\documentclass[letterpaper,12pt]{article}

\usepackage{amsfonts,amssymb,amsmath,amsthm,txfonts,pxfonts,mathrsfs,fullpage,enumitem}
\usepackage{xcolor}

\pagestyle{empty}
\begin{document} 

\begin{center}\textbf{Problem Set 9 \hspace{.7in} Foundations \hspace{.7in} Due 4/11/22} \end{center}

\noindent Four of these problems will be graded, 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 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 sum 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 a function being injective in terms of ordered pairs.

\end{enumerate}

\noindent We say that two vertices $v_1$ and $v_2$ of a graph $G$ are \textbf{adjacent} $\Leftrightarrow$ $\exists$ an edge in $G$ containing $v_1$ and $v_2$.

\begin{enumerate}[resume]

\item The relation of being adjacent in a graph is reflexive.

\item The relation of being adjacent in a graph is symmetric.

\item The relation of being adjacent in a graph is transitive.

\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}

\end{document}