From a175aa45128ac9728b39955a353d77de01f7adbe Mon Sep 17 00:00:00 2001 From: JirR02 Date: Sun, 2 Mar 2025 08:16:16 +0100 Subject: [PATCH] Analysis Chapter 1.1 Sumarized Chapter 1.1 of Analysis --- fs25/analysis_I_II/grundlagen/logik.tex | 84 +++++++++++++++++- .../notizen_analysis_ruh_jirayu.pdf | Bin 151137 -> 196486 bytes 2 files changed, 82 insertions(+), 2 deletions(-) diff --git a/fs25/analysis_I_II/grundlagen/logik.tex b/fs25/analysis_I_II/grundlagen/logik.tex index 5d69865..e075c81 100644 --- a/fs25/analysis_I_II/grundlagen/logik.tex +++ b/fs25/analysis_I_II/grundlagen/logik.tex @@ -1,5 +1,7 @@ \section{Logik} +\subsection{Grundlagen} + In der Logik werden (mathematische) Aussagen untersucht. Eine Aussage ist eine Äusserung, die entweder wahr oder falsch ist. \cite{Ziltner2024} (Wahr oder Falsch). \\ \\ @@ -11,7 +13,7 @@ In der mathematischen Logik gelten die folgenden Sätze. \end{itemize} \cite{Ziltner2024} \nt{ - Es gibt gewisse Aussagen, als logische Aussage gelten könnte aber nicht zulässig ist. Solche Aussagen sind meisten rückbezügliche Äusserungen und sind deswegen keine sinnvollen Aussagen. (Siege Lügner-Paradox) + Es gibt gewisse Aussagen, als logische Aussage gelten könnte aber nicht zulässig ist. Solche Aussagen sind meisten rückbezügliche Äusserungen und sind deswegen keine sinnvollen Aussagen. (Siehe Lügner-Paradox) } Aussagen können verneint und miteinander verknüpft werden. @@ -40,7 +42,7 @@ Für Verknüpfungen verwenden wir folgende Notationen. \end{tabular} \end{center} -Die Wahrheitstablelle der vorher erwähnten Verknüpfungen sind wie folgt. +Die Wahrheitstabelle der vorher erwähnten Verknüpfungen sind wie folgt. \begin{center} \begin{tabular}{| c | c | c | c | c | c | c |} @@ -52,3 +54,81 @@ Die Wahrheitstablelle der vorher erwähnten Verknüpfungen sind wie folgt. T & T & T & T & F & T & T \\ \end{tabular} \end{center} + +Aus der Tabelle kann man die Zusammenhänge der Verknüpfungen erkennen. + +\nt{ + Wir unterscheiden zwischen dem inklusiven Oder und dem exklusiven Oder. Beim inklusiven Oder können beide Aussagen warh sein während beim exklusiven oder nur einer der beiden Aussagen wahr sein kann. +} + +\nt{ + Verknüpfende Aussagen brauchen inhaltlich nicht zusammenzuhängen. +} + +\subsection{Äquivalenz} + +\thm{Äquivalenz}{ + Seien $P$ und $Q$ Aussagen. Wenn $P$ und $Q$ die gleichen Aussagen haben, so nennen wir sie logisch Äquivalent. + + \begin{equation} + P \equiv Q + \end{equation} +} + +Sobald 2 Aussagen äquivalent sind, so ist ihre Implikation, sowie ihr Kontraponiertes logisch äquivalent. + +\thm{Kontraponiertes}{ + Das Kontraponierte zur Implikation $A \Rightarrow B$ ist + + \begin{equation} + \neg B \Rightarrow \neg A + \end{equation} + + Dabei gilt + + \[ + A \Rightarrow B \equiv \neg B \Rightarrow \neg A + .\] +} + +Die Äquivalenz $A \Leftrightarrow B$ ist nur wahr, wenn die Implikationen $A \Rightarrow B$ und $B \Rightarrow A$ beide wahr sind. + +\[ + A \Leftrightarrow B \equiv (A \Rightarrow B) \land (B \Rightarrow A) + .\] + +\subsection{Axiome, Sätze und Beweise} + +In der Mathematik sind Axiome von grosser Bedeutung. Sie sind das Fundament der Mathematik. In der Analysis werden wir jedoch Sätze verwenden, welche durch Axiome bewiesen worden sind. +\\ +\\ +Um Aussagen zu Beweisen, verwenden wir in der Logik den Modus Ponens. + +\dfn{Modus Ponens}{ + Ein Beweis einer Aussage $A$ ist eine sukzessive Herleitung von $A$ aus dem Axiomen, in der logische Schlussregeln angewendet werden. Eine solche Regel ist der Modus Ponens. + + \begin{tabular}{ c } + $A$ \\ + $A \Rightarrow B$ \\ + \hline + B \\ + \end{tabular} + + $A$ ist die Prämise, $B$ die Konklusion. +} + +Aus dem Modus Ponens können wir schliessen, dass wenn $A$ und $A \Rightarrow B$ gilt, so gilt $B$. Der Modus Ponens ist die Basis eines Beweises. Wir werden später sehen, dass wir den Modus Ponens im Hintergrund verwenden. + +\nt{ + Wir können auch Beweise durchführen durch die Kontraposition. +} + +In der Analysis werden wir auch mit indirekten Beweisen arbeiten. Dabei nehmen wir an, dass eine Aussage falsch ist, woraus wir eine falsche Aussage herleiten. Dies nennen wir auch den Beweis mittels Widerspruch. Es lohnt sich aber oft, einen Widerspruchsbeweis als direkten Beweis umzuschreiben, da aus eine falsche oder einer wahren Aussage eine beliebige wahre Aussage hergeleitet werden kann. + +\thm{Prinzip der vollständigen Induktion}{ + Nehmen wir an das die Funktion $P(0)$ gilt. 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