From 42905b4df93e376b2632d8a639db8f14e5a6afbe Mon Sep 17 00:00:00 2001 From: Konstantin Chukharev Date: Fri, 27 Sep 2024 23:49:50 +0300 Subject: [PATCH] Refine and add some tasks --- hw2.tex | 122 +++++++++++++++++++++++++++++++++++++++++++++----------- 1 file changed, 99 insertions(+), 23 deletions(-) diff --git a/hw2.tex b/hw2.tex index 53d5f61..697f63a 100644 --- a/hw2.tex +++ b/hw2.tex @@ -18,6 +18,7 @@ %% Add custom setup below +%% Macros from physics \usepackage{physics} %% Math enquote @@ -34,13 +35,19 @@ %% 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). @@ -127,30 +134,37 @@ \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". @@ -168,29 +182,84 @@ 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. @@ -212,8 +281,15 @@ \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.