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Refine and add some tasks
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@Lipen
Lipen committed Sep 27, 2024
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%% Add custom setup below

%% Macros from physics
\usepackage{physics}

%% Math enquote
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%% Definitions
\declaretheorem[numbered=unless unique]{definition}

%% Algorithms
\usepackage[ruled,linesnumbered,vlined]{algorithm2e}


\begin{document}
\selectlanguage{english}

\setlength{\epigraphwidth}{0.4\textwidth}
\epigraph{\textpzc{In der Mathematik ist die Kunst Fragen zu stellen wertvoller als Probleme zu lösen}}{--- Georg Cantor}

\begin{tasks}
%% Task: Check properties of relations.
\item For each given relation $R_i \subseteq {M_i}^2$, determine whether it is \textit{reflexive, irreflexive, coreflexive, symmetric, antisymmetric, asymmetric, transitive, antitransitive, semiconnex, connex, left/right Euclidean}.
\item For each given relation $R_i \subseteq {M_i}^2$, determine whether it is \textit{reflexive, irreflexive, coreflexive, symmetric, antisymmetric, asymmetric, transitive, left/right Euclidean, connex}.
Provide a counterexample for each non-complying property (\eg \enquote{transitivity does not hold for $x,y,z = (3,1,2)$}).
Organize your answer in a table (\eg columns \--- relations, rows \--- properties).

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\item Let $R_{\theta}$ be a relation of $\theta$-similarity (clearly, $\theta \in [0; 1] \subseteq \Real$) of finite non-empty sets defined as follows: a set~$A$ is said to be \textit{$\theta$-similar} to~$B$ \textit{iff} the Jaccard index $\Jac(A,B) = \frac{\card{A \intersection B}}{\card{A \union B}}$ for these sets is at least~$\theta$, \ie $\Pair{A, B} \in R_{\theta} \iff \Jac(A,B) \geq \theta$.

\begin{subtasks}
\item Determine whether $\theta$-similarity is a tolerance relation.
\item Determine whether $\theta$-similarity is a tolerance relation\footnote{A tolerance relation is a \textit{reflexive} and \textit{symmetric} relation.}.
\item Determine whether $\theta$-similarity is an equivalence relation.
\item Draw the graph of a relation $R_{\theta} \subseteq \Set{A_i}^2$, where $\theta = 0.25$, $A_1 = \Set{1,2,5,6}$, $A_2 = \Set{2,3,4,5,7,9}$, $A_3 = \Set{1,4,5,6}$, $A_4 = \Set{3,7,9}$, $A_5 = \Set{1,5,6,8,9}$.
\end{subtasks}


% Task: Explore the characteristic function.
\item The characteristic function~$f_S$ of a set~$S$ is defined as follows:
\[
f_S(x) = \begin{cases}
1 &\text{if } x \in S \\
0 &\text{if } x \notin S
\end{cases}
\]
% % Task: Explore the characteristic function.
% \item The characteristic function~$f_S$ of a set~$S$ is defined as follows:
% \[
% f_S(x) = \begin{cases}
% 1 &\text{if } x \in S \\
% 0 &\text{if } x \notin S
% \end{cases}
% \]

Let~$A$~and~$B$ be finite sets.
Show that for all $x \in \universalset$:
% Let~$A$~and~$B$ be finite sets.
% Show that for all $x \in \universalset$:

\begin{subtasks}
\item $f_{\,\overline{A}} (x) = 1 - f_A(x)$
\item $f_{A \intersection B} (x) = f_A(x) \cdot f_B(x)$
\item $f_{A \union B} (x) = f_A(x) + f_B(x) - f_A(x) \cdot f_B(x)$
\item $f_{A \xor B} (x) = f_A(x) + f_B(x) - 2 f_A(x) \cdot f_B(x)$
\end{subtasks}
% \begin{subtasks}
% \item $f_{\,\overline{A}} (x) = 1 - f_A(x)$
% \item $f_{A \intersection B} (x) = f_A(x) \cdot f_B(x)$
% \item $f_{A \union B} (x) = f_A(x) + f_B(x) - f_A(x) \cdot f_B(x)$
% \item $f_{A \xor B} (x) = f_A(x) + f_B(x) - 2 f_A(x) \cdot f_B(x)$
% \end{subtasks}


% Task: Explore the Boolean product of matrices.
\item Any binary relation $R \subseteq M^2$ can be represented as a zero-one matrix~$\relmatrix{R} = [r_{ij}]$, where the element~$r_{ij}$ is equal to~1 if $\Pair{m_i, m_j} \in R$ and 0~otherwise.
Boolean product of two square matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ is a matrix $C = A \odot B = [c_{ij}]$ defined as follows: $c_{ij} = \biglornolim_{k} (a_{ik} \land b_{kj})$.
A~composition of relations $R$ and~$S$ is a relation $S \circ R$ defined as follows: $\Pair{a,b} \in S \circ R \iff \exists c : \Pair{a,c} \in R \land \Pair{c,b} \in S$.
Show that the matrix representation of the composition of relations $R$ and~$S$ is equal to the Boolean product of the corresponding matrices, \ie $\relmatrix{S \circ R} = \relmatrix{R} \odot \relmatrix{S}$.


% Task: Find the error in the "proof".
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transitive closure of~$R$ is not transitive.


% % Task: Explore the naive algorithm for transitive closure.
% \item Given a relation $R$, the transitive closure $R^{+}$ can be computed using the following algorithm.

% \begin{minipage}{0.8\textwidth}
% \begin{algorithm}[H]
% \caption{Na\"ive algorithm for computing the transitive closure $R^{+}$}
% \DontPrintSemicolon
% \SetKwInput{Input}{Input}
% \SetKwInput{Output}{Output}
% \Input{Zero-one matrix $M = \relmatrix{R}$ of size $n \times n$.}
% \Output{Zero-one matrix $B$ for the transitive closure $R^{+}$.}
% \BlankLine
% $A \gets M$\;
% $B \gets A$\;
% \For{$i = 2$ \KwTo $n$}{
% $A \gets A \odot M$\;
% $B \gets B \lor A$\;
% }
% \Return $B$
% \end{algorithm}
% \end{minipage}


% \item Given a relation $R \subseteq M^2$, the transitive closure $R^{+}$ can be computed using the Roy\--Warshall algorithm, which is presented below.

% \begin{minipage}{0.8\textwidth}
% \begin{algorithm}[H]
% \caption{Roy\--Warshall algorithm}
% \DontPrintSemicolon
% \SetKwInput{Input}{Input}
% \SetKwInput{Output}{Output}
% \Input{Zero-one matrix $M = \relmatrix{R}$ of size $n \times n$.}
% \Output{Zero-one matrix $W = [w_{ij}]$ for the transitive closure $R^{+}$.}
% \BlankLine
% $W \gets M$\;
% \For{$k = 1$ \KwTo $n$}{
% \For{$i = 1$ \KwTo $n$}{
% \For{$j = 1$ \KwTo $n$}{
% $w_{ij} \gets w_{ij} \lor (w_{ik} \land w_{kj})$\;
% }
% }
% }
% \Return $W$
% \end{algorithm}
% \end{minipage}


% Task: Explore the inverse of a composition.
\item Consider two relations $R \subseteq A \times B$ and $S \subseteq B \times C$.
Prove that $(S \circ R)^{-1} = R^{-1} \circ S^{-1}$.


% Task: Composition of injections and surjections.
\item Prove or disprove the following statements about the functions $f$ and $g$:

\begin{subtasks}
\item If $f$ and $g$ are injections, then $g \circ f$ is also an injection.

\item If $f$ and $g$ are surjections, then $g \circ f$ is also a surjection.

\item If $f$ and $f \circ g$ are injections, then $g$ is also an injection.

\item If $f$ and $f \circ g$ are surjections, then $g$ is also a surjection.
\end{subtasks}


%% Task: Explore the divisibility relation.
\item Let $H = \Set{1, 2, 4, 5, 10, 12, 20}$.
Consider a divisibility relation $R \subseteq H^2$ defined as follows: $x \!\rel\!\nobreak y \iff y \divby\nobreak x$.
Consider a divisibility relation $R \subseteq H^2$ defined as follows: $x \rel y \iff y \divby x$.

\begin{subtasks}
\item Sort $R$ (as a set of pairs) lexicographically\footnote{Lexicographic order for pairs: $\Pair{a,b} \preceq \Pair{a',b'} \iff (a < a') \lor ((a = a') \land (b \leq b'))$}.
\item Sort $R$ (as a set of pairs) lexicographically\footnote{Lexicographic order for pairs: $\Pair{a,b} \preceq \Pair{a',b'} \iff (a < a') \lor ((a = a') \land (b \leq b'))$. For example, $\Pair{1,2} \preceq \Pair{1,3} \preceq \Pair{2,1}$.}.

\item Show that $R$ is a partial order.

\item Determine whether $R$ is a linear (total) order.

\item Draw the Hasse diagram for a graded poset $\Triple{H, R, \rho}$, where $\rho : H \to \NaturalZero$ is a grading function which maps a number $n \in H$ to the sum of all exponents appearing in its prime factorization, \eg $\rho(20) = \rho(2^{\mathcolor{my-green}{2}} \cdot 5^{\textcolor{my-blue}{1}}) = \mathcolor{my-green}{2} + \textcolor{my-blue}{1} = 3$, so the number 20 would have the 3rd rank (bottom-up).
\item Draw the Hasse diagram for a graded poset $\Triple{H, R, \rho}$, where $\rho \colon H \to \NaturalZero$ is a grading function which maps a number $n \in H$ to the sum of all exponents appearing in its prime factorization, \eg $\rho(20) = \rho(2^{\mathcolor{my-green}{2}} \cdot 5^{\textcolor{my-blue}{1}}) = \mathcolor{my-green}{2} + \textcolor{my-blue}{1} = 3$, so the number 20 would have the 3rd rank (bottom-up).

\item Find the minimal, minimum (least), maximal and maximum (greatest) elements in the poset~$\Pair{H, R}$.
If there are multiple or none, explain why.
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\end{definition}


% Task: Dedekind-infinite set.
\item Prove that a set $S$ is infinite if and only if there is a proper subset $A \subset S$ such that there is a one-to-one correspondence (bijection) between $A$~and~$S$.
% % Task: Dedekind-infinite set.
% \item A set $A$ is \emph{finite} if there is a bijection between $A$ and a set of the form $\Set{1, 2, \dots, n}$ for some~$n \in \Natural$.
% A~set is called \emph{infinite} if it is not finite.
% A~set $S$ is \emph{Dedekind-infinite} if there is a proper subset $B \subset S$ such that there is a bijection between $S$ and~$B$.
%
% Prove that any infinite set is Dedekind-infinite.
% What about the converse?
%
% Prove that a set $S$ is infinite if and only if there is a proper subset $A \subset S$ such that there is a one-to-one correspondence (bijection) between $A$~and~$S$.


% Task: Refinement relation over partitions is a lattice.
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