From 60688821acd4a94cd8d45424f6490f6771cdce6e Mon Sep 17 00:00:00 2001 From: angerstoner Date: Sun, 14 Jan 2018 23:03:07 +0100 Subject: [PATCH] mafia update regarding chapter #2 --- MafIA1/mafia.pdf | Bin 209311 -> 235740 bytes MafIA1/mafia.tex | 69 +++++++++++++++++- .../GrundoperationenAussagenlogik.png | Bin 0 -> 7427 bytes env/commands.tex | 7 +- 4 files changed, 73 insertions(+), 3 deletions(-) create mode 100644 MafIA1/ressources/GrundoperationenAussagenlogik.png diff --git a/MafIA1/mafia.pdf b/MafIA1/mafia.pdf index ddff71e7577518fc8872ef987724ed7cfa266f26..9a3eabeb7a3c3d5f3a966dca1d962374f27481ab 100644 GIT binary patch delta 143103 zcmb@t1yr2PvM8G1PH+#d!QI{6U4jL-!SxFe+@0V~aCaxc-QC?GXwWzL_dffbd(V4s zt-IEnHM6GYtM01msxIl8?(7|;)fyCPN@a0L1{OvR1j@s-k zxgxK{uS_o|nOn*>6*~=U_0Zeg#f6f)16y25)}HmD9yh0>t3uXn@?)OC`{eax6Q5FJun4o> zL-nIHj!w4lGo|JE)VxMl{Tk-}6QWPG1(skP&o za>EUM3r!_)Rq973EP!YGkY)^tO&?>uO+3saf#zcqj@Hjt-uN&m!e6*y(uC@&@6e)M 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Sollten Fehler gefunden werden, bitte per Mail darüber informieren, dann können diese hier berichtigt werden. Auch Issues im Git sind gerne gesehen. Die Quelldateien sind öffentlich unter \url{https://git.webschneider.org/uni/sammlung} einsehbar. Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! @@ -177,7 +177,8 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! Da $f$ surjektiv und injektiv ist, folgt auch die \emph{Surjektivität} für $f$. - \subsubsection{Satz: Regeln für Abbildungen} + \subsubsection{Regeln für Abbildungen} + \ther \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 @@ -191,6 +192,70 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \[ R := \{x \in \Omega | x \not \in \{x\}\} \subset \Omega \] \[ R \in R \Lrarr R \not \in \{R\}\] +\section{Sprache und Logik} + \subsection{Grundlagen} + \subsubsection{Zeichen, Alphabete, Worte, Sprachen} + \cdef + \begin{enumerate} + \item Ein Zeichen ist ein Symbol (\ex $ x $ oder $\in$) + \item Eine Zeichenkette ist eine Aneinanderreihung von Zeichen (\ex $ x \in M $ oder ''diesisteineZeichenkette'') + \item Ein Alphabet ist eine endliche Menge von Zeichen + \item Ein Wort oder Satz das Länge $ n $ über einem Alphabet $ A $ ist eine Verkettung von $ n $ Zeichen aus $ A $. Das leere Word wird mit $ \epsilon $ bezeichnet.\\ + $ A^n $ ist die Menge aller Wörter der Länge $ n $\footnote{nicht zu verwechseln mit einem Produktraum von Alphabeten ($ A \times A \times A \times \dots$)} + \item Die freie Sprache $ S $ über einem Alphabet $ A $ ist eine Teilmenge von $ A* $ + \end{enumerate} + \subsubsection{Wahrheitswerte} + \cdef Die Menge $ B $ der Wahrheitstabelle ist wie folgt definitiert: + \begin{align} + B := \{\text{wahr, falsch}\} = \{\text{true, false}\} = \{W, F\} = \{1,0\} + \end{align} + \cdef Sei $ S $ eine Sprache über einem Alphabet $ A $. Sei $ T $ eine Teilmenge von $ S $ und es gebe eine Abbildung $ I: T \rarr B $. Dann heißen die Elemente von $ T $ logische \idx{Aussagen} und die die Abbildung $ I $ heißt \idx{Interpretation}\footnote{Anschaulicher: Aussagen sind Sprachsätze, die unter einer gegeben Interpretation einen Wahrheitswert haben}\\ + \ex $ 2 $ ist kleiner als $ 7 $\\ + \cdef Ein $n$-stelliges Prädikat auf $ M $ ist eine Abbildung von $ M^n \rarr B $. \\ + \[ \Rarr \text{Relation } r: M^n \rarr B , r (x) := \begin{cases} & \text{wahr, wenn } x\in R\\ & \text{falsch, wenn } x \not \in R \end{cases} \] + + \subsubsection{Grundoperationen der Aussagenlogik} + \begin{enumerate} + \item Negation $ \lnot A $ : es ist nicht wahr, dass $ A $ wahr ist + \item Konjunktion $ A \land B: A $ und $ B $ + \item Disjunktion $ A \lor B: A $ oder $ B $ + \item Implikation $ A \Rarr B $ : Aus $ A $ folgt $ B $ + \item Äquivalenz $ A \Lrarr B $: $ A $ genau dann wenn $ B $ + \end{enumerate} + \anm Etwas übersichtlicher sind die Operationen in Tabellenform: + + \begin{figure}[tbh] + \centering + \includegraphics[width=0.7\linewidth]{ressources/GrundoperationenAussagenlogik} + \caption{Grundoperationen der Aussagenlogik} + \label{fig:GrundoperationenAussagenlogik} + \end{figure} + + \ther + \begin{enumerate} + \item $p \land q = q \land p$ (Kommutativität) + \item $p \lor q = q \lor p$ (Kommutativität) + \item $(p \land q) \land r = p \land (q \land r)$ (Assoziativität) + \item $(p \lor q) \lor r = p \lor (q \lor r)$ (Assoziativität) + \item $(p \land q) \lor r = (p \lor r) \land (q \lor r)$ (Distributitivät von $\land$ und $\lor$) + \item $(p \lor q) \land r = (p \land r) \lor (p \land r)$ (Distributitivät von $\lor$ und $\land$) + \item $p \land p = p$ (Idempotenz) + \item $p \lor p = p $ (Idempotenz) + \item $\neg (p \land q) = (\neg p) \lor (\neg q)$ (Morgensche Regel) + \item $\neg (p \lor q) = (\neg p) \land (\neg q)$ (Morgensche Regel) + \item $\neg (\neg(q)) = q$ + \end{enumerate} + + \subsubsection{Beweistechnik} + Es gibt verschiedene Wege eine Behauptung zu beweisen. Drei wurden in der Vorlesung behandelt: + \begin{enumerate} + \item Direkter Beweis $p \rightarrow q$: Sei $p$, dann zeigen wir, dass q wahr ist + \item indirekter oder Widerspruchsbeweis (Kontrapositiv) + \item Beweis durch vollstaendige Induktion + \end{enumerate} + + + \section{Zahlen}\label{zahlen} \subsection{Natürliche Zahlen} \subsubsection{Peano-Axiome} diff --git a/MafIA1/ressources/GrundoperationenAussagenlogik.png b/MafIA1/ressources/GrundoperationenAussagenlogik.png new file mode 100644 index 0000000000000000000000000000000000000000..d412b38b1d787fc74d1564b1fe12b33067073351 GIT binary patch literal 7427 zcmb7Jc_5U1*ParJq)|vRa_gZGm87y2_hid53~i!9lu-7a%5o1`CLwz$#uAYwyCQ~S z$eJZf_H`!v65o&Nd7j?yeV_OH{xD<4?cC>_-?^@HopTd3)fHKo_A;SRC>CWUSPO;P z6c2v)+{yr6KM};JDAc|hW%w-KA!;J}>iK=GU6oZ{pXS~TrpJY9@smbLZ(bR*hb=r1 zG8fhAYbR=7R(a;ZBjx{)nGi<0g*%fnoRVX-Rg^cf$BUGyg1;Qr7wWt9^QJse-L2QZ zdc)`9*r3DecTNSVEx3JQt|%Gb&Ma7 z*51kdA&fMwV%I@2v8rGVS%Mgs%HM}!X|Vf_BK03$1=qQkPZc*LB8gCi(BoRFV? zs&Tp}WLx0rJm;8`1eN?vlhpwiEIz?DD+~9|tEqw5R{A(gk8}m-z};)B^Tgob*S)sW z)6)wpLvn$5Epq9*5Qfg{>qlTP6Bd<$+i!_{eNLW{VWNNAk98$@M zxxU+?@e26AyyL9;;Z%}!;SeF1^n|-7X53*w01L~=$Pkkvm6erGx-7gJ$xVZnz@3ql zloXZ>Gjnt7jT;aH14Hh(B0c(ZH6%%+A&CQ|{svY(+#?S%k8> zy81_oI;zZdzRM>$`H-r0ZwR4lBSe285Mm7oF~HB%%}rLQP0W4q1Np-T@vaywJXJKk zVu9hKA?Ob*fowelA=E3&ZrP%>)n%Nk3#jSPA?UW+%Nv5h;AkDBF#t+?(xq&Mg9apR zs2?lTqo+@5TwG`fbj!XTdL>T90u?(hyU;_o@03-=%!CfP7KbhdrWQst-1pArJ!E8s z{`lj)4X+5~G}EBiOLK@{&CSmCbpJkP(3WS(>jr8te9}(`t_t9JSoQVuHzj4|x1dT& zS(bZp4zP>(EX|#+pN9oFd*kOtyNY{Nuft;KiZ$K| z=xMTNNUOdx3q@u=YQRpYpMm;ea=0NOG-iMl6qx`y^$`T)F)Vatpq>IKE`#D1FJC^^ z%;<+qmMmM8Z_YS+?AT_tMnF?HQ^Xo52on?25n0)iUR&vJhV1CTOkT%+B1zG?Z1nYm zv^0^A6v~x|K$1kx=D_obNlBTO-6f<_9bMf%4fmzsfko;^&zw3(qZ>Y6~i zFO)8YBxmBuEIq+pw69gfpdb$rX6wKpf`o>J-CLmcvKaW>F*o;0yp<@|W1<6#700QoX`_ilV#B!EZCJM}_FgayBpq=%(lM$tDWAz|Un zH*ZXxT-dFwVR&L<;^L4j8=SIavfTuSOwD{yx1U+qWs$QT3#-V3Z(eh*=)j|4ccz9+LM^7xF z`7|eypcAm~)PprZIiVfok1Rd0UG?=rFV-lanC@=$A1^PjqZ#hgwUU5b*HNea{Svz2 z+qYv{+S;_>*p-UAT~)m9j{wR64CpVU>mBV%GDR1!E{);V#CoG++0nGU%Zo#?Rnru? 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