From 25eab64aa1a92d6c78393860127cd1244eb33bc0 Mon Sep 17 00:00:00 2001 From: Robert Jeutter Date: Mon, 14 Sep 2020 16:08:38 +0200 Subject: [PATCH] created GudS short paper --- .gitignore | 2 + ...n und Diskrete Strukturen - Cheatsheet.tex | 44 +- ...dlagen und Diskrete Strukturen - short.pdf | Bin 0 -> 193904 bytes ...dlagen und Diskrete Strukturen - short.tex | 408 ++++++++++++++++++ README.md | 17 +- 5 files changed, 442 insertions(+), 29 deletions(-) create mode 100644 Grundlagen und Diskrete Strukturen - short.pdf create mode 100644 Grundlagen und Diskrete Strukturen - short.tex diff --git a/.gitignore b/.gitignore index 859f705..3df46a1 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,5 @@ +.vscode + ## Core latex/pdflatex auxiliary files: *.aux *.lof diff --git a/Grundlagen und Diskrete Strukturen - Cheatsheet.tex b/Grundlagen und Diskrete Strukturen - Cheatsheet.tex index 9df6c9f..1782751 100644 --- a/Grundlagen und Diskrete Strukturen - Cheatsheet.tex +++ b/Grundlagen und Diskrete Strukturen - Cheatsheet.tex @@ -85,16 +85,18 @@ Seien p und q Aussagen, dass sind folgende Sätze auch Aussagen - $p \leftrightarrow q$ "genau dann wenn" \paragraph{Wahrheitswerteverlauf} -| p | q | $p\wedge q$ | $p\vee q$ | $\neg q$ | $p\rightarrow q$ | $p\leftrightarrow q$ | -| -- | -- | -- | -- | -- | -- | -- | -| f | f | f | f | w | w | w | -| f | w | f | w | w | w | f | -| w | f | f | w | f | f | f | -| w | w | w | w | f | w | w | +\begin{tabular}{ l | c | c | c | c | c | c } + p & q & $p\wedge q$ & $p\vee q$ & $\neg q$ & $p\rightarrow q$ & $p\leftrightarrow q$ + \hline + f & f & f & f & w & w & w + f & w & f & w & w & w & f + w & f & f & w & f & f & f + w & w & w & w & f & w & w +\end{tabular} \begin{description} \item[Aussagenlogische Variablen] Variable die den Wert w oder f annimmt - \item[Aussagenlogische Formel] Verknüpfung aussagenloser Variablen nach obrigen Muster + \item[Aussagenlogische Formel] Verknüpfung aussagenloser Variablen nach obigen Muster \item[Belegung] Zuordnung von w/f an jede Variable einer aussagenlogischer Formel \item[Wahrheitswerteverlauf] Wahrheitswert der Aussagenformel in Abhängigkeit von der Belegung der Variable \item[Tautologie] Formel deren Wahrheitswerteverlauf konstant w ist @@ -185,20 +187,20 @@ Ist A=B, so heißt R auch binäre Relation auf A Sei $R\in AxA$ binäre Relation auf A \begin{itemize} \item Reflexiv $\leftrightarrow$ xRx $\forall x \in A$ - \item Symetrisch $\leftrightarrow xRy \rightarrow yRx$ - \item Antisymetrisch $\leftrightarrow xRy \wedge yRx \rightarrow x=y$ + \item symmetrisch $\leftrightarrow xRy \rightarrow yRx$ + \item Antisymmetrisch $\leftrightarrow xRy \wedge yRx \rightarrow x=y$ \item Transitiv $\leftrightarrow xRy \wedge yRz \rightarrow xRz$ \item totale Relation $\leftrightarrow xRy \vee yRx \forall x,y \in A$ \end{itemize} \begin{itemize} - \item R heißt Äquivalenzrelation $\leftrightarrow$ R reflexiv, symetrisch und transitiv - \item R heißt Ordnung $\leftrightarrow$ R reflexiv, antisymetrisch und transitiv + \item R heißt Äquivalenzrelation $\leftrightarrow$ R reflexiv, symmetrisch und transitiv + \item R heißt Ordnung $\leftrightarrow$ R reflexiv, antisymmetrisch und transitiv \item R heißt Totalordnung $\leftrightarrow$ R Ordnung und total \item R heißt Quasiordnung $\leftrightarrow$ R reflexiv und transitiv \end{itemize} \paragraph{Äqivalenzrelation} -Sei A Menge, $C\wp (A)$ Menge von teilmengen von A. C heißt Partition von A, falls gilt: +Sei A Menge, $C\wp (A)$ Menge von Teilmengen von A. C heißt Partition von A, falls gilt: 1. $UC=A$ d.h. jedes $x\in A$ liegt in (wenigstens) einem $y\in C$ 2. $\emptyset \not \in C$ d.h. jedes $y\in C$ enthält (wenigstens) ein Element von A 3. $X \cap Y = \emptyset$ f.a. $X\not \in Y$ aus C @@ -212,7 +214,7 @@ Ein Graph $G=(V,E)$ ist ein Paar bestehend aus einer Menge V und $E\subseteq (x, Zu $a,b\in V$ heißt eine Folge $P=x_1,..,x_n$ von paarweise verschiedenen Ebenen mit $a=x_0, b=x_j; x_{j-1},x_i \in E{a*i \in b*j}$ ein a,b-Weg der Länge l oder Weg a nach b. Durch $a\sim b$ gibt es einen a,b-Weg in G, wird eine Äquivalenzrelation auf V definiert, denn: \begin{itemize} \item "$\sim$ reflexiv": es ist $x\sim x$, denn $P=x$ ist ein x,x-Weg in G - \item "$\sim$ symetrisch": aus $x\sim y$ folgt, es gibt einen x,y-Weg $\rightarrow$ es gibt einen y,x-Weg $y\sim x$ + \item "$\sim$ symmetrisch": aus $x\sim y$ folgt, es gibt einen x,y-Weg $\rightarrow$ es gibt einen y,x-Weg $y\sim x$ \item "$\sim$ transitiv": aus $x\sim y$ und $y\sim x$ folgt, es gibt einen x,y-Weg und einen y,x-Weg \end{itemize} Die Äquivalenzklassen von $\sim _G$ erzeugen die Zusammenhangskomponenten von G @@ -229,13 +231,13 @@ Sei also $leq$ eine Ordnung auf X. Seo $A\subseteq X, b\in X$ \item b untere Schranke von A $\leftrightarrow b\leq c f.a. c\in A$ \item b obere Schranke von A $\leftrightarrow c\leq b f.a. c\in A$ \item b kleinste obere Schranke von A $\leftrightarrow$ b ist kleinstes Element von $(b'\in X: \text{b' obere Schranke von A})$ auch Supremum von A: $\lor A = b$ - \item b größte untere Schranke von A $\leftrightarrow$ b ist das größte Element von $(b'\in X: \text{ b' untere Schranke von A} )$ auch Infinium von A; $\land A = b$ + \item b größte untere Schranke von A $\leftrightarrow$ b ist das größte Element von $(b'\in X: \text{ b' untere Schranke von A} )$ auch Infinum von A; $\land A = b$ \end{itemize} kleinstes und größtes Element sind jew. eindeutig bestimmt (falls existent) Satz: Sei X Menge. $\subseteq$ ist Ordnung auf $\wp(X)$. Ist $O\subseteq \wp(X)$, so ist $sup O=\bigcup O$ und $inf O=\bigcap O$ -Satz: Die Teilbarkeitsrelation | ist Ordnung auf den natürlichen Zahlen $\mathbb{N}$. Es gibt $sup(a,b)=kgV(a,b)$ (kleinstes gemeinsames Vielfaches) und $inf(a,b)=ggT(a,b)$ (größtes gemeinsames Vielfaches) +Satz: Die Teilbarkeitrelation | ist Ordnung auf den natürlichen Zahlen $\mathbb{N}$. Es gibt $sup(a,b)=kgV(a,b)$ (kleinstes gemeinsames Vielfaches) und $inf(a,b)=ggT(a,b)$ (größtes gemeinsames Vielfaches) \paragraph{Hesse Diagramm} Darstellung einer Ordnung $\subseteq$ auf X @@ -356,7 +358,7 @@ Satz: $m+n=n+m$ f.a. $m,n\in\mathbb{N}$ (Beweis induktiv über m) Satz: $l+(m+n)=(l+m)+n$ f.a. $l,m,n\in\mathbb{N}$ (Klammern sind neutral bzgl +) -Satz (Streichungsregel): aus $a+n=b+n$ folgt $a=b$ f.a. $a,b,n\in\mathbb{N}$ +Satz (Streichungregel): aus $a+n=b+n$ folgt $a=b$ f.a. $a,b,n\in\mathbb{N}$ \paragraph{Analog: Multiplikation} $*: \mathbb{N} x \mathbb{N} \rightarrow \mathbb{N}$ wird definiert durch: @@ -585,7 +587,7 @@ Satz: Sind $(\Omega, p_1),...,(\Omega, p_m)$ Wahrscheinlichkeitsräume so ist du $(\Omega, p)$ heißt Produktraum von $(\Omega_1, p_1),...$. $(\Omega, p)$ Wahrscheinlichkeitsraum; $A,B\in \Omega$ heißen (stochastisch) unabhängig, falls $p(A\cap B) = p(A)*p(B)$. -Bespiel: $p(A\cap B) = p({i,j}) =p_1{i}*p_2{j} = p(A)*p(B)$ für das Ereignis "der 1. Würfel zeigt i, der 2. Würfel zeigt j" +Beispiel: $p(A\cap B) = p({i,j}) =p_1{i}*p_2{j} = p(A)*p(B)$ für das Ereignis "der 1. Würfel zeigt i, der 2. Würfel zeigt j" \paragraph{Bedingte Wahrscheinlichkeiten} $(\Omega, p)$ Wahrscheinlichkeitsraum, $B\subseteq \Omega$ ("bedingtes Ereignis") mit $p(B)>0$, dann ist $p_B:B\rightarrow [0,1]; p_B(\omega)=\frac{p(\omega)}{p(B)}$ eine Verteilung auf B, denn $\sum p_b(\omega)=\sum \frac{p(\omega)}{p(B)}=\frac{1}{p(B)} \sum p(\omega)= \frac{1}{p(B)} p(B)= 1$. @@ -597,7 +599,7 @@ Satz (Totale Wahrscheinlichkeit): Seien $A_1, ...,A_k$ paarweise disjunkt, $\big Satz (Bayer, erweitert): $A_1,...,A_k,B$ wie eben, $p(B)>0$. Für $i\in {1,...,k}$ gilt $p(A_i|B)=\frac{p(B|A_i)*p(A_i)}{\sum p(B|A_j)*p(A_j)}$ -Bespiel: In einem Hut liegen drei beidseitig gefärbte Karten. Jemand zieht ("zufällig") eine Karte und leg sie mit einer ("zufälligen") Seite auf den Tisch. Karten rot/rot, rot/blau und blau/blau. Gegeben er sieht rot, wie groß ist die Wahrscheinlichkeit, dass die andere Seite auch rot ist? +Beispiel: In einem Hut liegen drei beidseitig gefärbte Karten. Jemand zieht ("zufällig") eine Karte und leg sie mit einer ("zufälligen") Seite auf den Tisch. Karten rot/rot, rot/blau und blau/blau. Gegeben er sieht rot, wie groß ist die Wahrscheinlichkeit, dass die andere Seite auch rot ist? p(unten rot | oben rot) = p(unten rot und oben rot)/p(oben rot) = $\frac{p\binom{r}{r}}{p(\binom{r}{r}\binom{r}{b})}=\frac{\frac{2}{6}}{\frac{3}{6}}=\frac{2}{3}$ Eine Funktion $X:\Omega \rightarrow \mathbb{R}$ heißt (reellwertige) Zufallsvariable. Weil $\Omega$ endlich ist, ist auch $X(\Omega)={X(\omega): \omega \in \Omega}\subseteq \mathbb{R}$ endlich. Durch $p_x(x):=p(X=x):=p({\omega \in \Omega: X(\omega)=x})$ wird ein Wahrscheinlichkeitsraum $(X(\Omega),p_x)$ definiert; denn $\sum p_x(x)=p(\Omega)=1$. $p_x$ heißt die von X induzierte Verteilung. $X(\Omega)$ ist meist erheblich kleiner als $\Omega$. @@ -638,7 +640,7 @@ Für $n\in \mathbb{N}$ sei $n!=n(n-1)(n-2)...*3*2*1=\prod i$; für $n\in\mathbb{ Satz: $\binom{n}{0}=\binom{n}{n}=1$ für jedes $n\in\mathbb{N}$, $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$ für $k\geq 1$ und $k\leq n-1$. Zweiter Teil: $[\binom{n-1}{k}]+[\binom{n-1}{k-1}]=\frac{n!}{k!(n-k)!} = [\binom{n}{k}]$. Also stimmen die Rekursionsgleichungen von $\binom{n}{k}$ und $[\binom{n}{k}]$ überein sowie $\binom{n}{k}=[\binom{n}{k}]$. Folglich ist die Anzahl k-elementiger Teilmengen eine n-elementige Menge gleich $\frac{n!}{k!(n-k)!}$. -Seien $X_1,...,X_n$ unabhängige ZVAen, alle $X_i$ seien Bernoulli-Verteilt im Parameter $p[0,1]$, d.h. $p(X_1=1)=p$, $p(X_i=0)=(1-p)$. Dann ist $X_i=X_1+X_2+...+X_n$ ebenfalls reelwertige ZVA. Im Fall $X_i:\Omega\rightarrow {0,1}$ ist $X:\Omega\rightarrow {0,1,...,n}$. Die Verteilung von X ergibt sich wie folgt, für $k\in {0,1,...,n}$: $p(X=k)=\binom{n}{k}*p^k(1-p)^{n-k}$ +Seien $X_1,...,X_n$ unabhängige ZVAen, alle $X_i$ seien Bernoulli-Verteilt im Parameter $p[0,1]$, d.h. $p(X_1=1)=p$, $p(X_i=0)=(1-p)$. Dann ist $X_i=X_1+X_2+...+X_n$ ebenfalls reellwertige ZVA. Im Fall $X_i:\Omega\rightarrow {0,1}$ ist $X:\Omega\rightarrow {0,1,...,n}$. Die Verteilung von X ergibt sich wie folgt, für $k\in {0,1,...,n}$: $p(X=k)=\binom{n}{k}*p^k(1-p)^{n-k}$ Eine ZVA heißt binominalverteilt in den Parametern n und p falls gilt: $p(X=k)=\binom{n}{k}p^k (1-p)^{n-k}$ für $k\in{0,1,...,n}$; schreibe $X\sim L(n,p)$. Sonst ist X Bernoulliverteilt (genau dann wenn $X\sim L(1,p)$). @@ -660,7 +662,7 @@ $E(X)=\sum_{x=0}^M \frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}=...=n*\frac $Var(X)=E(X^2)-E(X)^2 =...= n*\frac{M}{N}(1-\frac{M}{N})(\binom{N-n}{N-1})$ \section{Elementare Graphentheorie} -$G=(V,E)$ heißt Graph mit Eckenmenge $V(G)=V$ und Kantenmenge $E(G)=E\subseteq {{x,y}:x\not=y \in V}$. Veranschaulichung als Punkte in der Ebene (V) mit "Verknüpfungslinien" von x nach y. Bsp $G=({1,2,3,4},{12,13,14,15,16})$. +$G=(V,E)$ heißt Graph mit Eckenmenge $V(G)=V$ und Kantenmenge $E(G)=E\subseteq {{x,y}:x\not=y \in V}$. Veranschaulichung als Punkte in der Ebene (V) mit "Verknüpfunglinien" von x nach y. Bsp $G=({1,2,3,4},{12,13,14,15,16})$. $P=x_0,...,x_e$ Folge pw verschiedener Ecken mit $x_{i-1},...,x_i \in E(k)$ für $i\in{1,...,l}$ heißt ein Weg von $x_0$ nach $x_e$ der Länge l. Für $(a,b)\in V(G)$ heißt $d_G(a,b)=min(l: \text{ es gibt einen a,b-Weg der Länge l} )$ Abstand von a nach b. Falls es keinen a,b-Weg gibt, definiere $d_G(a,b)=+\infty$. @@ -751,7 +753,7 @@ Eliminiere Mehrfachnennungen in der Folge. Gibt es $i\not= j$ mit $x_j=x_i$ so s G Graph, $k\geq 0$. Eine Funktion $f:V(G)\rightarrow C$ mit $|C|\leq k$ heißt k-Färbung, falls $f(x)\not = f(y)$ für $xy\in E(G)$. G heißt k-färbbar, falls G eine k-Färbung besitzt. Das kleinste $k\geq 0$ für das G k-färbbar ist heißt dramatische Zahl von G, Bezeichnung $X(G)$. -Satz (Tuga): Sei $k\geq 2$ und G ein Graph ohne Kreise eine Lösung $l\equiv 1 mod k$, dann ist G k-faltbar. G 2-färbbar $\leftrightarrow$ G hat keine Kreise ungerader Länge. Ein Graph heißt bipartit mit den Klassen A,B falls $(x\in A \wedge y\in B)\vee (x\in B \wedge y\in A)$ für alle $xy \in E(G)$ gilt. Genau dann ist G bipoartit mit gewissen Klassen A,B wenn G 2-färbbar ist. +Satz (Tuga): Sei $k\geq 2$ und G ein Graph ohne Kreise eine Lösung $l\equiv 1 mod k$, dann ist G k-faltbar. G 2-färbbar $\leftrightarrow$ G hat keine Kreise ungerader Länge. Ein Graph heißt bipartit mit den Klassen A,B falls $(x\in A \wedge y\in B)\vee (x\in B \wedge y\in A)$ für alle $xy \in E(G)$ gilt. Genau dann ist G bipartit mit gewissen Klassen A,B wenn G 2-färbbar ist. Satz (Hall): Sei G bipartit mit Klassen A,B. 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z9Gn648X)cWs*1CNnXBnL5H!7`nfW^jz`@GJ&dUDYBJS$)J`zB$=3oMPpFkeu>hca4 z{obJNYGMsCb@_ML|KDlo8UD5A_xFDee=q&#Qs@{uF?Cks;@0ZBAc{zBP*nxEH>~!z_K0BEG|GH>6 z8#_9J%-(UYjcvgoBsLWdX>C>+S6f>XV|)95#Gqkm1qQr7{}Kl?KojH)erFBfWc`O9 zKahu;1<1<7@*iRXa{PxF0g6^`AizJ1Vv_=ySQ*uN0~(cv6f@zaFDae44tgQp;6v;~=eg(1S$W z?SLzn34o{1Op&YzSqsM=7Q#)(RsbAgLs(BUwjo%Y+;qtU>R^)yp_>2#&-BcwfoTl< zZZi!We#h$(x*u`Hgpgn%)L6vj+;5>msBdLBms~=yu+%nN5a}h}{s>&LldrUw^+9O^ z)0aEO)b_!#&FHo2guAF1{cC)zO7fSSz3=gSFfSwR155J_0j1-pyke8ENnuqiK2a`Fs5bU%p09mgh+iowtCb)Ttr?PeIFTCE%V?xN0XX-l#5Hh&H3;Ms zxOf0m5pe++0JMlCtxGqnnK+Ma53u4LfLRi4M*`y#K#r7)ahr!1`9)&w^%Xf$3_+d( z1@sqp$QYe57pp9n$oeay=2ztOjSTtqO5bh48F67QF)A)e7+nC8XfEzjE~!ASI*pt0 z`Ee-OL5Rpf7!*Np4tpWin}sq;dFHOp_Y!@XN|`U&%N2bZLFZciU#R>mXf?Kt%jIUL zZOR#DAz8EaM=w8Sqg%VN%XM6j*)+jNF|T^PGcUefY*lAM#y1M9v$eBAB-9s!PQq|} z`-CYLYT0t0w7WVn7W~0$ lässt sich bis auf die Reihenfolge der Faktoren eindeutig als Produkt von Primzahlen darstellen. + +\paragraph{Konstruktion von rationalen Zahlen aus $\mathbb{Z}$} +Sei $M=\mathbb{Z} x(\mathbb{Z} /{0})$ die Menge von Brüchen. Durch $(a,b)\sim (c,d)\leftrightarrow ad=bc$ wird Äquivalenzrelation auf M durchgeführt. Definiere Operationen +,* auf $\mathbb{Q}$ wie folgt: +\begin{itemize} + \item $\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{b*d}$ (wohldefiniert) + \item $\frac{a}{b}*\frac{c}{d} = \frac{a*c}{b*d}$ +\end{itemize} +Satz: $\mathbb{Q}$ mit +,* ist ein Körper. + +\paragraph{Ring der formalen Potenzreihe} +Sei k ein Körper. Eine Folge $(a_0, a_1,...,a:n)\in K^{\mathbb{N}}$ mit Einträgen aus K heißt formale Potenzreihe $K[[x]]$. Die Menge aller Polynome wird mit $K[x]$ bezeichnet. $K[[x]]$ wird mit +,* zu einem kommutativen Ring mit $1\not=0$ +\begin{itemize} + \item +: $(a_0,a_1,...) + (b_0,b_1,...) = (a_o+b_0, a_1+b_1, ...)$ + \item *: $(a_0,a_1,...) + (b_0,b_1,...) = (c_0, c_1,...)$ mit $c_K=\sum_{j=a}^{k} a_j*b_{k-j}$ +\end{itemize} + + +B mit $\vee, \wedge, \bar{ }$ seien boolesche Algebren. Sie heißen isomorph, falls es einen Isomorphismus von B nach $\dot{B}$ gibt, d.h. eine Bijektion $\phi: B \rightarrow \dot{B}$ mit: +\begin{itemize} + \item $\phi(a\vee b) =\phi(a)\dot{\vee}\phi(b)$ + \item $\phi(a\wedge b)=\phi(a)\dot{\wedge}\phi(b)$ + \item $\phi(\bar{a}) = \dot{\bar{\phi(a)}}$ +\end{itemize} + +Lemma: Sei B mit $\vee, \wedge, \bar{}$ eine boolesche Algebra, dann gilt: +\begin{itemize} + \item $a\vee T = T$ f.a. $a\in B$ + \item $a\wedge \perp = \perp$ f.a. $a\in B$ + \item $a\vee b$ ist obere Schranke von ${a,b}$, d.h. $a\leq a\vee b$, dann $a\vee(a\vee b)=a\vee b$ + \item $a\vee b$ ist kleinste obere Schranke, d.h. $a\leq z$ und $b\leq z$ folgt $a\vee b \leq z$ +\end{itemize} + + +\section{Diskrete Wahrscheinlichkeitsräume} +Ein Wahrscheinlichkeitsraum ist ein Paar $(\Omega, p)$ bestehend aus einer endlichen Menge $\Omega$ und einer Funktion $p:\Omega \rightarrow [0,1]\in \mathbb{R}$ mit $\sum_{\omega \in \Omega} p(\omega)=1$. Jeder derartige p heißt Verteilung auf $\Omega$. Die Elemente aus $\Omega$ heißen Elementarereignis, eine Teilmenge A von $\Omega$ heißt ein Ereignis; seine Wahrscheinlichkeit ist definiert durch $p(A):=\sum_{\omega in A} p(\omega)$.\ +$A=\emptyset$ und jede andere Menge $A\subseteq \Omega$ mit $p(A)=0$ heißt unmöglich (unmögliches Ereignis).\ +$A=\Omega$ und jede andere Menge $A\subseteq \Omega$ mit $p(A)=1$ heißt sicher (sicheres Ereignis).\ +Es gilt für Ereignisse $A,B,A_1,...,A_k$: +\begin{itemize} + \item $A\subseteq B \rightarrow p(A)\leq p(B)$ + \item $p(A\cup B) \rightarrow p(A)+p(B)-p(A\cap B)$ + \item disjunkt($A_i \cap A_J=\emptyset$ für $i\not =j$) so gilt $p(A_1 \cup ... \cup A_k)= p(A_1)+...+p(A_k)$ + \item $p(\Omega / A):=$ Gegenereignis von $A=1-p(A)$ + \item $p(A_1,...,A_k) \leq p(A_1)+...+p(A_k)$ +\end{itemize} +$(\Omega, p)$ heißt Produktraum von $(\Omega_1, p_1),...$.\ +$A,B\in \Omega$ heißen (stochastisch) unabhängig, falls $p(A\cap B) = p(A)*p(B)$. + +\paragraph{Bedingte Wahrscheinlichkeiten} +$B\subseteq \Omega$ ("bedingtes Ereignis") mit $p(B)>0$, dann ist $p_B:B\rightarrow [0,1]; p_B(\omega)=\frac{p(\omega)}{p(B)}$ eine Verteilung auf B. Für $A\subseteq \Omega$ gilt $p_B(A\cap B)=\sum p_B(\omega)=\sum\frac{p(\omega)}{p(B)}= \frac{p(A\cap B)}{p(B)}:= p(A|B)$ bedingte Wahrscheinlichkeit von A unter B. $p(A|B)=\frac{p(B|A)*p(A)}{p(B)}$\ + + +\paragraph{Erwartung, Varianz, Covarianz} +Erwartungswert $E(X) = \sum_{\omega \in \Omega} X(\omega)p(\omega)$\ +Linearität von E: $E(x+y)=E(x)+E(y)$ und $E(\alpha x)=\alpha E(x)$.\ +Varianz von X: $Var(X)=E((X^2)-E(X))^2)$\ +Covarianz: $Cov(X,Y)=E((X-E(X))*(Y-E(Y)))$\ +Verschiebungssatz: $Cov(X,Y)=E(X*Y)-E(X)*E(Y)$\ +$Var(X)=Cov(X,X)=E(X*X)-E(X)E(X)=E(X^2)-(E(X))^2$\ +Sind X,Y stochastisch unabhängig ZVA, so ist $E(X)*E(Y)=E(X*Y)$; folglich $Cov(X,Y)=0$\ +Bernoulliverteilt falls $p(X=1)=p$ und $p(X=0)=1-p$, $p\in [0,1]$. $E(X)=\sum x*p(X=x)= 1*p(X=1)=p$ + +\paragraph{Binominalkoeffizienten} +N sei Menge, dann ist $\binom{N}{k} := (x \subseteq N: \text{x hat genau k Elemente } (|x|=k) )$ für $k\in \mathbb{N}$. +$\binom{N}{0}=(\emptyset)$, $\binom{N}{n}={N}\rightarrow \binom{n}{0}=\binom{n}{n}=1$ $\binom{n}{0}=1$, $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}=\frac{n!}{k!(n-k)!}$ + +\paragraph{Hypergeometrische Verteilung} +Beispiel: Urne mit zwei Sorten Kugeln; N Gesamtzahl der Kugeln, M Gesamtzahl Kugeln Sorte 1, N-M Gesamtzahl Kugeln Sorte 2, $n\leq N$ Anzahl Elemente einer Stichprobe. X Anzahl der Kugeln Sorte 1 in einer zufälligen n-elementigen Stichprobe. +$p(X=k)=\frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}$\ +$E(X)=\sum_{x=0}^M \frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}=...=n*\frac{M}{N}$\ +$Var(X)=E(X^2)-E(X)^2 = n*\frac{M}{N}(1-\frac{M}{N})\binom{N-n}{N-1}$ + +\section{Elementare Graphentheorie} +$G=(V,E)$ heißt Graph mit Eckenmenge $V(G)=V$ und Kantenmenge $E(G)=E\subseteq {{x,y}:x\not=y \in V}$.\ +Für $(a,b)\in V(G)$ heißt $d_G(a,b)=min(l: \text{ es gibt einen a,b-Weg der Länge l} )$ Abstand von a nach b.\ +G heißt zusammenhängend, wenn G höchstens eine Komponente besitzt. +\begin{itemize} + \item $d_G(x,y)=0 \leftrightarrow x=y$ f.a. $x,y \in V(G)$ + \item $d_G(x,y)=d_G(y,x)$ f.a. $x,y\in V(F)$ + \item $d_G(x,z)\leq d_G(x,y) + d_G(y,z))$ f.a. $x,y,z \in V(G)$ +\end{itemize} + +$\leq$ ist Ordnung, denn: +\begin{itemize} + \item $G\leq G$ + \item $H\leq G \wedge G\leq H \rightarrow H=G$ + \item $H\leq G \wedge G=L \rightarrow H\leq L$ +\end{itemize} + +Ein Teilgraph H des Graphen G heißt aufspannend, falls $V(H)=V(G)$. Weiter $N_G(x):={x\in V(G): xy \in E(G)}$ die Menge der nachbarn von x in G. Hier gilt: $|N_G(x)=d_G(x)|$. In jedem Graph G gilt $\sum_{x\in V(G)} d_G(x)=2|E(G)|$. Der Durchschnittsgrad von G ist somit $\bar{d(G)}=\frac{1}{|V(G)|}\sum d_G(x)=\frac{2|E(G)|}{|V(G)|}$. + +Ein Graph ist ein Baum wenn "G ist minimal zusammenhängend und kreisfrei" +\begin{itemize} + \item G ist kreisfrei und zusammenhängend + \item G kreisfrei und $|E(G)|=|V(G)|-1$ + \item G zusammenhängend und $|E(G)|=|V(G)|-1$ +\end{itemize} +Breitensuchbaum von G falls $d_F(z,x)=d_G(z,x)$ f.a. $z\in V(G)$.\ +Tiefensuchbaum von G falls für jede Kante zy gilt: z liegt auf dem y,x-Weg in T oder y liegt auf dem z,t-Weg in T. + +Satz: Sei G zusammenhängender Graph $x\in V(G)$. +(X) sind $x_0,...,x_{e-1}$ schon gewählt und gibt es ein $+ \in (0,..., e-1)$ so, dass $x_{+}$ einen Nachbarn y in $V(G)\ (x_0,...,x_{e-1} )$, so setze $x_e=y$ und $f(e):=t$; iteriere mit $e+1$ statt e. +Dann ist $T:=({x_0,...,x_e},{x_j*x_{f(j)}: j\in {1,...,e}})$ ein Spannbaum +\begin{itemize} + \item f(e) wird in + stets kleinstmöglich gewählt, so ist T ein Breitensuchbaum + \item f(e) wird in + stets größtmöglich gewählt, so ist T ein Tiefensuchbaum +\end{itemize} + +\paragraph{Spannbäume minimaler Gewichte} +Sei G zuständiger Graph, $\omega:E(G)\rightarrow \mathbb{R}$; Setze $F=\emptyset$. Solange es eine Kante $e\in E(G)/F$ gibt so, dass $F \vee (e)$ kreisfrei ist, wähle e mit minimalem Gewicht $\omega(e)$, setzte $F=F\vee {e}$, iterieren. Das Verfahren endet mit einem Spannbaum $T=G(F)$ minimalen Gewichts. + +\paragraph{Das Traveling Salesman Problem} +Konstruiere eine Folge$x_0,...,x_m$ mit der Eigenschaft, dass jede Kante von T genau zweimal zum Übergang benutzt wird, d.h. zu $e\in E(T)$ existieren $i\not = j$ mit $e=x_i x_{i+1}$ und $e=x_j x_{j+1}$ und zu jedem k existieren $e\in E(T)$ mit $e=x_k x_{k+1}$. Das Gewicht dieser Folge sei $\sum \omega(x_i x_{i+1})= 2\omega(T)$. +Eliminiere Mehrfachnennungen in der Folge. Durch iteration erhält man einen aufspannenden Kreis mit $\omega(X) \leq 2 \omega(T)$. + +\paragraph{Färbung \& bipartit} +Eine Funktion $f:V(G)\rightarrow C$ mit $|C|\leq k$ heißt k-Färbung, falls $f(x)\not = f(y)$ für $xy\in E(G)$. +Ein Graph heißt bipartit mit den Klassen A,B falls $(x\in A \wedge y\in B)\vee (x\in B \wedge y\in A)$. Mit Bipartitheit gilt G hat ein Matching von A $\leftrightarrow |N_G(X)|\leq |X|$ für alle $X\subseteq A$. + +\end{multicols} +\end{document} \ No newline at end of file diff --git a/README.md b/README.md index e541b5e..bf72609 100644 --- a/README.md +++ b/README.md @@ -5,8 +5,9 @@ Unterlagen zu Informatik Vorlesungen der TU Ilmenau bisher: - [Algorithmen und Datenstrukturen](Algorithmen%20und%20Datenstrukturen.md) - [Einführung in die Medizinische Informatik](Einführung%20in%20die%20Medizinische%20Informatik.md) -- [Grundlagen und Diskrete Strukturen](Grundlagen%20und%20Diskrete%20Strukturen.md) - - [Grundlagen und Diskrete Strukturen - Cheatsheet](Grundlagen%20und%20Diskrete%20Strukturen%20-%20Cheatsheet.pdf) +- [Grundlagen und diskrete Strukturen](Grundlagen%20und%20Diskrete%20Strukturen.md) + - [GudS - Cheatsheet](Grundlagen%20und%20Diskrete%20Strukturen%20-%20Cheatsheet.pdf) + - [GudS - Short Paper](Grundlagen%20und%20Diskrete%20Strukturen%20-%20short.pdf) - [Neurobiologische Informationsverarbeitung](Neurobiologische%20Informationsverarbeitung.md) - [Neuroinformatik](Neuroinformatik.md) - [Programmierparadigmen](Programmierparadigmen.md) @@ -22,12 +23,12 @@ Dieses Repo ist zum Selbststudium und erlernen neuen Wissens gedacht. Kein Inhal Wir freuen uns über jeden der mitmacht. ## Mitmachen -- Erstelle ein 'issue'. Gehe zuerst die offene 'issues' durch. (Oder Überlege dir welches Fach noch fehlt) -- Klone das repository oder erstelle einen Fork. -- Erstelle deinen eigenen feature branch: git checkout -b my-new-feature -- Committe deine Änderungen: git commit -m 'Add some feature' -- Push auf deinen Branch: git push origin your-new-feature -- Erstelle ein pull request damit deine Änderungen in diesem Repository übernommen werden. +- Erstelle ein 'issue'. Gehe davor die offene 'issues' durch. (Oder überlege dir welches Fach noch fehlt) +- Klone das repository oder erstelle einen Fork: ```git clone https://github.com/wieerwill/Informatik.git``` +- Erstelle deinen eigenen feature branch: ```git checkout -b my-new-feature``` +- Stelle deine Änderungen bereit: ```git commit -m "Add some feature"``` (in "" kommt dein Kommentar) +- Lade auf deinen Branch hoch: ```git push origin your-new-feature``` +- Erstelle ein 'pull request' damit deine Änderungen in diesem Repository übernommen werden. ## Bild- und Textrechte Der Inhalt aller Dokumente hier ist die Mitschrift aus besuchten Vorlesungen. Es werden keine Bücher kopiert oder anderweitig Copyright verletzt. Die Verletzung des Copyright oder anderer Rechte Dritte wird mit einem Ausschluss aus dem Repository gehandelt. Sollte Ihnen ein Verstoß auffallen geben Sie uns bitte umgehend bescheid, wir werden jedem Fall nachgehen.