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\title{
    cmpe220 hw \#6
    \\
    Fall 2018
}
\author{Haluk Bingol}
\date{\today}

\begin{document}
\maketitle


We discussed relations on a set
and used graph representations of relations 
although we did not formally defined graphs. 
Now we cover graphs.
Actually they are closely related. 

\begin{defn}
    Let $R$ be a relation on a set $A$.
    The  \hDefined{connectivity relation} 
    $R^{*}$ consists of 
    the pairs  $(a, b)$ such that there is a path of length at least one from $a$ to $b$ in $R$.
\end{defn}

\begin{enumerate}

    \item 
    % source: Rosen5e 499
    Prove that
    \[
        R^{*} = \bigcup_{n = 1}^{\infty} R^{n}.
    \]
    
    \item
    What is the corresponding expression in terms of 
    matrices 
    $M_{R}$ and
    $M_{R^{*} }$
    of relations 
    $R$ and 
    $R^{*}$, 
    respectively?

    \item
    Let $R^{t}$ be the transitive closer of $R$.
    What is th erelation between $R^{t}$ and $R$?
%    What can you say about $R^{t}$?

    \item 
    % source: Rosen5e 499 Example 4
    Let $P$ be all the people on Tweeter.
    Define relation $R$ as $(a, b) \in R$  
    iff
    $a$ follows $b$.
    What is $R^{n}$,
    where $n$ is a positive integer greater than one?
    What is $R^{*}$?
    
\end{enumerate}

\end{document}