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\begin{center}\textbf{Problem Set \color{blue}8\color{black} \hspace{.7in} Foundations \hspace{.7in} Due 11/11/19} \end{center}

\noindent Five of these problems will be graded, with each problem worth 4 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item Let $S = \{a,b,c,d,e\}$ and $\Pi = \{\{a\},\{b,d\},\{c,e\}\}$.  Write the relation $R$ corresponding to the partition $\Pi$.

\item Let $S$ be a set and $\Pi$ a partition of $S$ defined by $a \sim b \Leftrightarrow \exists P \in \Pi$ for which $a, b \in P$.  Then $\Pi$ is a reflexive relation.

\item Let $S$ be a set and $\Pi$ a partition of $S$ defined by $a \sim b \Leftrightarrow \exists P \in \Pi$ for which $a, b \in P$.  Then $\Pi$ is a symmetric relation.

\item Let $S$ be a set and $\Pi$ a partition of $S$ defined by $a \sim b \Leftrightarrow \exists P \in \Pi$ for which $a, b \in P$.  Then $\Pi$ is a transitive relation.

\end{enumerate}

\noindent We say that two vertices $v_1$ and $v_2$ of a graph $G$ are \textbf{in the same component of $G$} $\Leftrightarrow$ $\exists$ a walk from $v_1$ to $v_2$.

\begin{enumerate}[resume]

\item The relation of being in the same component of a graph is reflexive.

\item The relation of being in the same component of a graph is symmetric.

\item The relation of being in the same component of a graph is transitive.

\end{enumerate}

\noindent We say that two vertices $v_1$ and $v_2$ of a graph $G$ are \textbf{on a common cycle of $G$} $\Leftrightarrow \exists$ a cycle including $v_1$ and $v_2$.

\begin{enumerate}[resume]

\item The relation of being on a common cycle of a graph is reflexive.

\item The relation of being on a common cycle of a graph is symmetric.

\item The relation of being on a common cycle of a graph is transitive.

\end{enumerate}

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