From cc25a5011a9364db1cdc2a2a3a836fc8f2468406 Mon Sep 17 00:00:00 2001 From: henrydatei Date: Wed, 10 Oct 2018 12:17:32 +0200 Subject: [PATCH] steinaltes ANAG 1 entfernt, wir haben ja alles im 2. Semester --- ...rundbegriffe_aus_Mengenlehre_und_Logik.tex | 178 ---------- .../chapter02_Aufbau_einer_math_Theorie.tex | 227 ------------- .../ANAG/TeX_files/chapter03_nat_Zahlen.tex | 111 ------- .../chapter04_ganze_u_rat_Zahlen.tex | 178 ---------- .../TeX_files/chapter05_reelle_Zahlen.tex | 56 ---- .../TeX_files/chapter06_komplexe_zahlen.tex | 27 -- .../chapter07_grundl_ungleichungen.tex | 132 -------- .../ANAG/TeX_files/chapter08_metr_raeume.tex | 306 ------------------ .../ANAG/TeX_files/chapter09_konvergenz.tex | 23 -- .../ANAG/TeX_files/chapter10_vollst.tex | 2 - .../ANAG/TeX_files/chapter11_kompaktheit.tex | 2 - .../ANAG/TeX_files/chapter12_reihen.tex | 2 - .../ANAG/TeX_files/chapter13_funktionen.tex | 3 - 1. Semester/ANAG/Vorlesung_ANAG.pdf | Bin 345120 -> 0 bytes 1. 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Semester/ANAG/TeX_files/chapter01_Grundbegriffe_aus_Mengenlehre_und_Logik.tex +++ /dev/null @@ -1,178 +0,0 @@ -\part{Grundlagen der Mathematik} - -Mathematik besitzt eine Sonderrolle unter den Wissenschaften, da -\begin{compactitem} - \item Resultate nicht empirisch gezeigt werden müssen - \item Resultate nicht durch Experimente widerlegt werden können -\end{compactitem} - -\paragraph{Literatur} -\begin{compactitem} - \item Forster: Analysis 1 + 2, Vieweg - \item Königsberger: Analysis 1 + 2, Springer - \item Hildebrandt: Analysis 1 + 2, Springer - \item Walter: Analysis 1 + 2, Springer - \item Escher/Amann: Analysis 1 + 2, Birkhäuser - \item Ebbinghaus: Einfühung in die Mengenlehre, BI-Wissenschaftsverlag - \item Teubner-Taschenbuch der Mathematik, Teubner 1996 - \item Springer-Taschenbuch der Mathematik, Springer 2012 -\end{compactitem} - -\chapter{Grundbegriffe aus Mengelehre und Logik} -\textbf{Mengenlehre:} Universalität von Aussagen \\ -\textbf{Logik:} Regeln des Folgerns, wahre/falsche Aussagen - -\begin{mydef}[Definition Aussage] - Sachverhalt, dem man entweder den Wahrheitswert "wahr" oder "falsch" zuordnen kann, aber nichts anders. -\end{mydef} - -\begin{exmp} - \item 5 ist eine Quadratzahl $\to$ falsch (Aussage) - \item Die Elbe fließt durch Dresden $\to$ wahr (Aussage) - \item Mathematik ist rot $\to$ ??? (keine Aussage) -\end{exmp} - -\begin{mydef}[Menge] - Zusammenfassung von bestimmten wohlunterscheidbaren Objekten der Anschauung oder des Denkens, welche die Elemente der Menge genannt werden, zu einem Ganzen.\\ (\textsc{Cantor}, 1877) -\end{mydef} - -\begin{exmp} - \item $M_1 :=$ Menge aller Städte in Deutschland - \item $M_2 := \{1;2;3\}$ -\end{exmp} - -Für ein Objekt $m$ und eine Menge $M$ gilt stets $m \in M$ oder $m \notin M$ \\ -Für die Mengen $M$ und $N$ gilt $M=N$, falls dieselben Elemente enthalten sind -$\{1;2;3\} = \{3;2;1\} = \{1;2;2;3\}$ \\ -- $N \subseteq M$, falls $n \in M$ für jedes $n \in N$ \\ -- $N \subset M$, falls zusätzlich $M \neq N$ \\ - -\begin{mydef}[Aussageform] - Sachverhalt mit Variablen, der durch geeignete Ersetzung der Variablen zur Aussage wird. -\end{mydef} - -\begin{exmp} - \begin{itemize} - \item $A(X) := $ Die Elbe fließt durch X - \item $B(X;Y;Z) := X + Y = Z$ - \item aber $A(Dresden) ,B(2;3;4)$ sind Aussagen, $A(Mathematik)$ ist keine Aussage - \item $A(X)$ ist eine Aussage f\"u jedes $X \in M_1$ $\to$ Generalisierung von Aussagen durch Mengen - \end{itemize} -\end{exmp} - -\section*{Bildung und Verknüpfung von Aussagen} - -\begin{tabular}{|c|c|c|c|c|c|c|} - \hline - $A$ & $B$ & $\lnot A$ & $A \land B$ & $A \lor B$ & $A \Rightarrow B$ & $A \iff B$\\ - \hline - w & w & f & w & w & w & w\\ - \hline - w & f & f & f & w & f & f\\ - \hline - f & w & w & f & w & w & f\\ - \hline - f & f & w & f & f & w & w\\ - \hline -\end{tabular} - -\begin{exmpn} - \begin{itemize} - \item $\lnot$(3 ist gerade) $\to$ w - \item (4 ist gerade) $\land$ (4 ist Primzahl) $\to$ f - \item (3 ist gerade) $\lor$ (3 ist Primzahl) $\to$ w - \item (3 ist gerade) $\Rightarrow$ (Mond ist Würfel) $\to$ w - \item (Die Sonne ist heiß) $\Rightarrow$ (es gibt Primzahlen) $\to$ w - \end{itemize} -\end{exmpn} - -\noindent Auschließendes oder: (entweder $A$ oder $B$) wird realisiert durch $\lnot(A \iff B)$.\\ -Aussageform $A(X)$ sei f\"ur jedes $X \in M$ Aussage: neue Aussage mittels Quantoren - -\begin{compactitem} - \item $\forall$: "für alle" - \item $\exists$: "es existiert" -\end{compactitem} - -\begin{exmpn} - $\forall n \in \mathbb{N}: n$ ist gerade $\to$ f\\ - $\exists n \in \mathbb{N}: n$ ist gerade $\to$ w -\end{exmpn} - -\begin{mydef}[Tautologie bzw. Kontraduktion/Widerspruch] - Zusammengesetzte Aussage, die unabhängig vom Wahrheitsgehalt der Teilaussagen stest wahr bzw. falsch ist. -\end{mydef} - -\begin{exmpn} - \begin{itemize} - \item Tautologie (immer wahr): - $(A) \lor (\lnot A), \lnot (A \land (\lnot A)), (A \land B) \Rightarrow A$ - \item Widerspruch (immer falsch): $A \land (\lnot A), A \iff \lnot A$ - \item besondere Tautologie: $(A \Rightarrow B) \iff (\lnot B \Rightarrow \lnot A)$ - \end{itemize} -\end{exmpn} - -\begin{satz}[Morgansche Regeln] - Folgende Aussagen sind Tautologien: - \begin{itemize}{ } - \item $\lnot(A \land B) \iff \lnot A \lor \lnot B$ - \item $\lnot(A \lor B) \iff \lnot A \land \lnot B$ - \end{itemize} -\end{satz} - -\section*{Bildung von Mengen} -Seien $M$ und $N$ Mengen -\begin{compactitem} - \item Aufzählung der Elemente: $\{1;2;3\}$ - \item mittels Eigenschaften: $\{X \in M \mid A(X)\}$ - \item $\emptyset:=$ Menge, die keine Elemente enthält - \begin{compactitem} - \item leere Menge ist immer Teilmenge jeder Menge $M$ - \item \textbf{Warnung:} $\{\emptyset\} \neq \emptyset$ - \end{compactitem} - \item Verknüpfung von Mengen wie bei Aussagen -\end{compactitem} - -\begin{mydef}[Mengensystem] - Ein Mengensystem $\mathcal M$ ist eine Menge, bestehend aus anderen Mengen. - \begin{compactitem} - \item $\bigcup M := \{X \mid \exists M \in \mathcal M: X \in M\}$ (Vereinigung aller Mengen in - $\mathcal M$) - \item $\bigcap M := \{X \mid \forall M \in \mathcal M: X \in M\}$ (Durchschnitt aller Mengen in - $\mathcal M$) - \end{compactitem} -\end{mydef} - -\begin{mydef}[Potenzmenge] - Die Potenzmenge $\mathcal P$ enth\"alt alle Teilmengen einer Menge $M$. \\ - $\mathcal P(X) := \{\tilde M \mid \tilde M \subset M\}$ -\end{mydef} - -Beispiel: -\begin{compactitem} - \item $M_3 := \{1;3;5\}$ \\ - $\to \mathcal P(M_3) = \{\emptyset, \{1\}, \{3\}, \{5\}, \{1;3\}, \{1;5\}, \{3;5\}, \{1;3;5\}\}$ -\end{compactitem} - -\begin{framed} - \textbf{Satz (de Morgansche Regeln f\"ur Mengen):} - \begin{compactitem} - \item $(\mathop{\bigcup}_{N \in \mathcal N} N)^C = \mathop{\bigcap}_{N \in \mathcal N} N^C$ - \item $(\mathop{\bigcap}_{N \in \mathcal N} N)^C = \mathop{\bigcup}_{N \in \mathcal N} N^C$ - \end{compactitem} -\end{framed} - -\begin{mydef}[Kartesisches Produkt] - $M \times N := \{m,n \mid m \in M \land n \in N\}$ \\ - $(m,n)$ hei{\ss}t geordnetes Paar (Reihenfolge wichtig!) \\ - allgemeiner: $M_1 \times ... \times M_k := \{(m_1,...,m_k) \mid m_j \in M_j, j=1, .., k\}$ \\ - $M^k := M \times ... \times M := \{(m_1,...,m_k) \mid m_j \in M_j, j=1, .., k\}$ -\end{mydef} - -\begin{satz}[Auswahlaxiom] - Sei $\mathcal M$ ein Mengensystem nichtleerer paarweise disjunkter Mengen $M$. - \begin{compactitem} - \item Es existiert eine Auswahlmenge $\tilde M$, die mit jedem $M \in \mathcal M$ genau 1 Element gemeinsam hat. - \item beachte: Die Auswahl ist nicht konstruktiv! - \end{compactitem} -\end{satz} \ No newline at end of file diff --git a/1. 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Semester/ANAG/TeX_files/chapter02_Aufbau_einer_math_Theorie.tex +++ /dev/null @@ -1,227 +0,0 @@ -\chapter{Aufbau einer mathematischen Theorie} -Axiome $\to$ Beweise $\to$ Sätze ("neue" wahre Aussagen) \\ -$\to$ ergibt Ansammlung (Menge) wahrer Aussagen - -\paragraph*{Formulierung mathematischer Aussagen} -\begin{compactitem} - \item typische Form eines mathematischen Satzes: "Wenn A gilt, dann gilt auch B." - \item formal: $A \Rightarrow B$ bzw. $A(X) \Rightarrow B(X)$ ist stets wahr (insbesondere falls - A wahr ist) -\end{compactitem} - -Beispiel -\begin{compactitem} - \item $X \in \mathbb N$ und ist durch 4 teilbar $\Rightarrow X$ ist durch 2 teilbar - \item beachte: Implikation auch wahr, falls $X = 5$ oder $X =6$, dieser Fall ist aber - uninteressant - \item genauer meint man sogar $A \land C \Rightarrow B$, wobei $C$ aus allen bekannten wahren - Aussagen besteht - \item man sagt: $B$ ist \textbf{notwendig} f\"ur $A$, da $A$ nur wahr sein kann, wenn $B$ - wahr ist - \item man sagt: $A$ ist \textbf{hinreichend} f\"ur $B$, da $B$ stets wahr ist, wenn $A$ wahr ist -\end{compactitem} - -\paragraph{Mathematische Beweise} -\begin{compactitem} - \item \textbf{direkter Beweis:} finde Zwischenaussagen $A_1,...,A_k$, sodass f\"ur $A$ auch wahr: \\ - $(A \Rightarrow A_1) \land (A_1 \Rightarrow A_2) \land ... \land (A_k \Rightarrow B)$ - \item Beispiel: Zeige $x > 2 \Rightarrow x^2-3x+2>0$ \\ - $(x>2) \Rightarrow (x-2>0) \land (x-1>0) \Rightarrow (x-2) \cdot (x-1) \Rightarrow x^2-3x+2>0$ - \item \textbf{indirekter Beweis:} auf Grundlage der Tautologie $(A \Rightarrow B) \iff - (\lnot B \Rightarrow \lnot A)$ f\"uhrt man direkten Beweis $\lnot B \Rightarrow \lnot A$ (das - hei{\ss}t angenommen $B$ falsch, dann auch $A$ falsch) - \item praktisch formuliert man das auch so: $(A \land \lnot B) \Rightarrow ... \Rightarrow (A - \land \lnot A)$ - \item Beispiel: Zeige $x^2-3x+2 \le 0$ sei wahr \\ - $\lnot B \Rightarrow (x-2) \cdot (x-1) \le 0 \Rightarrow 1 \le x \le 2 \Rightarrow x \le 2 - \Rightarrow \lnot A$ -\end{compactitem} - -\section{Relationen und Funktionen} -\begin{mydef}[Relation] - Seien $M$ und $N$ Mengen. Dann ist jede Teilmenge $R$ von - $M \times N$ eine Relation. \\ - $(x,y) \in R$ hei{\ss}t: $x$ und $y$ stehen in Relation zueinander -\end{mydef} - -\begin{exmp} - $M$ ist die Menge aller Menschen. Die Liebesbeziehung $x$ liebt $y$ sieht als geordnetes Paar - geschrieben so aus: $(x,y)$. Das hei{\ss}t die Menge der Liebespaare ist das: $L := \{(x,y) \mid - x \; liebt \; y\}$. Und es gilt: $L \subset M \times M$. -\end{exmp} - -Die Relation $R \subset M \times N$ hei{\ss}t \textbf{Ordnungsrelation} (kurz. Ordnung) auf M, falls f\"ur alle $a,b,c \in M$ gilt: -\begin{compactitem} - \item $(a,a) \in R$ (reflexiv) - \item $(a,b),(b,a) \in R$ (antisymetrisch) - \item $(a,b), (b,c) \in R \Rightarrow (a,c) \in R$ (transitiv) - \item z.B. $R = \{(X,Y) \in \mathcal P(Y) \times \mathcal P(Y) \mid X \subset Y\}$ -\end{compactitem} - -$\newline$ -Eine Ordnungsrelation hei{\ss}t \textbf{Totalordnung}, wenn zus\"atzlich gilt: $(a,b) \in R \lor -(b,a) \in R$ \\ - -$\newline$ -Beispiel \\ -Seien $m$, $n$ und $o$ natürliche Zahlen, dann ist $R = \{(m,n) \in \mathbb{N} \times \mathbb{N} -\mid x \le y\}$ eine Totalordnung, da -\begin{compactitem} - \item $m \le m$ (reflexiv) - \item $(m \le n \land n \le m) \Rightarrow m=n$ (antisymetrisch) - \item $(m \le n \land n \le o) \Rightarrow m \le o$ (transitiv) - \item $m \le n \lor n \le m$ (total) -\end{compactitem} - -$\newline$ -Eine Relation auf $M$ heißt \textbf{Äquivalenzrelation}, wenn für alle $a,b,c \in M$ gilt: -\begin{compactitem} - \item $(a,a) \in R$ (reflexiv) - \item $(a,b),(b,a) \in R$ (symetrisch) - \item $(a,b), (b,c) \in R \Rightarrow (a,c) \in R$ (transitiv) -\end{compactitem} - -$\newline$ -Obwohl Ordnungs- und Äquivalenzrelation die gleichen Eigenschaften haben, haben sie -unterschiedliche Zwecke: Ordnungsrelationen ordnen Elemente in einer Menge (z.B. das Zeichen $\le$ -ordnet die Menge der natürlichen Zahlen), während Äquivalenzrelationen eine Menge in disjunkte -Teilmengen (Äquivalenzklassen) ohne Rest aufteilen. \\ -$\newline$ - -Wenn $R$ eine Ordnung auf M ist, so wird häufig geschrieben: \\ -\noindent\hspace*{5mm} $a \le b$ bzw. $a \ge b$ falls $(a,b) \in \mathbb R$ \\ -\noindent\hspace*{5mm} $a < b$ bzw. $a > b$ falls zus\"atzlich $a \neq b$ \\ - -\begin{mydef}[Abbildung/Funktion] - Eine Funktion $F$ von $M$ nach $N$ - (kurz: $F: M \to N$), ist eine Vorschrift, die jedem Argument/Urbild $m \in M$ genau einen - Wert/Bild $F(m) \in N$ zuordnet. \\ - $D(F) := M$ heißt Definitionsbereich/Urbildmenge \\ - \noindent\hspace*{15mm} $N$ heißt Zielbild \\ - $F(M') := \{n \in N \mid n=F(m)$ f\"ur ein $m \in M' \}$ ist Bild von $M' \subset M$ \\ - $F^{-1}(N') := \{m \in M \mid n=F(m)$ für ein $N' \}$ ist Urbild von $N' \subset N$ \\ - $R(F) := F(M)$ heißt Wertebereich/Bildmenge \\ - $graph(F) := \{(m,n) \in M \times N \mid n=F(m)\}$ heißt Graph von $F$ \\ - $F_{\mid M'}$ ist Einschränkung von $F$ auf $M' \subset M$ -\end{mydef} - -Unterschied Zielmenge und Wertebereich: $f(x) = \sin(x):$ \\ -\noindent\hspace*{5mm} Zielmenge: $\mathbb R$ \\ -\noindent\hspace*{5mm} Wertebereich: $[-1;1]$ \\ - -Funktionen $F$ und $G$ sind gleich, wenn -\begin{compactitem} - \item $D(F) = D(G)$ - \item $F(m) = G(m) \quad \forall m \in D(F)$ -\end{compactitem} - -\noindent Manchmal wird auch die vereinfachende Schreibweise benutzt: -\begin{itemize}[label={-}] - \item $F: M \to N$, obwohl $D(F) \subsetneq M$ (z.B. $\tan: \mathbb R \to \mathbb R$, Probleme - bei $\frac{\pi}{2}$) - \item gelegentlich spricht man auch von "Funktion $F(m)$" statt Funktion $F$ -\end{itemize} - -\begin{lem}[Komposition/Verknüpfung] - Die Funktionen $F: M \to N$ und $G: N \to P$ - sind verknüpft, wenn \\ - $F \circ G: M \to P$ mit $(F \circ G)(m) := G(F(m))$ -\end{lem} - -\textbf{Eigenschaften von Funktionen:} \\ -\begin{compactitem} - \item injektiv: Zuordnung ist eineindeutig $\to F(m_1) = F(m_2) \Rightarrow m_1=m_2$ - \item Beispiel: $x^2$ ist nicht injektiv, da $F(2)=F(-2)=4$ - \item surjektiv: $F(M) = N \quad \forall n \in N \; \exists m \in M: F(m)=n$ - \item Beispiel: $\sin(x)$ ist nicht surjektiv, da es kein $x$ für $y=27$ gibt - \item bijektiv: injektiv und surjektiv -\end{compactitem} -$\newline$ -Für bijektive Abbildung $F: M \mapsto N$ ist Umkehrabbildung/inverse Abbildung $F^{-1}: N \mapsto M$ -definiert durch: $F^{-1}(n) = m \iff F(m)=n$ \\ -Hinweis: Die Notation $F^{-1}(N')$ f\"ur Urbild bedeutet nicht, dass die inverse Abbildung $F^{-1}$ -existiert. - -\begin{satz} - Sei $F: M \to N$ surjektiv. Dann existiert die Abbildung $G: N \to M$, - sodass $F \circ G = id_N$ (d.h. $F(G(n))=n \quad \forall n \in N$) -\end{satz} - -\begin{mydef}[Rechenoperation/Verknüpfung] - Eine Rechenoperation auf einer Menge $M$ ist - die Abbildung $*: M \times M \to M$ d.h. $(m,n) \in M$ wird das Ergbnis $m*n \in M$ zugeordnet. -\end{mydef} - -\textbf{Eigenschaften von Rechenoperationen:} -\begin{compactitem} - \item hat neutrales Element $e \in M: m*e=m$ - \item ist kommutativ $m*n=n*m$ - \item ist assotiativ $k*(m*n)=(k*m)*n$ - \item hat ein inverses Element $m' \in M$ zu $m \in M: m*m'=e$ -\end{compactitem} -$e$ ist stets eindeutig, $m'$ ist eindeutig, wenn die Operation $*$ assoziativ ist. \\ - -Beispiele: -\begin{compactitem} - \item Addition $+$: $(m,n) \mapsto m+n$ Summe, neutrales Element heißt Nullelement, inverses - Element $-m$ - \item Multiplikation $\cdot$: $(m,n) \mapsto m \cdot n$ Produkt, neutrales Element Eins, inverses - Element $m^{-1}$ -\end{compactitem} -Addition und Multiplikation sind distributiv, falls $k(m+n) = k \cdot m + k \cdot n$ - -\begin{mydef}[Körper] - Eine Menge $M$ ist ein Körper $K$, wenn man auf $K$ eine Addition - und eine Multiplikation mit folgenden Eigenschaften durchführen kann: - \begin{compactitem} - \item es gibt neutrale Elemente 0 und 1 $\in K$ - \item Addition und Multiplikation sind jeweils kommutativ und assoziativ - \item Addition und Multiplikation sind distributiv - \item es gibt Inverse $-k$ und $k^{-1} \in K$ \\ - $\to$ die reellen Zahlen sind ein solcher Körper - \end{compactitem} -\end{mydef} - -Eine Menge $M$ habe die Ordnung ``$\leq$'' und diese erlaubt die Addition und Multiplikation, wenn -\begin{compactitem} - \item $a \le b \iff a+c \le b+c$ - \item $a \le b \iff a \cdot c \le b \cdot c \quad c > 0$ \\ - $\to$ Man kann die Gleichungen in gewohnter Weise umformen. -\end{compactitem} -$\newline$ - -Ein Körper $K$ heißt angeordnet, wenn er eine Totalordnung besitzt, die mit Addition -und Multiplikation verträglich ist. \\ -$\newline$ - -\textbf{Isomorphismus} bezüglich einer Struktur ist die bijektive Abbildung $I: M_1 -\mapsto M_2$, die die vorhandene Struktur auf $M_1$ und $M_2$ erhält, z.B. -\begin{compactitem} - \item Ordnung $\le_1$ auf $M_1$, falls $a \le_1 b \iff I(a) \le_2 I(b)$ - \item Abbildung $F_i: M_i \to M_i$, falls $I(F_1(a)) = F_2(I(a))$ - \item Rechenoperation $*_i: M_i \times M_i \to M_i$, falls $I(a*_1b) = I(a) *_2 I(b)$ - \item spezielles Element $a_i \in M_i$, falls $I(a_1) = a_2$ -\end{compactitem} -$\newline$ - -\textit{"Es gibt 2 verschiedene Arten von reellen Zahlen, meine und Prof. Schurichts. Wenn wir einen - Isomorphismus finden, dann bedeutet das, dass unsere Zahlen strukturell die selben sind."}\\ - -Beispiele: $M_1 = \mathbb N$ und $M_2 = \{$gerade Zahlen$\}$, jeweils mit Addition, Multiplikation -und Ordnung \\ -$\to I: M_2 \to M_2$ mit $I(k)=2k \quad \forall k \in \mathbb N$ \\ -$\to$ Isomorphismus, der die Addition, Ordnung und die Null, aber nicht die Multiplikation erh\"alt - -\subsection*{Bemerkungen zum Fundament der Mathematik} -Forderungen an eine mathematische Theorie: -\begin{compactitem} - \item widerspruchsfrei: Satz und Negation nicht gleichzeitig herleitbar - \item vollständig: alle Aussagen innerhalb der Theorie sind als wahr oder falsch beweisbar -\end{compactitem} -$\newline$ - -zwei Unvollständigkeitssätze: -\begin{compactitem} - \item jedes System ist nicht gleichzeitig widerspruchsfrei und vollständig - \item in einem System kann man nicht die eigene Widerspruchsfreiheit zeigen -\end{compactitem} \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter03_nat_Zahlen.tex b/1. Semester/ANAG/TeX_files/chapter03_nat_Zahlen.tex deleted file mode 100644 index ae9ab36..0000000 --- a/1. Semester/ANAG/TeX_files/chapter03_nat_Zahlen.tex +++ /dev/null @@ -1,111 +0,0 @@ -\part{Zahlenbereiche} -\chapter{Natürliche Zahlen} -$\mathbb N$ sei diejenige Menge, die die \textbf{Peano-Axiome} erfüllt, das heißt -\begin{compactitem} - \item $\mathbb N$ sei induktiv, d.h. es existiert ein Nullelement und eine injektive Abbildung - $\mathbb N to \mathbb N$ mit $\nu(n) \neq 0 \quad \forall n$ - \item Falls $N \subset \mathbb N$ induktiv in $\mathbb N$ (0, $\nu(n) \in N$ falls $n \in N - \Rightarrow N = \mathbb N$ -\end{compactitem} -$\to \mathbb N$ ist die kleinste induktive Menge \\ -$\newline$ - -Nach der Mengenlehre ZF (Zermelo-Fraenkel) existiert eine solche Menge $\mathbb N$ der natürlichen -Zahlen. Mit den üblichen Symbolen hat man: -\begin{compactitem} - \item $0 := \emptyset$ - \item $1 := \nu(0) := \{\emptyset\}$ - \item $2 := \nu(1) := \{\emptyset, \{\emptyset\}\}$ - \item $3 := \nu(2) := \{\emptyset, \{\emptyset, \{\emptyset\}\}\}$ -\end{compactitem} -Damit ergibt sich in gewohnter Weise $\mathbb N = \{1; 2; 3; ...\}$ \\ -anschauliche Notation $\nu(n) = n+1$ (beachte: noch keine Addition definiert!) \\ - -\begin{theorem} - Falls $\mathbb N$ und $\mathbb N'$ die Peano-Axiome erfüllen, sind sie - isomorph bez\"uglich Nachfolgerbildung und Nullelement. Das hei{\ss}t alle solche $\mathbb N'$ - sind strukturell gleich und k\"onnen mit obigem $\mathbb N$ identifiziert werden. -\end{theorem} - -\begin{satz}[Prinzip der vollständigen Induktion] - Sei $\{A_n \mid n \in N\}$ eine Menge - von Aussagen $A_n$ mit der Eigenschaft: - \begin{enumerate}[ ] - \item IA: $A_0$ ist wahr - \item IS: $\forall n \in \mathbb N$ gilt $A_n \Rightarrow A_{n+1}$ - \end{enumerate} - $A_n$ ist wahr für alle $n \in \mathbb N$ -\end{satz} - -\begin{lem} Es gilt: - \begin{enumerate} - \item $\nu(n) \cup \{0\} = \mathbb N$ - \item $\nu(n) \neq n \quad \forall n \in \mathbb N$ - \end{enumerate} -\end{lem} - -\begin{satz}{(rekursive Definition/Rekursion)} Sei $B$ eine Menge und $b \in B$. Sei $F$ eine - Abbildung mit $F: B \times \mathbb N \mapsto B$. Dann liefert nach Vorschrift: $f(0):= b$ und - $f(n+1) = F(f(n),n) \quad \forall n \in \mathbb N$ genau eine Abbildung $f: \mathbb N \mapsto B$. - Das heißt eine solche Abbildung exstiert und ist eindeutig. -\end{satz} -$\newline$ - -\textbf{Rechenoperationen:} -\begin{compactitem} - \item Definition Addition '$+$': $\mathbb N \times \mathbb N \mapsto \mathbb N$ auf $\mathbb N$ - durch $n+0:=n$, $n+\nu(m):=\nu(n+m) \quad \forall n,m \in \mathbb N$ - \item Definition Multiplikation '$\cdot$': $\mathbb N \times \mathbb N \mapsto \mathbb - N$ auf $\mathbb N$ durch $n \cdot 0 := 0$, $n \cdot \nu(m) := n \cdot m + n \quad \forall - n,m \in \mathbb N$ -\end{compactitem} -Für jedes feste $n \in \mathbb N$ sind beide Definitionen rekursiv und eindeutig definiert. \\ -$\forall n \in \mathbb N$ gilt: $n+1=n+\nu(0)=\nu(n+0) = \nu(n)$ - -\begin{satz} - Addition und Multiplikation haben folgende Eigenschaften: - \begin{compactitem} - \item es existiert jeweils ein neutrales Element - \item kommutativ - \item assoziativ - \item distributiv - \end{compactitem} -\end{satz} - -Es gilt $\forall k,m,n \in \mathbb N$: -\begin{compactitem} - \item $m \neq 0 \Rightarrow m+n \neq 0$ - \item $m \cdot n = 0 \Rightarrow n=0$ oder $m=0$ - \item $m+k=n+k \Rightarrow m=n$ (Kürzungsregel der Addition) - \item $m \cdot k=n \cdot k \Rightarrow m=n$ (Kürzungsregel der Multiplikation) -\end{compactitem} - -Ordnung auf $\mathbb N:$ Relation $R := \{(m,n) \in \mathbb N \times \mathbb N \mid m \le n\}$ \\ -wobei $m \le n \iff n=m+k$ f\"ur ein $k \in \mathbb N$ \\ - -\begin{satz} - Es gilt auf $\mathbb N:$ - \begin{compactitem} - \item $m \le n \Rightarrow \exists ! k \in \mathbb N: n=m+k$, nenne $n-m:=k$ (Differenz) - \item Relation $R$ (bzw. $\le$) ist eine Totalordnung auf $\mathbb N$ - \item Ordnung $\le$ ist vertr\"aglich mit der Addition und Multiplikation - \end{compactitem} -\end{satz} - -\begin{proof} - \item Sei $n=m+k=m+k' \Rightarrow k=k'$ - \item Sei $n=n+0 \Rightarrow n \le n \Rightarrow$ reflexiv \\ - sei $k\le m, m \le n \Rightarrow \exists l,j: m=k+l, n=m+j=(k+l)+j=k+(l+j) \Rightarrow - k \le n \Rightarrow$ transitiv \\ - sei nun $m \le n und n \le m \Rightarrow n=m+j=n+l+j \Rightarrow 0=l+j \Rightarrow j=0 - \Rightarrow n=m \Rightarrow$ antisymmetrisch \\ - Totalordnung, d.h. $\forall m,n \in \mathbb N: m\le n$ oder $n\le m$ \\ - IA: $m=0$ wegen $0=n+0$ folgt $0 \le n \forall n$ \\ - IS: gelte $m\le n$ oder $n \le m$ mit festem $m$ und $\forall n \in \mathbb N$, dann \\ - falls $n \le m \Rightarrow n \le m+1$ \\ - falls $m < n \Rightarrow \exists k \in \mathbb N: n=m+(k+1)=(m+)1+k \Rightarrow m+1 \le n$ \\ - $m\le n$ oder $n \le m$ gilt für $m+1$ und $\forall n \in \mathbb N$, also $\forall n,m \in - \mathbb N$ - \item sei $m \le n \Rightarrow \exists j: n=m+j \Rightarrow n+k=m+j+k \Rightarrow m+k \le n+k$ - \QEDA -\end{proof} \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter04_ganze_u_rat_Zahlen.tex b/1. Semester/ANAG/TeX_files/chapter04_ganze_u_rat_Zahlen.tex deleted file mode 100644 index 2d88a79..0000000 --- a/1. Semester/ANAG/TeX_files/chapter04_ganze_u_rat_Zahlen.tex +++ /dev/null @@ -1,178 +0,0 @@ -\chapter{Ganze und rationale Zahlen} -\textbf{Frage:} Existiert eine natürliche Zahl $x$ mit $n=n'+x$ für ein gegebenes $n$ und $n'$? \\ -\textbf{Antwort:} Das geht nur falls $n \le n'$, dann ist $x=n-n'$ \\ -\textbf{Ziel:} Zahlenbereichserweiterung, sodass die Gleichung immer lösbar ist. Ordne jedem Paar -$(n,n') \in \mathbb N \times \mathbb N$ eine neue Zahl als L\"osung zu. Gewisse Paare liefern die -gleiche L\"osung, z.B. $(6,4),(5,3),(7,5)$. Diese m\"ussen mittels Relation identifiziert werden. \\ -$\newline$ - -$\mathbb Q := \{(n_1,n_1'),(n_2,n_2') \in (\mathbb N \times \mathbb N) \times (\mathbb N \times -\mathbb N) \mid n_1+n_2'=n_1'+n_2\}$ \\ -$\newline$ - -\begin{mydef} - $\mathbb Q$ ist die Äquivalenzrelation auf $\mathbb N \times \mathbb N$. -\end{mydef} -$\newline$ - -\begin{exmp} - $(5,3) \sim (6,4) \sim (7,5)$ bzw. $(5-3) \sim (6-4) \sim (7-5)$\\ - $(3,6) \sim (5,8)$ bzw. $(3-6) \sim (5-8)$ -\end{exmp} - -\begin{proof} - % find a way to give an example a better formating!!! - offenbar $((n,n'),(n,n')) \in \mathbb Q \Rightarrow$ reflexiv\\ - falls $((n_1,n_1'),(n_2,n_2')) \in \mathbb Q \Rightarrow (n_2,n_2'),(n_1,n_1')) \in - \mathbb Q \Rightarrow$ symmetrisch\\ - sei $((n_1,n_1'),(n_2,n_2')) \in \mathbb Q$ und $((n_2,n_2'),(n_3,n_3')) \in \mathbb Q - \Rightarrow n_1+n_2'=n_1'+n_2, n_2+n_3'=n_2'+n_3 \Rightarrow n_1+n_3'=n_1'+n_3 \Rightarrow - ((n_1,n_1'),(n_3,n_3')) \in \mathbb Q \Rightarrow$ transitiv.\QEDA -\end{proof} - -\noindent setze $\overline{\mathbb{Z}} := \{[(n,n')] \mid n,n' \in \mathbb{N}\}$ Menge der ganzen Zahlen, -[ganze Zahl] \\ -Kurzschreibweise: $\overline m := [(m,m')]$ oder $\overline n := [(n,n')]$ \\ - -\begin{satz} - Sei $[(n,n')] \in \overline{\mathbb{Z}}$. Dann existiert eindeutig $n* \in \mathbb N$ mit $(n*,0) \in [(n,n')]$, falls $n \ge n'$ bzw. $(0,n*) \in [(n,n')]$ falls $n < n'$. -\end{satz} - -\begin{proof} - $n \ge n' \Rightarrow \exists ! n* \in \mathbb N: n=n'+n* \Rightarrow (n*,0) \sim (n,n')$\\ - $n < n' \Rightarrow \exists ! n* \in \mathbb N: n+n*=n' \Rightarrow (0,n*) \sim (n,n')$\QEDA -\end{proof} - -\noindent\textbf{Frage:} Was hat $\overline{\mathbb{Z}}$ mit $\whole$ zu tun?\\ -\textbf{Antwort:} identifiziere $(n,0)$ bzw. $(n-0)$ mit $n \in \natur$ und identifiziere $(0,n)$ -bzw. $(0-n)$ mit Symbol $-n$ \\ -$\Rightarrow$ ganze Zahlen kann man eindeutig den Elementen folgender Mengen zuordnen: $\mathbb Z := -\natur \cup \{(-n) \mid n \in \natur\}$ \\ -$\newline$ - -\textbf{Rechenoperationen auf $\overline{\whole}$:} \\ -\begin{itemize} - \item Addition: $\overline m + \overline n = [(m,m')]+[(n,n')]=[(m+n,m'+n')]$ - \item Multiplikation: $\overline m \cdot \overline n = [(m,m')] \cdot [(n,n')]=[(mn+m'n',mn'+m'n)]$ -\end{itemize} - -\begin{satz} - Addition und Multiplikation sind eindeutig definiert, d.h. unabhängig von Repräsentant bezüglich $\mathbb Q$ -\end{satz} -\begin{proof} - Sei $(m_1,m_1') \sim (m_2,m_2'), (n_1,n_1') \sim (n_2,n_2')\\ - \Rightarrow m_1+m_2'=m_1'+m_2, n_1 - +n_2'=n_1'+n_2\\ - \Rightarrow m_1+n_1+m_2'+n_2'=m_1'+n_1'+m_2+n_2\\ \Rightarrow (m_1,m_1')+(n_1,n_1') \sim (m_2,m_2')+(n_2,n_2')$\QEDA -\end{proof} - -\begin{satz} - Für Addition und Multiplikation auf $\mathbb Z$ gilt $\forall\;\overline m, - \overline{n} \in \overline{\whole}$: - \begin{enumerate} - \item es existiert eine neutrales Element: $0:=[(0,0)]$, $1:=[(1,0)]$ - \item jeweils kommutativ, assoziativ und gemeinsam distributiv - \item $- \overline{n} := [(n',n)] \in \whole$ ist invers bezüglich der Addition zu - $[(n,n')] = \overline n$ - \item $(-1) \cdot \overline n = - \overline n$ - \item $\overline m \cdot \overline n = 0 \iff \overline m =0 \lor \overline n=0$ - \end{enumerate} -\end{satz} - -\begin{proof} - \begin{enumerate}[label={\arabic*)}, nolistsep] - \item offenbar $\overline n +0=0+\overline n=\overline n$ und $\overline n \cdot 1 = 1 \cdot - \overline n = \overline n$ - \item Fleißarbeit $\to$ SeSt - \item offenbar $\overline n+(- \overline n) = (- \overline n)+\overline n=[(n+n',m+m')]=0$ - \item $(-1)\cdot \overline n = [(0,1)]\cdot [n,n']=[n',n]=-\overline n$ - \item ÜA \QEDA - \end{enumerate} - -\end{proof} - -\begin{satz} - Für $\overline m, \overline n \in \mathbb Z$ hat die Gleichung $\overline m=\overline n + \overline x$ die Lösung $\overline x=\overline m+(-\overline n)$. -\end{satz} - -\noindent Ordnung auf $\overline{\whole}:$ betrachte Relation $R := \{(\overline{m},\overline{n}) \in -\overline{\whole} \times \overline{\whole} \mid \overline{m} \le \overline{n}\}$ - -\begin{satz} - $R$ ist Totalordnung auf $\whole$ und verträglich mit Addition und - Multiplikation -\end{satz} - -\noindent Ordnung verträglich mit Addition: $\overline n < 0 \iff 0=\overline n+(-\overline n) < -\overline n -= (-1) \cdot \overline n$ \\ - -\noindent \textbf{beachte:} $\mathbb Z := \mathbb N \cup \{(-n) \mid n \in \mathbb N_{>0}\}$ \\ - -\begin{satz} - $\whole$ und $\overline{\whole}$ sind isomorph bezüglich Addition, Multiplikation und Ordnung. -\end{satz} - -\begin{proof} - betrachte Abbildung $I: \mathbb Z \to \overline{\whole} $ mit $I(k):=[(k,0)]$ und $I(-k):=[(0,k)] \quad \forall k \in \natur \Rightarrow$ ÜA \QEDA -\end{proof} - -\noindent Notation: verwende stets $\mathbb Z$, schreibe $m,n,...$ statt $\overline m, \overline n,...$ für -ganze Zahlen in $\mathbb Z$ \\ - -\noindent \textbf{Frage:} Existiert eine ganze Zahl mit $n=n' \cdot x$ f\"ur $n,n' \in \mathbb Z, n' \neq 0$ \\ -\textbf{Antwort:} im Allgemeinen nicht -\textbf{Ziel:} Zahlbereichserweiterung analog zu $\mathbb N \to \mathbb Z$ \\ -ordne jedem Paar $(n,n') \in \mathbb Z \times \mathbb Z$ neue Zahl $x$ zu \\ -schreibe $(n,n')$ auch als $\frac{n}{n'}$ oder $n:n'$ \\ -identifiziere Paare wie z.B. $\frac 4 2, \frac 6 3, \frac 8 4$ durch Relation \\ -$\mathbb Q := {(\frac{n_1}{n'_2}, \frac{n_2}{n'_2}) \in (\mathbb Z \times \mathbb Z_{\neq 0}) - \times (\mathbb Z \times \mathbb Z_{\neq 0}) \mid n_1n'_2=n'_1n_2}$ \\ -$\Rightarrow \mathbb Q$ ist eine Äquivalenzrelation auf $\mathbb Z \times \mathbb Z_{\neq 0}$ \\ - -\noindent setze $\mathbb Q := {[\frac{n}{n'}] \mid (n,n') \in \mathbb Z \times \mathbb Z_{\neq 0}}$ Menge der -rationalen Zahlen \\ -beachte: unendlich viele Symbole $\frac{n}{n'}$ für gleiche Zahl $[\frac{n}{n'}]$ \\ -wir schreiben später $\frac{n}{n'}$ für die Zahl $[\frac{n}{n'}]$ \\ -\noindent offenbar gilt die Kürzungsregel: $[\frac{n}{n'}]=[\frac{kn}{kn'}] \quad \forall k \in -\mathbb Z_{\neq 0}$ \\ - -\noindent \textbf{Rechenoperationen auf $\mathbb Q$:} \\ -\begin{compactitem} - \item Addition: $[\frac{m}{m'}]+[\frac{n}{n'}] := [\frac{mn'+m'n}{m'n'}]$ - \item Multiplikation: $[\frac{m}{m'}] \cdot [\frac{n}{n'}] := [\frac{mn}{m'n'}]$ -\end{compactitem} - -\begin{satz} - Mit Addition und Multiplikation ist $\mathbb Q$ ein Körper mit\\ - neutralen Elementen: $0=[\frac{0_{\mathbb Z}}{1_{\mathbb Z}}]= - [\frac{0_{\mathbb Z}}{n_{\mathbb Z}}], 1:=[\frac{1_{\mathbb Z}}{1_{\mathbb Z}}]=[\frac n n] \neq 0$\\ - inversen Elementen: $-[\frac{n}{n'}]=[\frac{-n}{n}], [\frac{n}{n'}]^{-1}=[\frac{n'}{n}]$\\ -\end{satz} - -\noindent Ordnung auf $\mathbb Q:$ f\"ur $[\frac{n}{n'}] \in \mathbb Q$ kann man stets $n'>0$ annehmen \\ -Realtion: $R:=\{([\frac{m}{m'}],[\frac{n}{n'}]) \in \mathbb Q \times \mathbb Q \mid mn' \le m'n, -m',n' > 0\}$ gibt Ordnung $\le$ \\ - -\begin{satz} - $\mathbb Q$ ist ein angeordneter K\"orper (d.h. $\le$ ist eine Totalordnung undv erträglich mit Addition und Multiplikation). -\end{satz} - -Notation: schreibe vereinfacht nur noch $\frac{n}{n'}$ für die Zahl $[\frac{n}{n'}] \in \mathbb Q$ und verwende auch Symbole $p,q,...$ für Elemente aus $\mathbb Q$ \\ - -Gleichung $p \cdot x = q$ hat stets eindeutige Lösung: $x=q \cdot p^{-1}$ ($p,q \in \mathbb Q, p \neq 0$) \\ - -\textbf{Frage:} $\mathbb N \subset \mathbb Z \to \mathbb Z \subset \mathbb Q$? -\textbf{Antwort:} Sei $\mathbb Z_{\mathbb Q} := {\frac n 1 \in \mathbb Q \mid n \mathbb Z}, I: -\mathbb Z \to \mathbb Z_{\mathbb Q}$ mit $I(n)=\frac n 1$ \\ -$\Rightarrow I$ ist Isomorphismus bez\"uglich Addition, Multiplikation und Ordnung. \\ -In diesem Sinn: $\mathbb N \subset \mathbb Z \subset \mathbb Q$ \\ - -\begin{folg} - Körper $\mathbb Q$ ist archimedisch angeordnet, d.h. f\"ur alle $q \in \mathbb Q \exists n \in \mathbb N: q<_{\mathbb Q} n.$ -\end{folg} - -\begin{proof} - Sei $q = [\frac{k}{k'}]$ mit $k'>0$ \\ - $n := 0$ falls $k<0 \Rightarrow q=[\frac{k}{k'}] < [\frac{0}{k'}]=0=n$ \\ - $n := k+1$ falls $k \ge 0 \Rightarrow q=[\frac{k}{k'}] < [\frac{k+1}{k'}]=n$ \QEDA -\end{proof} \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter05_reelle_Zahlen.tex b/1. Semester/ANAG/TeX_files/chapter05_reelle_Zahlen.tex deleted file mode 100644 index 1353e8d..0000000 --- a/1. Semester/ANAG/TeX_files/chapter05_reelle_Zahlen.tex +++ /dev/null @@ -1,56 +0,0 @@ -\chapter{Reelle Zahlen} -\begin{description} - \item[Frage:] Frage: algebraische Gleichung $a_0+a_1x+\dots+a_x^k=0\;(a_j\in \whole)$\\ - i.A nur für $k=1$ lösbar (d.h. lin. Gl.) -\end{description} - -\begin{exmpn} - $x^2 - 2 = 0$ keine Lösung in $\ratio$. Angenommen es existiert eine Lösung $x = \frac{m}{n} \in \ratio$, o.B.d.A. höchstens eine der Zahlen $m,n$ gerade $\Rightarrow \frac{m^2}{n^2} = 2 \Rightarrow m^2 = 2n^2 \Rightarrow m$ gerade $\overset{m=2k}{\Rightarrow} 4k^2 = 2n^2 \Rightarrow 2n^2 \Rightarrow 2k^2 = n^2 \Rightarrow n$ gerade $\Rightarrow \lightning$.\QEDA -\end{exmpn} - -\noindent Offenbar $1,4^2 < 2 < 1,5^2,\; 1,41^2 < 2 < 1,42^2,\;\dots,$ falls es $\sqrt{2}$ gibt, kann diese in $\ratio$ beliebig genau approximiert werden. Es folgt, dass $\ratio$ anscheinend "`Lücken"' hat. -\textbf{Fläche auf dem Einheitskreis} kann durch rationale Zahlen beliebig genau approximiert werden. Falls "`Flächenzahl"' $\pi$ existiert, ist das \textbf{nicht} Lösung einer algebraischen Gleichung (Lindemann 1882).\\ - -\begin{description} - \item[Ziel:] Konstruktion eines angeordneten Körpers, der diese Lücken füllt. -\end{description} - -\section{Struktur von archimedisch angeordneten Körper (allg.)} -$\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$. -\begin{satz} - Sei $\field$ Körper. Dann gilt $\forall a,b \in \field$: - \begin{enumerate}[label={\arabic*)}, nolistsep] - \item $0,1, (-a), b^{-1}$ sind eindeutig bestimmt - \item $(-0) = 0$, $1^{-1} = 1$ - \item $-(-a) = a$, $(b^{-1})^{-1} = b$ $(b \neq 0)$ - \item $-(a + b) = (-a) + (-b)$, $(a^{-1}b^{-1}) = (a^{-1}b^{-1})$ $(a,\neq 0)$ - \item $-a = (-1)\cdot a$, $(-a)(-b)=ab$, $a \cdot 0 = 0$ - \item $ab=0 \iff a=0 \text{ oder } b=0$ - \item $a + x = b \text{ hat eindeutige Lösung } x = b + (-a) =:b-a$ Differenz\\ - $ax=b \text{ hat eindeutige Lösung } x = a^{-1}b:=\frac{b}{a}$ Quotient - \end{enumerate} -\end{satz} - -\begin{proof} - \begin{enumerate}[label={\arabic*)}] - \item vgl. lin. Algebra - \item betrachte $0 + 0 = 0$ bzw. $1 \cdot 1 = 1$ - \item $(-a) + a = 0 \overset{komm}{\Rightarrow} a = -(-a)$ Rest analog - \item $a+b = ((-a) + (-b)) \Rightarrow$ Behauptung, Addition und Multiplikation analog - \item $a\cdot 0 = 0$ vgl. lin. Algebra\\ - $1a + (-1)a = 0 \Leftrightarrow (1-1)a=0 \Rightarrow (-1)a=-1$, $(-a)(-b)=(-1)(-a)b\overset{3,5}{=}ab$ - \item ($\Leftarrow$): nach 5)\\ - ($\Rightarrow$) sei $a\neq0$ (sonst klar) $\Rightarrow 0 = a^{-1}\cdot 0 \overset{ab=0}{=} a^{-1}ab = b \Rightarrow$ Beh. - \item $a+x=b \Leftrightarrow x = (-a) + a \neq x = (-a) + b$, für $ax=b$ analog \QEDA - \end{enumerate} -\end{proof} - -Setze für alle $a, \dots a_k \in \field,n\in \natur_{\geq 1}$ -\begin{itemize} - \item[Vielfache] $n\cdot a$ (kein Produkt in $\field$!) - \item[Potenzen] $a^n=\prod_{k=1}^{n} a_k \text{für } n \in N_{\geq 1}$ damit $(-n)a:=n(-a) \text{, } 0_{\natur}a=0_{\natur} \text{ für } n\in\natur_{\geq1}\\ - a^{-n}=(a^-1)^n \text{, }a^{0_{\natur}}:=1_{\field} \text{ für } n \in \natur_{\geq 1}, a \neq 0\\ - beachte: 0^0 = (0_\natur)^{0_{\natur}} \text{ \emph{nicht} definiert!}$ - \item[Rechenregeln] $\forall\;a,b\in \field\text{, } m,n\in \whole \text{ (sofern Potenz definiert) } $ -\end{itemize} -%TODO \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter06_komplexe_zahlen.tex b/1. Semester/ANAG/TeX_files/chapter06_komplexe_zahlen.tex deleted file mode 100644 index dcf3d7b..0000000 --- a/1. Semester/ANAG/TeX_files/chapter06_komplexe_zahlen.tex +++ /dev/null @@ -1,27 +0,0 @@ -\chapter{Komplexe Zahlen (kurzer Überblick)} -\begin{description} - \item[Problem:] $x^2 = -1$ keine Lösung in $\real \Rightarrow$ Körpererweiterung $\real \to \comp$ - \item[Betrachte Menge der komplexen Zahlen] $\comp := \real \times \real = \real^2$ - \item mit Addition und Multiplikation:\\ - $(x,x^{'}) + (y,y^{'}) = (x+y, x^{'} + y^{'})$\\ - $(x,x^{'}) \cdot (y,y^{'}) = (xy - x^{'}y^{'}, xy^{'}+x^{'}y)$ - \item $\comp$ ist ein Körper mit (vgl. lin Algebra):\\ - $0_{\field} = (0,0)$, $1_{\field} = (1,0)$, $-(x,y) = (-x,-y)$ and $(x,y)^{-1} = \bigg(\frac{x}{x^2+y^2},\frac{-y}{x^2+y^2}\bigg)$\\ - mit imaginärer Einheit $\iota=(0,1)$\\ - $z=x+\iota y$ statt $z=(x,y)$ mit $x:=\Realz(z)$ Realteil von $z$, $y:= \Imag(z)$ Imaginärteil von $z$\\ - komplexe Zahl $z=x + \iota y$ wird mit reeller Zahl $x \in \real$ identifiziert\\ - offenbar $\iota^2=(-1,0)=-1$, d.h. $z=\iota \in \comp$ und löst die Gleichung $z^2=-1$ (nicht eindeutig, auch $(-\iota)^2 = -1$)\\ - Betrag $|\cdot|: \comp \to \real_{> 0}$ mit $|z|:= \sqrt{x^2+y^2}$ (ist Betrag/Länge des Vektors $(x,y)$)\\ - es gilt: - \begin{enumerate}[label={\alph*)}] - \item $\Realz(z) = \frac{z+\overline{z}}{2}, \Imag(z) = \frac{z+\overline{z}}{2\iota}$ - \item $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$, $\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$ - \item $|z| = 0 \iff z=0$ - \item $|\overline{z}| = |z|$ - \item $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$ - \item $|z_1 + z_2| \leq |z_1| + |z_2|$ (Dreiecks-Ungleichung: Mikoswski-Ungleichung) - \end{enumerate} -\begin{proof} - SeSt \QEDA -\end{proof} -\end{description} \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter07_grundl_ungleichungen.tex b/1. Semester/ANAG/TeX_files/chapter07_grundl_ungleichungen.tex deleted file mode 100644 index 80fab02..0000000 --- a/1. Semester/ANAG/TeX_files/chapter07_grundl_ungleichungen.tex +++ /dev/null @@ -1,132 +0,0 @@ -\part{Metrische Räume und Konvergenz} -\begin{description} - \item[Konvergenz:] grundlegender Begriff in Analysis %(benötigt Abstandsbegriff (Metrik)) -\end{description} -\chapter{Grundlegen Ungleichungen} - -\begin{satz}[Geometrisches und arithmetisches Mittel]\label{satz_7_1_geo_mittel} - Seien $x_1, \dots, x_n \in \real_{>0}$\\ -$\Rightarrow$ - \begin{tabular}{ccc} - $ \sqrt[n]{x_1, \dots, x_n}$ & $=$ & $\frac{x_1, \dots, x_n}{n}$ \\ - geoemtrisches Mittel & & arithmetisches Mittel \\ - \end{tabular}\\ -Gleichheit gdw $x_1 = \dots = x_n$. -\end{satz} - -\begin{proof} - Zeige zunächst mit vollständiger Induktion\\ - \begin{align} %% add /nonumber to have no numbering - \prod_{i=1}^{n}x_i= \Rightarrow \sum_{i=1}^{n} x_i \geq n \text{, mit } x_1=\dots=x_n \label{7_1_ind} - \end{align} - \begin{itemize} - \item (IA) $n = 1$ klar - \item (IS) (\ref{7_1_ind}) gelte für $n$, zeige (\ref{7_1_ind}) für $n+1$ d.h. $\prod_{i=1}^{n+1} = 1$, falls alle $x_i=1 \beha$ Sonst oBdA $x_n < 1$, $x_{n+1} > 1:$\\ mit $y_n:=x_n x_{n+1}$ gilt $x_1\cdot\dots\cdot x_{n-1}\cdot y_n=1$ - \begin{align*} - \Rightarrow x_1 + \dots + x_{n+1} &= \underbrace{x_1+\dots+x_{n-1}}_{\geq \text{ (IV)}} + y_n - y_n + x_n+x_{n+1}\\ - &\geq n + \underbrace{(x_{n+1} -1)}_{>n}\underbrace{(1-x_n)}_{>n}\\ - &\Rightarrow (\ref{7_1_ind}) \forall n \in \natur& \text{vollständige Induktion}\\ - \shortintertext{allgemein sei nun $g:=\big( \prod_{i=1}^{n} x_i \big)^{\frac{1}{n}} \Rightarrow \prod_{i=1}^{n} \frac{x_i}{g} = 1$} - &\Rightarrow \sum_{i=1}^{n} \frac{x_i}{g} \geq n \beha& \text{Satz \ref{7_1_ind}}\\ - \shortintertext{Aussage über Gleichheit nach nochmaliger Durchsicht.} - \end{align*} - \end{itemize} - \QEDA -\end{proof} - -\begin{satz}[allg. Bernoulli-Ungleichung] - Seien $\alpha, x \in \real$. Dann\\ - \begin{align*} - 1)\;(1+x)^{\alpha} &\geq 1 + \alpha x \; \forall x > -1, \alpha > 1\\ - 2)\; (1+x)^{\alpha} &\leq 1+\alpha x \; \forall x \geq -1, 0 < \alpha < 1 - \end{align*} -\end{satz} - -\begin{proof} % fix alignment - \begin{enumerate} - \item[2)] Sei $\alpha =\frac{m}{n} \in \ratio_{<1}\text{, d.h. } m\leq n$ - \begin{align*} - &\Rightarrow (1+x)^\frac{m}{n} = \sqrt[n]{(1+x)^m\cdot1^{n-m}}& \text{Definition} \\ - &\leq \frac{m(1+x)+(n-m)\cdot1}{n}&\\ - &=\frac{n + mx}{n} = 1 + \frac{m}{n}x \text{, für } \alpha \in \ratio \beha& - \shortintertext{Sei $\alpha \in \real$ angenommen $(1+x)^{\alpha} > 1 + \alpha x$ ($x\neq 0$ sonst klar!)} - & \Rightarrow \exists \in \ratio_{<1} - \begin{cases*} - x > 0&$\alpha1$ mit $\frac{1}{q} + \frac{1}{q} =1 \Rightarrow ab \leq \frac{a^p}{p}+\frac{b^q}{q}\;\forall a,b \geq 0$ (Gleichheit gdw $a^p = b^q$)\\ - Spezialfall($p=q=2$): $ab \geq \frac{a^2 + b^2}{2}$ gilt $\forall a,b \in \real$ (folgt direkt $0\leq (a-b)^2$) -\end{satz} - -\begin{proof} %fix formating - \begin{align*} - \shortintertext{Sei $a,b > 0$ (sonst klar!)} - &\Rightarrow \big(\frac{b^q}{a^p}\big)^{\frac{p}{q}} = \big(1+\big(\frac{b^q}{a^p} -1\big)\big)^{\frac{p}{q}}&\\ - &\leq 1+ \frac{1}{q}\big(\frac{b^q}{a^p} -1\big)& \text{Bernoulli-Ungleichung}\\ - &=\frac{1}{p}+\frac{1}{q}+\frac{1}{q}\frac{b^q}{a^p}-\frac{1}{q}\\ - &\Rightarrow a^p\frac{b}a^{\frac{p}{q}} = a^{p(1-\frac{1}{q})}b = ab \leq \frac{a^p}{p} + \frac{b^q}{q}& \cdot a^p - \end{align*}\QEDA -\end{proof} - -\begin{satz}[Höldersche Ungleichung] - Sei $p,q \in \real;\;p,q > 0$ mit $\frac{1}{q} + \frac{1}{p} = 1$\\ - $\Rightarrow \sum_{i=1}^{n} \vert x_i y_i\vert \leq \big( \sum_{i=1}^{n} \vert x_i \vert \big)^{\frac{1}{p}} \big( \sum_{i=1}^{n} \vert y_i \vert \big)^{\frac{1}{p}}\;\forall x,y \in \real$ -\end{satz} - -\begin{remark} - \begin{enumerate}[label={\arabic*)}] - \item Ungleichung gilt auch für $x_i,y_i \in \comp$ (nur Beträge gehen ein) - \item für $p=q=2$ heißt Ungleichung Cauchy-Schwarz-Ungleichung (Gleichheit gdw $\exists x \in \real x_i = \alpha y_i \text{ oder } y_i = \alpha x_i\;\forall i$) - \end{enumerate} -\end{remark} - -\begin{proof} - Faktoren rechts seien $\mathcal{X} \text{ und } \mathcal{Y}$ d.h. - \begin{align*} - \mathcal{X}^p &= \sum_{i=1}^{n} \vert x_i \vert^{\frac{1}{p}}, \mathcal{Y}^p = \sum_{i=1}^{n} \vert y_i \vert^{\frac{1}{q}}\text{, falls } \mathcal{X}=0&\\ &\Rightarrow x_i = 0\;\forall i \beha \text{, analog für } \mathcal{Y} =0&\\ - \shortintertext{Seien $\mathcal{X}, \mathcal{Y} > 0$} - &\Rightarrow \frac{\vert x_i y_i \vert}{\mathcal{XY}} \leq \frac{1}{p}\frac{\vert x_i \vert^p}{\mathcal{X}^p}+ \frac{1}{q}\frac{\vert y_i \vert^q}{\mathcal{Y}^p} \forall i& \text{Satz 7.3}\\ - &\Rightarrow \frac{1}{\mathcal{XY}}\sum_{i=1}^{n}\vert x_i y_i \vert \leq \frac{1}{p}\frac{\mathcal{X}^p}{\mathcal{X}^p}+\frac{1}{q}\frac{\mathcal{Y}^p}{\mathcal{Y}^p} = 1 \beha & \cdot \mathcal{XY} - \end{align*}\QEDA -\end{proof} - -\begin{satz}[Minkowski-Ungleichung] - Sei $p\in \real, p \geq 1 \Rightarrow \big(\sum_{i=1}^{n} \vert x_i + y_i \vert^p \big)^\frac{1}{p} \leq \big(\sum_{i=1}^{n} \vert x_i \vert^p \big)^\frac{1}{p} + \big(\sum_{i=1}^{n} \vert y_i \vert^p \big)^\frac{1}{p}\forall x,y\in \real$ -\end{satz} - -\begin{remark} - \begin{enumerate}[label={\arabic*)}] - \item Ungleichung gilt auch für $x_i, y_i \in \comp$ (vgl. Beweis) - \item ist $\Delta$-Ungleichung für $p$-Normen (vgl. später) - \end{enumerate} -\end{remark} - -\begin{proof} - $p=1$ Beh. folgt aus $\Delta$-Ungleichung $\vert x_i + y_i\vert \overset{Satz 5.5}{\leq} \vert x_i \vert + \vert y_i \vert \forall i$\\ $p>1$ sei $\frac{1}{p} + \frac{1}{q} = 1$, $z_i:=\vert x_i + y_i\vert^{p-1}\forall i$ - \begin{align*} - \mathcal{S}^p &= \sum_{i=1}^{n} \vert z_i \vert^q & q = \frac{p}{p-1}\\ - & = \sum_{i=1}^{n} \vert +x_i+y_i \vert\cdot\vert z_i \vert^q & \\ - & = \sum_{i=1}^{n} \vert x_i + y_i \vert + \sum_{i=1}^{n} \vert z_i \vert & \Delta\text{-Ungleichung}\\ - & \leq \big(\mathcal{X+Y}\big)\big(\sum_{i=1}^{n} \vert z_i\vert^q \big)^\frac{1}{p} & \text{Hölder-Ungleichung}\\ - & = \big(\mathcal{X+Y}\big)\mathcal{S}^\frac{p}{q} & \\ - & \beha & p=\frac{p}{q}+1 - \end{align*}\QEDA -\end{proof} - - -%continue-+ diff --git a/1. Semester/ANAG/TeX_files/chapter08_metr_raeume.tex b/1. 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Semester/ANAG/TeX_files/chapter08_metr_raeume.tex +++ /dev/null @@ -1,306 +0,0 @@ -\chapter{Metrische und normierte Räume} -\section{Metrische Räume} -\begin{mydefn}[Metrik] - Sei $X$ Menge und Abbildung $d: X \times X \to \real$ heißt \underline{Metrik} auf $X$ falls $\forall x,y,z \in X$ - \begin{enumerate}[label={\alph*)}] - \item $d(x,y) = 0 \Leftrightarrow x=y$ - \item $d(x,y) = d(y,x)$ (Symmetrie) - \item $d(x,z) \leq d(x,y) + d(y,z)$ ($\Delta$-Ungleichung) - \end{enumerate} - $(X,d)$ heißt metrischer Raum. -\end{mydefn} -Man hat $d(x,y) = 0 \forall x,y \in X$, dann - -\begin{align} - 0 &= d(x,x) = d(x,y) + d(y,x) & \text{a), c)}\nonumber\\ - & = 2d(x,y) \forall x,y & \text{b)}\nonumber\\ - \text{nach } & \text{b), c) } &\nonumber\\ - & \vert d(x,y) -d(z,y)\vert \leq d(x,y) \forall x,y,z \in X & -\end{align} - -\begin{exmpn}[Standardmetrik]\label{8_1_exmp_metrik} - $d(x,y) := \vert x-y\vert$ ist Metrik auf $X=\real$ bzw. $X=\comp$ - \begin{align*} - \text{Eig. a), b), c) klar}& &\\ - \text{c) } \vert x-z\vert& \vert (x+y)-(x-z)\vert &\\ - &\leq \vert x+y\vert + \vert y+z\vert & \Delta\text{-Ungleichung für }\real\text{, }\comp\text{-Betrag} - \end{align*} -\end{exmpn} - -\begin{exmpn}[diskrete Metrik] - Diskrete Metrik auf beliebiger Menge $X$.\\ - \[d(x,y) = - \begin{dcases*} - 0 & x = y\\ - 1 & $x \neq y$ - \end{dcases*}\] - ist offenbar eine Metrik. -\end{exmpn} - -\begin{exmpn}[induzierte Metrik] - Sei $(X,d)$ metrischer Raum, $Y \subset X$\\ - $\Rightarrow (Y,d)$ ist metrischer Raum mit \emph{induzierter Metrik} $\tilde{d}(x,y):=d(x,y)\forall x,y \in Y$ -\end{exmpn} - -\section{Normierte Räume} - -wichtiger Spezialfall: normierte Vektorraum(VR) - -\begin{mydefn}[Norm] - Sei $X$ Vektorraum über $K=\real$ oder $K=\comp$.\\ - Abbildung $\Vert \cdot \Vert: X \to \real$ heißt \emph{Norm} auf $X$ falls $\forall x,y \in X, \forall \lambda \in \real$ gilt: - \begin{enumerate}[label={\alph*)}] - \item $\Vert x\Vert = \Leftrightarrow x=0$ - \item $\Vert \lambda x\Vert = \vert \lambda \vert \Vert x\Vert$ (Homogenität) - \item $\Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert$ ($\Delta$-Ungleichung) - \end{enumerate} - $(X,\Vert \cdot\Vert)$ heißt \emph{normierter Raum}. -\end{mydefn} - -\begin{align*} - \text{Metrik} &\leftarrow \text{Norm}&\\ - \text{Abbildung} & \not \rightarrow \text{VR, Abstand } x,0\\ - \text{man hat } \Vert x \Vert &\leq 0 \forall x \in X \text{, denn } 0 = \Vert x-x\Vert \leq \Vert x\Vert + \Vert -x\Vert = 2\Vert x\Vert & \text{a), c), b)}\\ -\end{align*} -Analog Satz 5.5 folgt\\ -\begin{align} - \vert \Vert x \Vert - \Vert y \Vert\vert &\leq \Vert x-y\Vert \forall x,y \in X -\end{align} -$\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \emph{Halbraum} falls nur b), c) gelten analog Beispiel \ref{8_1_exmp_metrik} folgt. - -\begin{satz} - Sei $(X,\Vert\cdot \Vert)$ normierter Raum, dann $X$ metrischer Raum mit Metrik $d(x,y):=\Vert x-y \Vert\forall x,y \in X$. -\end{satz} - -\begin{exmpn}\label{8_5_exmp_Norm} - $X=\real^n$ ist Vektorraum über $\real$, Elemente in $\real^n$\\ $x=(x_1,\dots,x_n), y=(y_1, \dots, y_n)$,\\ man hat unter anderem folgende Normen auf $\real^n$ - \begin{align*} - p\text{-Norm}: \vert x \vert_p& := \Bigg( \sum_{i=0}^{n} \vert x_i \vert^p \Bigg)^{\frac{1}{p}} & (1\leq p < \infty)\\ - \text{Maximum-Norm}: \vert x \vert_p& := \max\{\vert x_i \vert \mid i=1,\dots n\} &\\ - \text{a), b) jeweils klar, c) für } & - \begin{cases*} - \vert \cdot \vert_p & \text{ist Minkowski-Ungleichung}\\ - \vert \cdot \vert_{\infty} & \text{wegen } $\vert x_i + y_i \vert \leq \vert x_i \vert + \vert y_i \vert \forall i$ - \end{cases*} - \end{align*} - Standardnorm in $\real^n$: - $\vert \cdot \vert = \vert \cdot \vert_{p=2}$ heißt \emph{eukldische Norm}.\\ -\end{exmpn} - -\begin{mydefn}[Skalarprodukt] - $\langle x,y \rangle = \sum_{i=1}^{n}$ heißt \emph{Skalarprodukt} (inneres Produkt) von $x,y \in \real^n$ offenbar $\langle x,y \rangle = \vert x \vert_2 \forall x \in comp$ nur für euklidische Räume gibt es Skalarprodukt (nur für euklische Norm!).\\ - Man hat $\vert \langle x,y\rangle \vert \leq \vert x \vert_2 \cdot \vert y \vert_2 \forall x,y \in \real^n$ Cauchy-Schwarsche Ungleichung (CSU), denn - \begin{align*} - \vert \langle x,z \rangle \vert &= \vert \sum_{i=1}^{n} x_i y_i \vert \leq \sum_{i=1}^{n}\vert x_i y_i\vert & \Delta\text{-Ungleichung in } \real\\ - & \leq \vert x \vert_2 \cdot\vert y \vert_2 & \text{Hölder-Ungleichung mit } p=q=2 - \end{align*} -\end{mydefn} - -\begin{exmpn} - $X=\comp^n$ ist Vektorraum über $\comp$, $x=(x_1,\dots,x_n) \in\comp^n, x_i \in \comp$\\ - analog zum Bsp. \ref{8_5_exmp_Norm} sind $\vert \cdot \vert_{p} \text{ und } \vert \cdot \vert_{\infty}$ Normen auf $\comp^n$\\ - $\langle x,y\rangle = \sum_{i=1}^{n} \bar{x}_i y_i\forall x_i, y_i \in \comp$ heißt \emph{Skalarprodukt} von $x,y \in \comp^n$ (beachte $\langle x,y\rangle \in \comp, \langle x,x \rangle=\vert x \vert^2$) \\ - $\overset{\text{wie oben}}{\Rightarrow} \vert \langle x,y\rangle \vert \leq \vert x \vert\cdot \vert y \vert \forall x,y \in \comp^n$ -\end{exmpn} - -\begin{mydefn}[Orthogonalität] - $x,y \in \real^n(\comp^n)$ heißen \emph{orthogonal} falls $\langle x,y\rangle =0$ -\end{mydefn} - -\begin{exmpn} - Sei $M$ beliebige Menge, $f: M \to \real$\\ - $\Vert f\Vert:= \sup\{\vert f(x) \vert \mid x\in M\}$. Dann ist \\ - \[\mathcal{B}(M):= \{f: M \to \real \mid \Vert f\Vert < \infty\}\] - \emph{Menge der beschränkte Funktionen} auf $M$\\ - $\mathcal{B}(M)$ ist Vektorraum auf $\real$ - \begin{enumerate}[label={\alph*)}] - \item $((f+g)(x) = f(x) + g(x)$ - \item $(\lambda f)(x) = \lambda f(x)$ - \item Nullelement ist Nullfunktion $f(x)=0 \forall x \in M$ - \end{enumerate} - \begin{align*} - \shortintertext{$\Vert \cdot\Vert$ ist Norm auf $\mathcal{B}(M)$, denn a), b) klar}\\ - \Vert f+g\Vert&:=\sup\{\vert f(x)+g(x) \vert\mid x \in M\}&\\ - &\leq \sup\{\vert f(x) \vert + \vert g(x) \vert\mid x\in M \} & \Delta\text{-Ungleichung in }\real\\ - &\leq \sup\{\vert f(x) \vert\mid x \in M \} + \sup\{\vert g(x) \vert\mid x \in M \} & \text{Übungsaufgabe}\\ - &=\Vert f\Vert + \Vert g\Vert - \end{align*} -\end{exmpn} - -\begin{exmpn} - $\Vert x \Vert:=\vert x_1 \vert$ auf $X=\real^n \to$ kein Nullvektor ``nur'' Halbnorm (später wichtige Halbnorm in Integraltheorie). Normen $\Vert \cdot \Vert_1,\;\Vert \cdot \Vert_2$ auf $X$ heißen äquivalent falls - \[ - \exists \alpha,\beta > 0\;\alpha \vert x \vert_1 \leq \vert x \vert_2 \leq \beta\vert x \vert_1\qquad\forall x \in X - \] - (Indizes entsprechen hier keinem p, sondern es sind hier nur beliebige unterschiedliche Normen gemeint.) -\end{exmpn} - -\begin{exmpn} - \[ - \vert x \vert_{\infty} \leq \vert x \vert_p \leq \sqrt[p]{n}\vert x \vert_{\infty}\qquad \forall x \in \real^n,\;p\geq 1\\ - \] - $\vert \cdot \vert_\infty$ und $\vert \cdot \vert_\infty$ sind äquivalent $\forall p \geq 1$ -\end{exmpn} - -\begin{proof} - \begin{align*} - \vert x \vert_{\infty} &=\big(\max \{ \vert x_j \vert, \vert \dots \}^p\big)^\frac{1}{p} \leq \bigg(\sum_{j=1}^{n} \vert x_j \vert^p \bigg)^\frac{1}{p} = \vert x \vert_p\\ - \vert x \vert_{\infty} &\leq \big( n\cdot \max\{ \vert x_j \vert, \vert \dots \}^p\big)^\frac{1}{p} \leq \bigg(\sum_{j=1}^{n} \vert x_j \vert^p \bigg)^\frac{1}{p} = \sqrt[p]{n}\vert x \vert_{\infty} - \end{align*}\QEDA -\end{proof} - -\begin{folg} - $\vert \cdot \vert_p,\;\vert \cdot \vert_q$ äquivalent auf $\real^n\;\forall p,q \geq 1$ (siehe Aufgabe 45b)) -\end{folg} - -\section{Begriffe im metrischen Raum} - -\begin{mydefn}[Kugel im metrischen Raum] - Sei $(X,d)$ metrischer Raum. - \begin{itemize} - \item $B_r(a):= \{ a \in X \mid d(a,x) 0$ - \item $B_r[a]:= \overline{B}_r(a) = \{ a \in X \mid d(a,x) \leq r \}$ heißt abgeschlossene \emph{Kugel} um $a$ mit Radius $r>0$ - \end{itemize} -\end{mydefn} -Hinweis: muss keine übliche Kugel sein z.B. $\{x\in \real^n \mid d(0,x) < 1\}$ ist Quadrat $B_r(0)$. - -\begin{mydefn} - \begin{itemize} - \item Menge $M\subset X$ \emph{offen} falls $\forall x \in M\;\exists \epsilon > 0\; B_{\epsilon}(x) \subset M$ - \item Menge $M$ offen falls $X\setminus M$ abgeschlossen - \item $U \subset X$ Umgebung von $M \subset X$ falls $\exists V \subset X$ offen mit $M \subset V \subset U$ - \item $x \in M$ \emph{innerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset M$ - \item $x \in M$ \emph{äußerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset X\setminus M$ - \item $x \in X$ \emph{Randpunkt} von $M$ falls $x$ weder innerer noch äußerer Punkt ist - \item $\inter M:=$ Menge der \emph{inneren} Punkte von $M$ heißen inneres von $M$ - \item $\ext M:=$ Menge der \emph{äußeren} Punkte von $M$ heißen äußeres von $M$ - \item $\partial M:=$ Menge der Randpunkte von $M$ heißt \emph{Rand} von $M$ - \item $\cl M:= \overline{M}:=\overline{\inter M} \cup \partial M$ heißt Abschluss von $M$ (closure) - \item $M \subset X$ \emph{beschränkt} falls $\exists a \in X, r >0\; M \subset B_r(a)$ - \item $x \in X$ \emph{Häufungskt (Hp)} von $M$ falls $\forall \epsilon > 0$ enhält \emph{$B_{\epsilon}(x)$ unendlich viele} Elemente aus $M$ - \item $x \in M$ \emph{isolierter} Punkt von $M$ falls $x$ kein Hp von $M$ - \end{itemize} -\end{mydefn} - -\begin{exmpn} - \begin{enumerate}[label={\alph*)}] - \item Sei $X=\real$ mit $d(x,y)=\vert x-y \vert$ - \begin{itemize} - \item $(a,b),(-\infty,a)$ offen - \item $[a,b], (-\infty, b]$ abgeschlossen - \item $[a,b)$ weder abgeschlossen noch offen, aber beschränkt - \end{itemize} - Es gilt: - \begin{itemize} - \item $\inter(a,b) = \inter[a,b] = (a,b)$ - \item $\ext(a,b) = \ext[a,b] = (-\infty,a) \cup (b, \infty)$ - \item $\partial(a,b) = \partial[a,b] = \{a,b\}$ - \item $\cl(a,b) = \cl[a,b]=[a,b]$ - \end{itemize} - Speziell: - \begin{itemize} - \item $\ratio$ weder offen noch abgeschlossen in $\real$, da $\inter \ratio =\emptyset, \ext \ratio = \emptyset, \partial \ratio = \real$ - \item $\real \setminus \emptyset$ ist offen - \item $\natur \text{ in } \real$ abgeschlossen und nicht beschränkt - \item $[0,3]$ ist Umgebung von $[1,2], B_r(a)$ ist Umgebung von $a$ (eigentlich $\{a\}$) - \item $a$ ist Hp von $(a,b),[a,b]$ für $a0,a\in X$ - \item $M\subset X$ beschränkt $\Rightarrow \forall a \in X \exists r>0\colon M\subset B_r(a)$ - \end{enumerate} -\end{lem} - -\begin{proof} - \begin{enumerate}[label={\arabic*)}] - \item Sei $b \in B_r(a),\epsilon := r - a-d(a,b)>0$, dann gilt für beliebige $x \in B_{\epsilon}(b)$ - \begin{align*} - d(a,x) &\leq d(a,b) + d(b,x) & \Delta\text{-Ungleichung mit } b\\ - & 0 \colon B_{\epsilon_i}(x) \subset U_i \forall i, \epsilon = \min\{\epsilon_1, \dots \epsilon_n\}$\\ - $\Rightarrow B_{\epsilon}(x) \in \bigcap_{i=0}^{n} U_i \beha$ - \item Sei $x \in \bigcup_{U\in\tau^{\prime}} U \Rightarrow \exists \tilde{U}\in \tau^{\prime}\colon x \in \tilde{U} \overset{\tilde{U} \text{ offen}}{\Rightarrow}\exists \epsilon > 0 \colon B_{\epsilon}(x) \subset \tilde{U} \in \bigcup_{U\in\tau^{\prime}} U \beha$. - \end{enumerate}\QEDA -\end{proof} - -Hinweis: Durchschnitt beliebiger vieler offener Menge ist nicht offen! - -\begin{exmp} - $\bigcap_{n\in \natur} (-\frac{1}{n},1+\frac{1}{n}) = [0,1]$ -\end{exmp} - -Komplementbildung im Satz \ref{8_13_satz_open_topo} liefert: - -\begin{folg}[Abgeschlossene Kugeln sind Topologie auf X] - Sei $(X,d)$ metrischer Raum und $\sigma :=\{ V \subset X \mid V \text{ abgeschlossen}\}$. Dann - \begin{enumerate}[label={\arabic*)}] - \item $X,\emptyset\in\sigma$ - \item $\bigcup_{i=1}^{n} U_i \in \sigma$ falls $U_i\in \sigma \text{ für } i = 1, \dots, n$ (endlich viele) - \item $\bigcap_{U\in\sigma^{\prime}} U \in \sigma$ falls $\sigma^{\prime} \subset \sigma$ (beliebig viele) - \end{enumerate} -\end{folg} - -\begin{mydefn}[Topologie] - Sei $X$ Menge und $\tau$ Menge von Teilmengen von $X$ (d.h. $\tau \in \powerset(X)$)\\ - $\tau$ ist Topologie und $(X, \tau)$ topologischer Raum, falls 1), 2), 3) aus Satz \ref{8_13_satz_open_topo} gelten. -\end{mydefn} -\begin{remark} - Menge $U\in \tau$ heißen dann offen (per Definition!). Folglich oben definierte offene Mengen in metrischen Räumen bilden ein Spezialfall für eine Topologie. Beachte! In metrischem Raum $(X,d)$ ist $\tilde{\tau} = \{\emptyset, X\}$ stets eine Topologie für beliebige Menge $X$). -\end{remark} - -\begin{satz} - Seinen $\Vert \cdot\Vert_1, \Vert \cdot\Vert_2$ äquivalente Normen auf $X$ und $U\subset X$. Dann\\ - $U$ offen bezüglich $\Vert \cdot\Vert_1 \Leftrightarrow U \text{ offen bezüglich } \Vert \cdot\Vert_2$. -\end{satz} - -\begin{proof} - Übungsaufgabe.\QEDA -\end{proof} - -\begin{satz} - Sei $(X,d)$ metrischer Raum und $M\subset X$. Dann - \begin{enumerate}[label={\arabic*)}] - \item $\inter M, \ext M$ offen - \item $\partial M, \inter M$ abgeschlossen - \item $M \leq \inter M$ falls $M$ offen, \\ - $M= \cl M$ falls $M$ abgeschlossen - \end{enumerate} -\end{satz} - -\begin{proof} - \begin{enumerate}[label={\arabic*)}] - \item Seien $x \in \inter M$, d.h. innere Punkte von $M \Rightarrow \exists \epsilon > 0 \colon B_{\epsilon}(x) \subset M$, da $B_{\epsilon}(x)$ offene Menge, ist jedes $y \in B_{\epsilon}(x)$ eine Teilemenge von $\inter M$ $\Rightarrow B_{\epsilon}(x) \subset M \beha$ ($\ext M$ analog) - \item $\partial X\setminus (\inter M \cup \ext M)$ ist abgeschlossen, $\cl M = X\setminus\ext M$ abgeschlossen - \item $M$ offen: es ist stets $\int M$ und da $M$ offen $M \subset \inter M \beha$ $\Rightarrow X\setminus M = \inter(X\setminus M) = \ext M = X \setminus \cl M \beha$. - ($M$ abgeschlossen analog) - \end{enumerate}\QEDA -\end{proof} \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex b/1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex deleted file mode 100644 index bd34413..0000000 --- a/1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex +++ /dev/null @@ -1,23 +0,0 @@ -\chapter{Konvergenz} -Sei $(X,d)$ metrischer Raum. - -\textbf{Ab jetzt alles ohne Bweise, folgen später.} - -\begin{mydef}[konvergente Folge, Grenzwert] - Folge $\{a_n\}_{n\in\natur}$ (d.h. $a_n \in X$) heißt konvergent falls $a\in X$ existiert mit $\forall \epsilon > 0\exists n_0 \in \natur\colon d(a_n,a) <\epsilon \quad \forall n \geq n_0$. Dann heißt $a$ Grenzwert (Limes).\\ Schreibe $a = \lim_{n\to \infty} a_n$ bzw. $a_n \longrightarrow a$ für $n \longrightarrow \infty$ oder $a_n \overset{n \to \infty}{\longrightarrow} a$. -\end{mydef} - -Sprich: ``'Für jede Kugel um Grenzwert befinden sich ab einem gewissen Index fasst alle FOlgenglieder innerhalb der Kugel.'' Folge $\{a_n\}$ heißt divergent, falls sie nicht konvergent ist. - -\begin{folg} - Für Folge $\{a_n\}$ gilt: $\forall > 0\quad a = \lim_{n\to \infty} a_n \Leftrightarrow$ jede Kugel $B_{\epsilon}(a)$ enthält fast alle Folgeglieder $a_n$, das heißt alle $a_n$ bis auf endlich viele. -\end{folg} - -\begin{exmp}[Konstante Folge] - Sei $\{a_n\} = \{a\}_{n\in \natur}$ (das heißt $a_n = a \forall n$) $\Rightarrow d(a_n,a) = d(a,a) = 0 < \epsilon \forall \epsilon > 0, n \in \natur \Rightarrow a = \lim_{n\to \infty} a_n$. -\end{exmp} - -\begin{exmp} - - $\forall \epsilon > 0 \exists n_0 \in \natur\colon \frac{1}{n} = \vert \frac{1}{n} - 0 \vert = d(\frac{1}{n},0)<\epsilon \forall n \geq n_0 \Rightarrow \lim_{n\to \infty} \frac{1}{n} = 0$. -\end{exmp} \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter10_vollst.tex b/1. Semester/ANAG/TeX_files/chapter10_vollst.tex deleted file mode 100644 index 833d882..0000000 --- a/1. Semester/ANAG/TeX_files/chapter10_vollst.tex +++ /dev/null @@ -1,2 +0,0 @@ -\chapter{Vollständigkeit} -%TODO \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter11_kompaktheit.tex b/1. Semester/ANAG/TeX_files/chapter11_kompaktheit.tex deleted file mode 100644 index 62cfcd3..0000000 --- a/1. Semester/ANAG/TeX_files/chapter11_kompaktheit.tex +++ /dev/null @@ -1,2 +0,0 @@ -\chapter{Kompaktheit} -%TODO \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter12_reihen.tex b/1. Semester/ANAG/TeX_files/chapter12_reihen.tex deleted file mode 100644 index c9e6eb1..0000000 --- a/1. Semester/ANAG/TeX_files/chapter12_reihen.tex +++ /dev/null @@ -1,2 +0,0 @@ -\chapter{Reihen} -%TODO \ No newline at end of file diff --git a/1. Semester/ANAG/TeX_files/chapter13_funktionen.tex b/1. Semester/ANAG/TeX_files/chapter13_funktionen.tex deleted file mode 100644 index 22340be..0000000 --- a/1. Semester/ANAG/TeX_files/chapter13_funktionen.tex +++ /dev/null @@ -1,3 +0,0 @@ -\part{Funktionen und Stetigkeit} -\chapter{Funktionen} -%TODO \ No newline at end of file diff --git a/1. Semester/ANAG/Vorlesung_ANAG.pdf b/1. 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Math Operators -\DeclareMathOperator{\inter}{int} % Set of inner points -\DeclareMathOperator{\ext}{ext} % Set of outer points -\DeclareMathOperator{\cl}{cl} % Closure - -% Hack page break on part page. - -\titleclass{\part}{top} -\titleformat{\part}[display] -{\normalfont\huge\bfseries}{\centering\partname\ \thepart}{20pt}{\Huge\centering} -\titlespacing*{\part}{0pt}{50pt}{40pt} -\titleclass{\chapter}{straight} -\titleformat{\chapter}[display] -{\normalfont\huge\bfseries}{\chaptertitlename\ \thechapter}{20pt}{\LARGE} -\titlespacing*{\chapter} {0pt}{50pt}{40pt} - -\setlength\parindent{0pt} % noindent whole file! - -\begin{document} - -\title{\textbf{Analysis 1. Semester (WS2017/18)}} -\author{Dozent: Prof. Dr. Friedemann Schuricht\\ - Kursassistenz: Moritz Schönherr} -\date{Stand: \today} - -\frontmatter -\maketitle -\tableofcontents - -\mainmatter -% PArt 1 Grundlagen der Mathematik -\include{./TeX_files/chapter01_grundbegriffe_aus_mengenlehre_und_logik} -\include{./TeX_files/chapter02_aufbau_einer_math_theorie} -% Part 2 Zahlenbereiche -\include{./TeX_files/chapter03_nat_zahlen} -\include{./TeX_files/chapter04_ganze_u_rat_zahlen} -\include{./TeX_files/chapter05_reelle_zahlen} -\include{./TeX_files/chapter06_komplexe_zahlen} -% Part 3 Metrische Räume und Konvergenz -\include{./TeX_files/chapter07_grundl_ungleichungen} -\include{./TeX_files/chapter08_metr_raeume} -\include{./TeX_files/chapter09_konvergenz} -\include{./TeX_files/chapter10_vollst} -\include{./TeX_files/chapter11_kompaktheit} -\include{./TeX_files/chapter12_reihen} -% Part 4 Funktionen und Stetigkeit -\include{./TeX_files/chapter13_funktionen} - -\backmatter -% bibliography, glossary and index would go here. - -\end{document} \ No newline at end of file