Keine Lust.
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}