1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/tex/theory1_ex9.tex	Mon Jul 18 20:17:25 2011 +0200
     1.3 @@ -0,0 +1,46 @@
     1.4 +\documentclass[a4paper, 10pt, pagesize, smallheadings]{article}  
     1.5 +\usepackage{graphicx}
     1.6 +%\usepackage[latin1]{inputenc}
     1.7 +\usepackage{amsmath, amsthm, amssymb}
     1.8 +\usepackage{typearea}
     1.9 +\usepackage{algorithm}
    1.10 +\usepackage{algorithmic}
    1.11 +\usepackage{fullpage}
    1.12 +\usepackage{mathtools}
    1.13 +\usepackage[all]{xy}
    1.14 +\title{Theory I, Sheet 9 Solution}
    1.15 +\author{Eugen Sawin}
    1.16 +\renewcommand{\familydefault}{\sfdefault}
    1.17 +\newcommand{\Pos}{\mathcal{P}os}
    1.18 +\newcommand{\E}{\mathcal{E}}
    1.19 +\newcommand{\D}{\mathcal{D}}
    1.20 +\newcommand{\J}{\mathcal{J}}
    1.21 +\include{pythonlisting}
    1.22 +
    1.23 +\pagestyle{empty}
    1.24 +\begin{document}
    1.25 +\maketitle
    1.26 +
    1.27 +\section*{Exercise 9.1.3}
    1.28 +\begin{align*}
    1.29 +ADT_{bool} = &(\Sigma_{bool}, \E)\\
    1.30 +\Sigma_{bool} = &\{true^{(0)}, false^{(0)}, not^{(1)}, and^{(2)}, or^{(2)}\}\\
    1.31 +\E = &\{true \approx not(false),\\
    1.32 +& false \approx not(true),\\
    1.33 +& not(not(x)) \approx x\\
    1.34 +& and(x, y) \approx not(or(not(x), not(y))),\\
    1.35 +& or(x, y) \approx not(and(not(x), not(y)))\}\\
    1.36 +\end{align*}
    1.37 +
    1.38 +\section*{Exercise 9.1.4}
    1.39 +\begin{align*}
    1.40 +\D_{bool} &= (M, \J)\\
    1.41 +M &= \{0, 1\}\\
    1.42 +\J(true) &= 1\\
    1.43 +\J(false) &= 0\\
    1.44 +\J(not)(x) &= 1 - x\\
    1.45 +\J(and)(x, y) &= min(x, y)\\
    1.46 +\J(or)(x, y) &= max(x, y)\\
    1.47 +\end{align*}
    1.48 +
    1.49 +\end{document}