From 09431c1925d0c99b26522f2fc0f4858bcd76ed1b Mon Sep 17 00:00:00 2001 From: angerstoner Date: Sun, 14 Jan 2018 00:50:01 +0100 Subject: [PATCH 1/2] mafia chapter 1 finished #1 --- MafIA1/mafia.idx | 3 +++ MafIA1/mafia.pdf | Bin 197862 -> 208715 bytes MafIA1/mafia.tex | 41 +++++++++++++++++++++++++++++++++++------ env/commands.tex | 1 + 4 files changed, 39 insertions(+), 6 deletions(-) diff --git a/MafIA1/mafia.idx b/MafIA1/mafia.idx index e69de29..7f839cd 100644 --- a/MafIA1/mafia.idx +++ b/MafIA1/mafia.idx @@ -0,0 +1,3 @@ +\indexentry{Urbildmenge|hyperpage}{5} +\indexentry{Bildmenge|hyperpage}{5} +\indexentry{Definitionsbereich|hyperpage}{5} diff --git a/MafIA1/mafia.pdf b/MafIA1/mafia.pdf index 51bd1e141c9854dc749b1aef4a6f57762ee1ed41..027a7b0f560ee173d33edf8ae036370c2b1b3c86 100644 GIT binary patch delta 175371 zcmZs?Q*@xgm%Sa^w%xI9+qRRA?Kie<+qP}n>7--3lXUv~&CIvf|6*pXx~S_qwRSz{ z>|Li(2s8T#G?dEXk_;@29B`D2t7DsRES$_F%p{Jcws8FXa7^+R4pwf~BrI&~tSO-M 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zA{4*rO$IC?%&)20OJ7j4sky18g0{iP+Uhl^QCKE4X<96#xBe|-cF{S;Ph9F_nP-!` zUs)hZ+Y{Xp-Ra-s`thr%RrZ0%SHrS6yo~x3GV0p!vdBvXqzF_V0?2^jM023tR_&&E zuEUoS=eH#K0*H$JQQA@3r>+IqqZt@2QR*;!;1E+)K(=Zb{mO9=QyH`{--xBM<%Kc= zIZIxdH9^V*1CVR4et|haxr{?ZbfLR6V@<~us=CIW>{CN>S4j(&{O$;>0H8>lHESV}!>Y?8NGl!Z_nB9q>1Gx;Y zll`qRKt5P6LI9-)dB^?vyCHkV1&pbkiL;B7siEzEKG_>#S;08F7&^IlIGLKkFmbVS OvT?$Yk%=jY!~6$SKRQhS diff --git a/MafIA1/mafia.tex b/MafIA1/mafia.tex index f8aa7c1..c75795a 100644 --- a/MafIA1/mafia.tex +++ b/MafIA1/mafia.tex @@ -84,7 +84,7 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! Eine Relation ist zunächst eine Teilmenge beliebiger Mengen $M, N$. Man schreibt für ein beliebiges Paar aus $(x,y) \in M \times N$ entweder $xRy$ oder seltener $(x,y) \in R$. - + \subsection{Eigenschaften}\label{relation:eigenschaften} Zur Beschreibung einer Relation gibt es folgende Eigenschaften, dazu betrachten wir eine zweistellige Relation $R$ auf $M$: @@ -100,15 +100,30 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \ex antisymmetrisch: $ 3 \le 3 \land 3 \ge 3 \Rightarrow 3 = 3$\\ \subsection{Äquivalenzrelation}\label{relationen:aequivalenz} - Eine Relation heißt Äquivalenzrelation, wenn die ersten drei Eigenschaften aus~\ref{relation:eigenschaften} (reflexiv, symmetrisch, transitiv) - zutreffen. + Eine Relation heißt Äquivalenzrelation, wenn die ersten drei Eigenschaften aus~\ref{relation:eigenschaften} (reflexiv, symmetrisch, transitiv) zutreffen. \subsection{Ordnungsrelation}\label{relationen:ordnung} - Eine Relation heißt Ordnungsrelation, wenn sie total, reflexiv, antisymmetrisch und transitiv ist.\\ - % TODO: halbe Ordnung/ganze Ordnung + Eine Relation heißt Ordnungsrelation ($ \prec $), wenn sie total, reflexiv, antisymmetrisch und transitiv ist.\\ + \ex Sei $ M = \R, \prec \rarr \le: x \le y $ + \subsubsection{Halbe Ordnung} + $ \prec $ heißt eine Halbordnung auf $ M $, wenn Reflexivität, Transitivität und Antisymmetrie gegeben sind für $ x \prec y $ für $ x,y \in M $ + \subsubsection{Ganze Ordnung} + $ \prec $ heißt eine ganze Ordnung, wenn alle $ x,y \in M $ entweder $ x \prec y $ oder $ y \prec x $ erfüllen. \section{Abbildungen} - \subsection{Eigenschaften}\label{abb:eigenschaften} + \subsection{Definition} + Eine Abbildung $ f $ einer Menge $ U \subset M $ auf eine Menge $ N $ ist eine Vorschrift, die jedem $ x \in M $ (Urbild) genau ein Element $ y \in N $ (Bild von $x$) zuordnet. + Dabei ist $ U $ die \idx{Urbildmenge} und $ N $ die \idx{Bildmenge}. Weiter ist $ U $ die Domain(\idx{Definitionsbereich}) von $ f $ in $ M $ ($ U = dom(f) $)\\\\ + Eine Abbildung $ f $ von einer Menge $ M $ in eine Menge $ Y \subset N $ ist gegeben durch eine Relation:\\ + $ R_f $ zwischen $ M $ und $ N $, bei der jedes $ x \in M $ mit genau einem $ y \in N $ in Relation steht:\\ + $ (x,y) \in R_f $ und $ (x,\tilde{y}) \in R_f \Rarr y = \tilde{y} $. Wir schreiben in diesem Fall $ f(x) = y $ + \subsubsection{Verknüpfte Abbildungen} + \begin{align} + (f \circ g)(x) = f(g(x)) \text{ wobei } + g: M \rarr N \text{ und } f: N \rarr P \Rarr f \circ g: M \rarr P + \end{align} + + \subsection{Eigenschaften}\label{abb:eigenschaften} Wir betrachten eine Abbildung $f: M \rightarrow N$. \begin{tabu}{rl} @@ -161,6 +176,20 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \end{proof} Da $f$ surjektiv und injektiv ist, folgt auch die \emph{Surjektivität} für $f$. + + \subsubsection{Satz: Regeln für Abbildungen} + \begin{enumerate}[a.)] + \item Sind $ f $ und $ g $ injektiv, so ist $ f \circ g $ injektiv + \item Sind $ f $ und $ g $ surjektiv, so ist $ f \circ g $ surjektiv + \item Sind $ f $ und $ g $ bijektiv, so ist $ f \circ g $ bijektiv + \item Ist $ f \circ g $ injektiv, so ist auch $ g $ injektiv + \item Ist $ f \circ g $ surjektiv, so ist auch $ f $ surjektiv + \item Ist $ f \circ g $ bijektiv, so ist $ f $ injektiv und $ g $ surjektiv + \end{enumerate} + Die Beweise zu a.) - f.) werden zur \prac gelassen. Teilweise wurden sie schon in der Vorlesung gezeigt.\\ + \anm: In der Vorlesung wurde noch kurz Russels Paradoxon\footnote{\url{https://de.wikipedia.org/wiki/Barbier-Paradoxon} bzw. \url{https://de.wikipedia.org/wiki/Russellsche_Antinomie}} angesprochen:\\ + \[ R := \{x \in \Omega | x \not \in \{x\}\} \subset \Omega \] + \[ R \in R \Lrarr R \not \in \{R\}\] \section{Zahlen}\label{zahlen} \subsection{Natürliche Zahlen} diff --git a/env/commands.tex b/env/commands.tex index 22cad69..8ad4106 100644 --- a/env/commands.tex +++ b/env/commands.tex @@ -18,6 +18,7 @@ \newcommand{\ex}{\emph{Beispiel: }} \newcommand{\anm}{\emph{Anmerkung: }} +\newcommand{\prac}{\textbf{\textcolor{red}{Übung}}} \newcommand{\linf}{\lim\limits_{n \rightarrow{} \infty}} From c93be0adeacebd90f04a18059f3cdbd931a69891 Mon Sep 17 00:00:00 2001 From: angerstoner Date: Sun, 14 Jan 2018 00:51:48 +0100 Subject: [PATCH 2/2] readme update --- README.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 4e1a4c3..8d9cc8e 100644 --- a/README.md +++ b/README.md @@ -56,5 +56,11 @@ Diese sind meist unvollständig und nur nach Lust und Laune erstellt, also Obach ### Diskrete Mathematik 2. Zählen und Kombinatorik - 2.1 Schubfachprinzip (In der Vorlesung wurde viele Beispiele genannt, die hier nicht aufgelistet sind) + 2.1 Schubfachprinzip (Weitere Beispiele wurden in der Vorlesung genannt) + +### Mathematik für Informatik-Anfänger: + 1. Mengen und Abbildungen + 1.1. Mengenlehre + 1.2. Relationen + 1.3. Funktionen/Abbildungen