\documentclass[11pt]{article}
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\begin{document}
\headers{Math303}{Quiz 1}{Fri. Aug. 28, 2026}

%\makeatletter
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\makeatletter
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\noindent
\hfill Name: \underline{\hspace{3.5in}} 

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%Name: \underline{\hspace{3.5in}} \\
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\def\columnseprule{0.4pt}

\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\st}{\colon \,}
\renewcommand{\emptyset}{\varnothing}

\noindent
\textbf{Directions:} Solve the following problems.  Give supporting work/justification where appropriate.

\begin{enumerate}
\item\ppoints{4}{1} Decide whether the following statements are true or false, writing the entire word ``true'' or ``false'' to indicate your answer.  Give brief explanations.
\newcommand{\myspc}{\vspace{2cm}\\~}
\begin{multicols}{2}
\begin{enumerate}
	\item $\{1,2,3\} = \{1,1,2,3,3\}$
	\myspc
	\item $\{1,2,3\} = \{3,2,1\}$
	\myspc

	\columnbreak
	\item $1 \in \{0, 0.9, 0.99, 0.999, 0.9999, \ldots\}$
	\myspc
	\item $\{\{1\},\{1,1\}\} = \{\{1\}\}$
	\myspc
\end{enumerate}
\end{multicols}

\item\ppoints{4}{1} Determine whether the following sets are infinite or finite.  If the set is finite, then determine its cardinality.
\begin{enumerate}
	\item $\{1, \{1\}, \{\{1\}\}, \{\{\{1\}\}\}, \ldots\}$
	\vfill
	
	\item $\{\RR\}$
	\vfill
	
	\item $\{x\in\RR\st x^2 = 1\}$
	\vfill
	
	\item $\{1, 2, 3, \{1,2\}, 1, \{2, 1, 2\}\}$
	\vfill
\end{enumerate}

\newpage

\item\ppoints{2}{1} Use set-builder notation to express the following sets compactly.
\begin{enumerate}
	\item The set of all points $(x,y)$ in the plane strictly between the lines $x=-1$ and $x=1$.
	\vspace{3cm}
	\item $\{0, 0.9, 0.99, 0.999, 0.9999, \ldots\}$
	\vspace{3cm}
\end{enumerate}

\item\points{2} Use set-builder notation to express the subset of $\RR^2$ displayed below.
\begin{center}
\begin{tikzpicture}[scale=1.1]
	\scriptsize

	\fill[black!25] (-1,0) -- (1,0) arc (0:180:1cm) ; 
	\begin{scope}[black!50]
		\draw[<->] (-2.3,0) -- (2.3,0) node[anchor=west] {$x$} ;
		\draw[<->] (0,-2.3) -- (0,2.3) node[anchor=south west] {$y$} ;
	
		\foreach \x in {-2, -1, 1, 2}
		{{
			\draw [-] (\x, 0.1) -- (\x, -0.1) node[anchor=north] {$\x$} ;
			\draw [-] (-0.1, \x) -- (0.1, \x) node[anchor=west] {$\x$} ;
		}}
	\end{scope}

	\clip (-2.4, -2.4) rectangle (2.4, 2.4) ;
	
	\begin{scope}[line width=1pt]

		\foreach \x/\n in {-1/L, 1/R}
		{{
			\node[coordinate] at (\x, 0) (\n) {} ;
		}}

		\draw[dashed] (L) -- (R) ;
		\draw[solid] (R) arc (0:180:1cm) ;

		\foreach \x/\n in {-1/L, 1/R}
		{{
			\node[solid, draw, circle, minimum size=2.5pt, inner sep=0,fill=white] at (\x, 0) (\n) {} ;
		}}
		
	\end{scope}
	
\end{tikzpicture}
\end{center}
\vspace{2cm}
\end{enumerate}

\end{document}