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\begin{center}\textbf{Problem Set 8 \hspace{.7in} Foundations \hspace{.7in} Due 4/8/23} \end{center}

\noindent Four of these problems will be graded (my choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but you  must write up your own final submission without reference to any sources other than the textbook and instructor. \\


\begin{enumerate}

\item Express the definition of the quotient of two functions in terms of ordered pairs.

\item Express the definition of the composition of two functions in terms of ordered pairs.

\item Express the definition of an injective function in terms of ordered pairs.

\item Any graph where every vertex has odd degree must have an even number of vertices.

\item Any graph where each vertex has degree at least 2 is connected.

\item Any graph where some vertex $v_0$ is joined to each other vertex in the graph by a walk is connected.

\item The maximum possible number of edges in a graph with $n$ vertices is \rule{1 in}{1 pt}.

\item We say that two vertices $v_1$ and $v_2$ of a graph $G$ are \textbf{adjacent} $\Leftrightarrow \exists$ an edge $\{v_1, v_2\}$ in $G$.  Determine whether the relation of being adjacent in a graph is reflexive, symmetric, and transitive.

\item 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$ in $G$.  Determine whether the relation of being in the same component of a graph is reflexive, symmetric, and transitive.

\item 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 $v_2$.  Determine whether the relation of being on a common cycle of a graph is reflexive, symmetric, and transitive.

\end{enumerate}



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