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\documentclass[a4paper, 10pt, pagesize, smallheadings]{article}
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\usepackage{graphicx}
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%\usepackage[latin1]{inputenc}
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\usepackage{amsmath, amsthm, amssymb}
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\usepackage{typearea}
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\usepackage{algorithm}
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\usepackage{algorithmic}
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\usepackage{fullpage}
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\usepackage{mathtools}
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\usepackage[all]{xy}
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\addtolength{\voffset}{-40pt}
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\title{CSP Exercise 01 Solution}
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\author{Eugen Sawin}
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\renewcommand{\familydefault}{\sfdefault}
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\newcommand{\E}{\mathcal{E}}
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\newcommand{\R}{\mathcal{R}}
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%\include{pythonlisting}
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\pagestyle{empty}
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\begin{document}
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\maketitle
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\section*{Exercise 1.1}
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\begin{tabular}{|c|c|c||c|c|c||c|c|c|}
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\hline
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5&3&6 &4&7&2 &8&1&9 \\ \hline
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4&2&1 &6&9&8 &5&7&3 \\ \hline
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7&8&9 &1&5&3 &4&6&2 \\ \hline\hline
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8&6&3 &9&2&4 &1&5&7 \\ \hline
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1&7&5 &8&3&6 &9&2&4 \\ \hline
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2&9&4 &5&1&7 &6&3&8 \\ \hline\hline
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3&1&8 &2&4&5 &7&9&6 \\ \hline
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9&4&7 &3&6&1 &2&8&5 \\ \hline
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6&5&2 &7&8&9 &3&4&1 \\ \hline
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\end{tabular} \\\\
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I have applied following strategies to solve the puzzle:
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\begin{itemize}
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\item look for most contrained blocks, rows and columns
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\item look for values with most resolved positions
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\item annotate cells with consistent values
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\item look for inconsistent connections between cell annotations
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\end{itemize}
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\section*{Exercise 1.2}
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(a) $R_{x,y} \bowtie S_{y,z} = \{(a,b,a),(a,b,c)\}$\\\\
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(b) $\sigma_{z=c}(R_{x,y} \bowtie S_{y,z}) = \{(a,b,c)\}$\\\\
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(c) $\pi_x(R_{x,y}) = \{(a)\}$\\\\
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(d) $R_{x,y} \circ S_{y,z} = \{(a,a),(a,c)\}$
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\section*{Exercise 1.3}
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(a) By definition holds
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\begin{align}
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R \circ (S \cup T) &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in R \text{ and } (y,z) \in (S \cup T)\}\\
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R \circ S &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in R \text{ and } (y,z) \in S\}\\
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R \circ T &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in R \text{ and } (y,z) \in T\}
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\end{align}\setcounter{equation}{0}\\
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The union of (2) and (3) gives us the set of all $(x,y) \in R$ with either $(y,z) \in S$ or $(y,z) \in T$, from which follows that $(y,z) \in (S \cup T)$, making it equal to (1). \qed\\\\
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(b) By definition holds
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\begin{align}
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-R &= \{(x,y) \in X^2 \mid (x,y) \notin R\}\\
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(-R)^{-1} &= \{(y,x) \in X^2 \mid (x,y) \in -R\}\\
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-(R^{-1}) &= \{(x,y) \in X^2 \mid (x,y) \notin R^{-1}\}\\
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-(R^{-1}) &= \{(x,y) \in X^2 \mid (y,x) \notin R\}
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\end{align}\setcounter{equation}{0}\\
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We've obtained (4) by applying the converse definition on (3). From the definition of $-R$ (1) follows that $(x,y) \in -R$ iff $(x,y) \notin R$, and therefore we see that (2) and (4) define equal sets. \qed\\\\
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(c) By definition holds
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\begin{align}
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(R \circ S)^{-1} &= \{(z,x) \in X^2 \mid \exists{y \in X}: (x,y) \in R \text{ and } (y,z) \in S\}\\
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S^{-1} \circ R^{-1} &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in S^{-1} \text{ and } (y,z) \in R^{-1}\}\\
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S^{-1} \circ R^{-1} &= \{(x,z) \in X^2 \mid \exists{y \in X}: (y,x) \in S \text{ and } (z,y) \in R\}
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\end{align}\setcounter{equation}{0}\\
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We've obtained (3) by applying the converse definition on (2). After some variable juggling we see that (1) and (3) define equal sets. \qed\\\\
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(d) By definition holds
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\begin{align}
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R \circ S &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in R \text{ and } (y,z) \in S\}\\
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(R \circ S) \cap T^{-1} &= (R \circ S) \cap \{(y,x) \in X^2 \mid (x,y) \in T\}\\
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(R \circ S) \cap T^{-1} &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in R \text{ and } (y,z) \in S \text{ and } (z,x) \in T\}\\
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(S \circ T) \cap R^{-1} &= \{(x,z) \in X^2 \mid \exists{y \in X}: (x,y) \in S \text{ and } (y,z) \in T \text{ and } (z,x) \in R\}
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\end{align}\\
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We've obtained (3) by directly applying intersection within the set comprehension of (2). It follows that\\
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\begin{align}
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(R \circ S) \cap T^{-1} = \emptyset &\iff \forall{x,y,z \in X}: (x,y) \notin R \text{ or } (y,z) \notin S \text{ or } (z,x) \notin T\\
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\text{ and } (S \circ T) \cap R^{-1} = \emptyset &\iff \forall{x,y,z \in X}: (x,y) \notin S \text{ or } (y,z) \notin T \text{ or } (z,x) \notin R
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\end{align}\\
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As we can see, the intersections form a ring-like relationship between the sets' tuples $(...,R, S, T, R,...)$. After some variable reordering, we see that both intersections are empty iff there is no ring-like relationship between any tuples of the three sets. \qed
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\end{document}
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