H8S~qNN@C8RGcj zsesQA0BOMSG4yt_{rC)86* z3TC-=g}<@j-ZX*Q9kfDO;ef&q?_B_#G=TG^mSzv5{_O)8XGFz?6PiWJr1Q5UMxge6PXW^+t zqIX$Tzmcdd?4(lf61dR%qn!rWud%LWeoEBH=ZwuiK}cFE+0SQ?^+W}l4c8ML{{+vV zI|N)KNO0QM3pxt{0gh!4ALL{AMl8K6BLCV-3}?%SzQ`t1p2b}mL@Cb$#mKty+R+N& zjI80qIH@tj6&zyi=P=FDH4f`^4eKPhET4$@J|7A!q2+{Ya=$oEba3MU6H+DlZm;6q zl|GB%`BLp+dkQ7NsRRo0-V&^m>xRPfiqZP33yBAgLjL26wRHKxli|Di%tutb&q=;3 z|0LL08V&`HHHq-8ZF}Pr7HoG*8O9XV&cxZp$<)vm#?;Ql-pC4unT;a}83q-Um4*Gk z|8s_{{RS73|IIf#ACJ&*_|S_XHL4-RYDK~*1C1_K0jMq+YyT_3hzX)toC3{5*3j*3+~MceQ$uW*Gk ziWnsI(X!Ni(5s=KD7bV5TD0Y-`T~|jCE)D(j#e#^$+{e<22C1B14$x&pL|G8sg%3T zBy@(ki9!K=05i_J$nbdvkE*27N~BSSr^EJzm(j9k(dR;L}yDV@~ZXm-V0q&~6x zVP*&ZGD@iqcM^9hfIDOUV0kU@M=3hhE<=$}!C~`g5-UxEJ81r+w-uR-r=PnvhIc%r z?e<+qhJwpeSVYBG!FxW?e?*?6900Ru&epzp6d}i)a2QHS%C;!JAq?rQ^Fc{2W2e59 zi?gP$8n#5RZVz1Z%&q&lr5!aM*)oJxwrPRn4S4~xfuW1Js!CxOPk}6=Qp;owHkw~f zYNZ#4Q$lZr)KhI~(rRe@VYsOBEnr#HTxvOyTOd!0{`tfIlgV;Tqv2Ic2Y9ei@7O>^ z82VtUrzvA*^V~Cb_8L){U-rbO=reN3w3LTil4+_hJWSTw=On6q-Wd3+0?`x>PDtd{ znWl|LHI^n+g&NP%JY{FmGUyYmj;VdG5rTmSLeD@AwTM`wZHB8&=Nqc6xc;I9`)zLJ zSRAY}t6Y@|1N^G;hvT$Y8(=b-*DJX1#Wg5P2;(EBnpEg)QC%bYY@a@WT+3}4Q)eeq zbW~YR%1B`THY)|x&HvD|TIr*C{IHPn#G_9_^ZS>HX@=rWqgEHaC^CC?yLj>!o8tc9 zHbnr1VZ@pl?#U7vV^axjmCY}@a|Xs{fEKqcFt$135rtyLtV3aX2_W!H&yFQz+}?Wt zTEJFr4q~c#06QDKTx1;He!y-anOPJbBWI94)EE(~Do~a9GNIJItU?VhZLx>V*2HZt z`DwAanfPXuCynj6q2IWq+6=+2rY?GgX-TQ@V_?XuDjl}L0OlH{!^L#CrVl`?Uu@|QnPk*5aINd z?S`##Y~y{N-QjE6^T@~;2LbA&0sgm5wO4KpMwFP?wB2{{D4@}Le0T4$*P#cIFd1^RMwl3;r1!P$w9!*oF8-{*tQ)U(}A z!bL^xos$N0TH|-7-?HGNb(=YY58bYi#tD|y6jRo9deR~{KgwKKE+^uWiTK7Fx`*EF ztMr-riBaBf?Fj_<0{3XRW^#7GX!UXj(!*x_pOAE#Tqgf&>HkMnIhdPrF<3xBI5_`R z*Q6utX%J49q=9$RG(}}#A5dma)+A&Lsia(7A`njIfBi{yEJ#VU7JMMgoK16V0N|u( z#;c?p0>Y+rrhDL|-)!(nT}03zO#g{=(hU&|;BPAQfxvz9Glk>Psv2HZVkOra8fgqw zZF#{osluBnz_-hBdVn}AnWE)mmnIAE5y-ayH%21J%kOFEr8?c7!Gf95;*JK;*WKyy zUE&5_CT3k5S+ya;ItWY8Za9Ue!8yD!&e`N1W!ur64O2vMdB0LsH-se}%?>v@XGDV! 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z>$ACG0)yM6w1xq2dh>7_@bFox`By5R=7Qz$l$epjaBOsNkNuFLWhnVU zh?gi>Fu!H;qKR^CILenC^6DkJxz!^5$MC2nWYN8N;o{MM-N0(Sbv<=8mp;oBo;VqR z%mgE(f%H|9iwU+-k~tH7Ka^VogQdq<1G!3)uLAZbIOw@ r \Rightarrow s \in x$ + \item[(d)] $\forall r \in x \exists s_0 \in x : s_0 < r$ +\end{itemize} + +Wir definieren + +\[ + \mathbb{R} := {\text{reelle Zahl}} = {\text{Dedekind-Schnitt}} + .\] +} + +\nt{ + Die Definition von Herr Ziltner ist eine alternative Definition. Normalerweise tut man die untere Hälfte definieren. Da die Rechenoperationen der unteren Hälfte aufwendiger zu definieren ist als die obere, definieren wir die untere Hälfte. +} + +In anderen Worten ist eine reelle Zahl eine Menge von rationalen Zahlen, welche in eine oberen und in einer unteren Hälfte unterteilt ist. Dies beide Hälften sind eine Teilmenge der rationalen Zahlen. Punkt (a) besagt, dass die untere Hälfte rationale Zahlen beinhalten muss und nicht die leere Menge sein darf. Zusätzlich darf die untere Hälfte nicht eine reelle Zahl sein da sonst die untere Hälfte die ganze Zahlenebene wäre. Dies besagt Punkt (b). Punkt (c) sagt aus, dass eine rationale Zahl $s$ gibt, welche kleiner ist als die Zahl $r$, welche sich in der unteren Hälfte befindet. Zusätzlich gilt laut (d), dass es kein grösstes Element $s_0$ gibt, welches grösser als $r$ ist. + +\nt{ + Damit es keine Verwirrung gibt zwischen der reellen Zahl $r = \sqrt{2}$ und der reellen Zahl als ein Intervall von einem Dedekind-Schnitt wird diese als $\mathbf{r}$ gekennzeichnet. (In der Vorlesung \framebox{r}) +} + +Formell beschreiben wir der Dedekind-Schnitt +\[ + \mathbf{r} := {s \in \mathbb{Q} | s < r} \in \mathbb{R} +\] +was nichts anderes Bedeutet als $\mathbf{r}$ ist die Menge aller rationalen Zahlen $s$, wobei $s$ kleiner als $r$, der Grenzwert vom Intervall ist. + +Da wir die reellen Zahlen als eine Menge definiert haben, kann man die üblichen Rechenoperationen nicht mehr wie bei ''normalen'' Zahlen verwenden. + +\dfn{Ordnung, Addition, Multiplikation reeller Zahlen \cite{Ziltner2024}}{ + \begin{itemize} + \item[(i)] (Ordnung:) Für $x, y \in \mathbb{R}$ definieren wir + \[ + x \boldsymbol{\leq} y :\leftrightarrow x \supseteq y d.h. y \subseteq x + .\] + \[ + x \boldsymbol{<} y :\leftrightarrow x \boldsymbol{\leq} y \land x \neq y + .\] + + \item[(ii)] (Addition:) Wir definieren die Addition reeller Zahlen als die Abbildung + \[ + \boldsymbol{+}: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} + .\] + \[ + x \boldsymbol{+} y := \boldsymbol{+}(x,y) := {r+s | r \in x, s \in y} + .\] + \item[(iii)] (Subtraktion:) Für jedes $x \in \mathbb{R}$ definieren wir $\mathbf{-}x$ als das eindeutige Element von $\mathbb{R}$, sodass + \[ + x \boldsymbol{+}(\boldsymbol{-}x) = \mathbf{0} + .\] + \item[(iv)] (Multiplikation:) Wir definieren die Multiplikation reeller Zahlen als die Abbildung $\mathbf{\cdot}: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ gegeben durch + \[ + x \mathbf{\cdot} y = + \begin{cases} + \{rs | r \in x, s \in y\}, & \text{falls } x, y \boldsymbol{\geq}\mathbf{0}. \\ + \boldsymbol{-} ((\boldsymbol{-}x)\boldsymbol{\cdot}y), & \text{falls } x \boldsymbol{<} \mathbf{0}, y \boldsymbol{\geq} \mathbf{0}. \\ + \boldsymbol{-}(x \boldsymbol{\cdot} (\boldsymbol{-}y)), & \text{falls } x \boldsymbol{\geq} 0, y \boldsymbol{<} 0. \\ + (\boldsymbol{-} x \boldsymbol{\cdot} (\boldsymbol{-} y)), & \text{falls } x,y \boldsymbol{<} \mathbf{0} \\ + \end{cases} + .\] + \end{itemize} +} + +Gehen wir die einzelnen Punkte durch. (i) besagt, dass die untere Hälfte $x \geq$ die andere untere Hälfte $y$ ist, genau dann wenn $x$ eine Teilmenge von $y$ ist. Zusätzlich gilt, dass $x > y$ ist wenn $x \leq y$ ist und $x \neq y$ ist. In einfachen Worten gesagt bedeutet dies, dass wenn der untere Grenzwert von $x$ kleiner ist als der untere Grenzwert von $y$, so ist $y$ entweder in $x$ enthalten da sich die zwei Mengen schneiden oder $x$ und $y$ gleich.\\ +(ii) sagt aus, dass wenn du ein beliebiges Element aus $x$ nimmst und ein beliebiges Element aus $y$ nimmst und die zusammen addierst, so erhältst du eine Zahl, welches grösser ist als $X + Y$, wobei $X$ die reelle Zahl ist, welche $x$ darstellen soll und $Y$ respektive die reelle Zahl ist, welche $y$ darstellen soll.\\ +(iii) ist hoffentlich klar.\\ +(iv) ist einfach eine komplizierte Art die Multiplikation zu definieren. Grundsätzlich sagt es aus, dass wenn du ein Element von $x$ nimmst und ein Element von $y$ und die miteinander multiplizierst, so erhältst du eine neue Menge welches die resultierende reelle Zahl aus Dedekind-Schmitte darstellt. + +\mlenma{Bernoullische Ungleichung \cite{Ziltner2024}}{ +Für alle $n \in \mathbb{N}_0$ und $x \in [-1, \infty)$ gilt +\[ + (1 + x) ^{n} \leq 1 + nx + .\] +} + +In einfachen Worten sagt die Bernoullische Ungleichung, dass exponentielles Wachstum stärker oder gleich stark ist wie lineares Wachstum. Diese Gleichung wird vor allem für Beweise von der Konvergenz von Reihen und Folgen verwendet. Meistens wird der Beweis mit Induktion durchgeführt. + +\dfn{b-adischer Bruch \cite{Ziltner2024}}{ + Sei $b \leq 2$. Ein b-adischer Bruch ist Abbildung $a : \mathbb{Z} \rightarrow \{ 0, ... , b - 1 \}$, oder das Negative einer solchen Abbildung, mit den folgenden Eigenschaften: + \begin{itemize} + \item[(a)] Es gibt eine Zahl $k \in \mathbb{Z}$, sodass für jedes $i > k$ gilt $a_i := a(i) = 0$. + \item[(b)] Es gibt keine Zahl $i \in \mathbb{Z}$, sodass für jedes $i \leq l$ gilt $a_i = b - 1$. + \end{itemize} + Wir definieren + \[ + R_b := \{ \text{b-adischer Bruch} \} + .\] +} + +Der b-adischer Bruch ist einfach gesagt eine Art, um eine reelle Zahl darzustellen. $b$ ist die Basis und zeigt mit welchen Zahlen die reelle Zahl dargestellt werden kann. (10 = Dezimalsystem) Dabei gilt, dass der $b$-adischer Bruch laut (a) nach links alle Elemente eventuell Null sein werden, jedoch laut (b) nach rechts nicht die gleiche Zahl wiederholen dürfen. Dadurch werden Zahlen eliminiert, welche einfach aufgerundet werden können.\\ +\\ + +\dfn{Ordnung von $b$-adischer Brüchen \cite{Ziltner2024}}{ + Wir definieren $<_b$ als die strikte lexikographische Ordnung auf $\mathbb{R}_b. \text{ d.h. für } a,a' \in \mathbb{R}_b$ definieren wir + \[ + a <_b a' :\rightarrow \exists n \in \mathbb{Z} (\forall i > n : a_i = a_i') \land a_n < a_n' + .\] + Wir definieren + \[ + a \leq_b a' :\rightarrow a = a' \lor a <_b a' + .\] +} + +Die obige definition sagt einfach aus, dass man die Zahlen einer reellen Zahl ausgedrückt als ein $b$-adischer Bruch von links nach rechts vergleicht. Sobald die Zahl $a$ kleiner ist als $a'$ so ist die Zahl grösser zur Basis $b$. Dabei ist einfach wichtig, dass die vorherigen Zahlen gleich sind. + +\dfn{Betrag \cite{Ziltner2024}}{ + Der (Absolut-) Betrag einer Zahl ist die Zahl + \[ + |x| := + \begin{cases} + x, & \text{falls } x \geq 0 \\ + -x, & \text{sonst} \\ + \end{cases} + .\] +} + +Hoffentlich ist der Betrag einer Zahl allen bekannt. Diese Definition ist einfach formell und sagt aus, dass wenn die Zahl $x$ positiv ist so ist deren Betrag die Zahl selbst und sonst ist es $-x$, da das negative einer negativen Zahl eine positive Zahl ergibt.\\ +Mit dem Betrag können 2 Sätze eingeführt werden, welche für Beweise sehr nützlich sind. + +\thm{Dreiecks-Ungleichung \cite{Ziltner2024}}{ + Für alle $x,y \in \mathbb{R}$ gilt + \[ + |x + y| \leq |x| + |y| + .\] +} + +\thm{Youngsche Ungleichung \cite{Ziltner2024}}{ + Es seien $x,y,c \in \mathbb{R}$, sodass $c > 0$. Dann gilt + \[ + 2|xy| \leq cx ^2 + \frac{y ^2}{c} + .\] +} diff --git a/fs25/analysis_I_II/zahlen_und_vektoren/supremum_infimum.tex b/fs25/analysis_I_II/zahlen_und_vektoren/supremum_infimum.tex new file mode 100644 index 0000000..b7ace04 --- /dev/null +++ b/fs25/analysis_I_II/zahlen_und_vektoren/supremum_infimum.tex @@ -0,0 +1,58 @@ +\section{Supremum und Infimum} + +\dfn{Schranke, Beschränkheit \cite{Ziltner2024}}{ + Sei $A \subset \mathbb{R}$. + + \begin{itemize} + \item Eine obere Schranke für $A$ ist eine Zahl $b \in \mathbb{R}$, sodass für jedes $a \in A$ gilt $a \leq b$. + \item $A$ heisst nach oben beschränkt genau dann, wenn es eine obere Schranke für $A$ gibt. + \item Die Begriffe untere Schranke und nach unten beschränkt sind analog definiert. + \item $A$ heisst beschrankt genau dann, wenn $A$ nach oben und unten beschränkt ist. + \end{itemize} +} + +Gehen wir die einzelnen Punkte der Definition durch. Der erste Punkt besagt, dass eine Menge von Zahlen $A$ eine Zahl hat, welche $\geq$ der grössten Zahl in der Menge ist. Dies bedeutet, dass die grösste Zahl der Menge diese Zahl ist oder sie annähert. Dieser wird ''obere Schranke'' genannt. Obwohl es mehrere Zahlen sein können ist es keine Menge.\\ +Punkt zwei sagt aus, dass eine Menge nach oben beschränkt ist, wenn es eine obere Schranke hat.\\ +Punkt drei definiert die obere Schranke gleich der unteren Schranke. Dies bedeutet, dass eine Menge von Zahlen $A$ eine Zahl hat, welche $\leq$ die kleinste Zahl der Menge ist. Dies bedeutet wiederum, dass die kleinste Zahl der Menge diese Zahl ist oder sie annähert. Auch hier gilt wieder, dass mehrere Zahlen die obere Schranke sein können, jedoch die obere Schranke keine Menge ist. +Der letzte Punkt definiert eine beschränkte Menge. Eine beschränkte Menge ist einfach eine Menge, welche nach unten und nach oben beschränkt ist. + +\thm{Vollständigkeit der reellen Zahlen \cite{Ziltner2024}}{ + \begin{itemize} + \item[(i)] Jede nicht leere, nach oben beschränkte Teilmenge $A \subset \mathbb{R}$ besitzt eine kleinste obere Schranke. (Damit meinen wir ein kleinstes Element der Menge $S := \{ -\text{obere Schranke von } A \}$.) + \item[(ii)] Jede nicht leere, nach unten beschränkte Teilmenge $A \subset$ besitzt eine grösste untere Schranke. + \end{itemize} +} + +Wie vorher erwähnt kann eine Menge $A$ mehrere obere oder untere Schranken haben. Das Theorem besagt, dass die Menge $A$, falls sie nicht leer ist eine kleinste obere Schranke haben muss (das grösste Element in der Menge $A$) und eine grösste untere Schranke. (das kleinste Element der Menge $A$) + +\dfn{Supremum, Infimum \cite{Ziltner2024}}{ + Sei $A \subset \mathbb{R}$. Wir definieren das Supremum von $A$ als + \[ + \text{sup} A := + \begin{cases} + \text{kleinste obere Schranke für } A, & \text{falls } A \neq \emptyset \text{nach oben beschränkt ist,} \\ + \infty, & \text{falls } A \text{ nicht nach oben beschränkt ist,} \\ + - \infty, & \text{falls } A = \emptyset. \\ + \end{cases} + \] + Wir definieren das Infimum von $A$ als + \[ + \text{inf} A := + \begin{cases} + \text{grösste untere Schranke für } A, & \text{ falls } A \neq \emptyset \text{ und } A \text{nach unten beschränkt ist,} \\ + \infty, & \text{falls } A \text{nicht nach oben beschränkt ist,} \\ + - \infty, & \text{falls } A = \emptyset . \\ + \end{cases} + \] +} + +Gehen wir nun die einzelnen Definitionen von Supremum und Infimum durch.\\ +Das Supremum ist im allgemeinen Fall die kleinste obere Schranke. Falls $A$ nicht beschränkt ist, so ist das Supremum $\infty$. Falls $A$ zusätzlich noch die leere Menge ist, so ist das Supremum von $A$ $- \infty$.\\ +Beim Infimum ist die grösste untere Schranke der allgemeine Fall. Falls $A$ nicht beschränkt ist, so ist das Supremum $- \infty$. Falls $A$ zusätzlich noch die leere Menge ist, so ist das Infimum von $A$ $\infty$.\\ +Grundsätzlich kann man einfach sagen, dass das Supremum und Infimum die Definitionen von Schranken erweitert. + +\dfn{Maximum, Minimum einer Teilmenge von $\mathbb{R}$ \cite{Ziltner2024}}{ + Sei $A \subset \mathbb{R}$. Ein Maximum von $A$ ist ein Element $a \in A$, sodass $a \geq b$, für jedes $b \in A$. Ein Minimum von $A$ ist ein Element $a \in A$, sodass $a \leq b$, für jedes $b \in A$. +} + +Einfach gesagt ist das Maximum bzw. das Minimum einer Menge das grösste, bzw. das kleinste Element von Menge. diff --git a/fs25/analysis_I_II/zahlen_und_vektoren/zahlen_und_vektoren.tex b/fs25/analysis_I_II/zahlen_und_vektoren/zahlen_und_vektoren.tex index f3bf29d..378dbeb 100644 --- a/fs25/analysis_I_II/zahlen_und_vektoren/zahlen_und_vektoren.tex +++ b/fs25/analysis_I_II/zahlen_und_vektoren/zahlen_und_vektoren.tex @@ -1,3 +1,10 @@ \chapter{Zahlen und Vektoren} +Neben Logik bilden Zahlen die Basis für die Analysis. + +\input{natuerlichen_ganzen_rationalen_zahlen.tex} +\input{reellen_zahlen.tex} +\input{supremum_infimum.tex} +\input{komplexe_zahlen} + \newpage