Commit d54f09e4 authored by Marius Frinken's avatar Marius Frinken

marius's stuff for tutorial 8 HAPPY HOLIDAYS :)

parent 9b9d40ed
\documentclass[a4paper]{article}
%% Language and font encodings
\usepackage[german,english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
%% Sets page size and margins
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
%% Useful packages
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}
\usepackage{url}
\usepackage[affil-it]{authblk}
\setlength{\parindent}{0em}
% use these fields for a title
\author{AUTHOR}
\title{TITLE}
\begin{document}
\maketitle
This is the full proof of the validity of $A \Rightarrow A$.\\
I do not expect you to do hand in your homework in such a complete form.
You may shorten the steps 1 -- 4, since they are trivial transformations. Also,you may do several steps at once, e.g. 5~\&~6 or 7 \& 8 and so on, as long as you don't make any mistakes!\\
\textbf{Generally:} You may also solve 8.3 without this specific notation, as long as you use \textbf{Interpretations}, \textbf{Value Functions} and \textbf{Assignments} as defined in the lecture.
\begin{align}
& \mathcal{M} \models^\varphi A \Rightarrow A \text{ for all Assignments } \varphi \\
%
\stackrel{valid}{\Leftrightarrow} & \text{ for any } \varphi: \mathcal{I}_\varphi(A \Rightarrow A) = T\\
%
\stackrel{\Rightarrow}{\Leftrightarrow} & \text{ for any } \varphi: \mathcal{I}_\varphi(\neg A \lor A) = T\\
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\stackrel{\lor}{\Leftrightarrow} & \text{ for any } \varphi: \mathcal{I}_\varphi(\neg ( \neg \neg A \land \neg A)) = T\\
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\stackrel{\mathcal{I_\varphi}(\neg)}{\Leftrightarrow} & \text{ for any } \varphi: \mathcal{I}(\neg) \mathcal{I}_\varphi(\neg \neg A \land \neg A) = T\\
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\stackrel{\mathcal{I}(\neg)}{\Leftrightarrow} & \text{ for any } \varphi: \mathcal{I}_\varphi(\neg \neg A \land \neg A) = F\\
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\stackrel{\mathcal{I_\varphi}(\neg)}{\Leftrightarrow} & \text{ for any } \varphi: \mathcal{I} (\land) [ \mathcal{I}_\varphi(\neg \neg A), \mathcal{I}_\varphi(\neg A) ] = F\\
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\stackrel{\mathcal{I}(\land)}{\Leftrightarrow} & \text{ for any } \varphi: \text{ either }\mathcal{I}_\varphi(\neg \neg A) = F \text{ or: } \mathcal{I}_\varphi( \neg A) = F \\
%
\stackrel{\mathcal{I_\varphi}(\neg) \text{ on both sides}}{\Leftrightarrow} & \text{ for any } \varphi: \text{ either } \mathcal{I}(\neg)\mathcal{I}_\varphi(\neg A) = F \text{ or: } \mathcal{I}(\neg) \mathcal{I}_\varphi(A) = F \\
%
\stackrel{\mathcal{I}(\neg) \text{ on both sides}}{\Leftrightarrow} & \text{ for any } \varphi: \text{ either } \mathcal{I}_\varphi(\neg A) = T \text{ or: } \mathcal{I}_\varphi(A) = T \\
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\stackrel{\mathcal{I_\varphi}(\neg) \text{on the left side}}{\Leftrightarrow} & \text{ for any } \varphi: \text{ either }\mathcal{I}(\neg)\mathcal{I}_\varphi(A) = T \text{ or: } \mathcal{I}_\varphi(A) = T \\
%
\stackrel{\mathcal{I}(\neg) \text{on the left side}}{\Leftrightarrow} & \text{ for any } \varphi: \text{ either }\mathcal{I}_\varphi(A) = F \text{ or: } \mathcal{I}_\varphi(A) = T \\
%
\stackrel{\text{def of } \varphi}{\Leftrightarrow} & \text{this holds always } \hspace{0.5cm} \square
\end{align}
\end{document}
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