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\begin{document}
\headers{Math303}{Quiz 2}{Wed Sep 9, 2026}

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

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\newcommand{\PS}{\mathcal{P}}

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

\begin{enumerate}

\item\ppoints{4}{1} Let $A=\{1, 2\}$, $B=\emptyset$, and $C=\{3\}$.  Find the following sets.
\begin{multicols}{2}
\newcommand{\myspc}{\vspace{3cm}}
\begin{enumerate}
	\item $\PS(A)$
	\myspc\\~
	\item $A^2$
	\myspc\\~
	
	\columnbreak
	\item $\PS(B^0)$
	\myspc\\~
	\item $A\times C\times A$
	\myspc\\~
\end{enumerate}
\end{multicols}

\item\points{1} Is it true that $\RR^3 = \RR^2 \times \RR$?  Explain why or why not.
\vfill

\item\points{1} Let $D=\{1,2,3,4\}$.  Express $\{X\subseteq D\st |D| = 2\}$ by listing the elements between braces.
\vfill


\newpage
\item\points{1} Draw a picture of $[1,2] \times (-1,3]$ in the plane.  Use solid lines to indicate boundaries in the set and dashed lines to indicate boundaries outside the set.
\vfill

\item\ppointsf{6}{0.5} True/False.  Write the entire word true or false.  No justification necessary.  We define:
\begin{align*}
A &= \{1,2,\{1,2\}, \{\emptyset\}\} & B &= \{1,\{1, 1\}, \emptyset\} & C &= \{\{1\}\}  &  D &= \{\emptyset\}
\end{align*}
\begin{multicols}{3}
\newcommand{\myspc}{\vspace{3cm}}
\begin{enumerate}
	\item $1 \in A$
	\myspc\\~
	\item $C \in A$
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	\columnbreak
	\item $1 \subseteq B$
	\myspc\\~
	\item $C \subseteq B$
	\myspc\\~
	
	\columnbreak
	\item $D \in B$
	\myspc\\~
	\item $D \subseteq C$
	\myspc\\~
\end{enumerate}
\end{multicols}


\end{enumerate}

\end{document}