1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/tex/theory1_ex11.tex	Wed Aug 03 21:39:33 2011 +0200
     1.3 @@ -0,0 +1,40 @@
     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 10 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 11.1.1}
    1.28 +We show $attr(\alpha) \subseteq Y \subseteq X \implies \pi[Y](\sigma[\alpha]r) \equiv \sigma[\alpha](\pi[Y]r)$:
    1.29 +\begin{align*}
    1.30 +\pi[Y](\sigma[\alpha]r) &= \{\mu \in Tup(Y) \mid \exists{\mu' \in \sigma[\alpha]r}: \mu = \mu'[Y]\}\\
    1.31 +\sigma[\alpha]r &= \{\mu \in Tup(X) \mid \mu \in r \land \mu \text{ fulfills } \alpha\}
    1.32 +\end{align*}
    1.33 +From $Y \subseteq X$ follows:
    1.34 +\begin{align*}
    1.35 +\pi[Y](\sigma[\alpha]r) &= \{\mu \in Tup(Y) \mid \mu \in r \land \mu \text{ fulfills } \alpha\}\\
    1.36 +\sigma[\alpha](\pi[Y]r) &= \{\mu \in Tup(Y) \mid \mu \in \pi[Y]r \land \mu \text{ fulfills } \alpha\}\\
    1.37 +\pi[Y]r &= \{\mu \in Tup(Y) \mid \exists{\mu' \in r}: \mu = \mu'[Y]\}\\
    1.38 +\end{align*}
    1.39 +Again from $Y \subseteq X$ follows:
    1.40 +\begin{align*}
    1.41 +\sigma[\alpha](\pi[Y]r) &= \{\mu \in Tup(Y) \mid \mu \in r \land \mu \text{ fulfills } \alpha\}\\
    1.42 +\end{align*}
    1.43 +\end{document}