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\headers{Math303}{Homework 4}{Due: Fri. Sep 18, 2026}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Unless the problem indicates otherwise, all problems require some justification; a correct answer without supporting reasoning is not sufficient.  See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item {} [BP 1.8.\{6,8,10\}] Recall that for real numbers $a$ and $b$, we have $[a,b] = \{x\in\RR\st a\le x\le b\}$.  Find the following sets.
\begin{multicols}{2}
\begin{enumerate}
	\item $\bigcup_{i\in\NN} [0,i+1]$
	\item $\bigcap_{i\in\NN} [0,i+1]$
	\item $\bigcup_{\alpha\in\RR} (\{\alpha\} \times [0,1])$
	\item $\bigcap_{\alpha\in\RR} (\{\alpha\} \times [0,1])$
	\item $\bigcup_{x\in [0,1]} ([x,1] \times [0,x^2])$
	\item $\bigcap_{x\in [0,1]} ([x,1] \times [0,x^2])$
\end{enumerate}
\end{multicols}

\item For each $k\in \NN$, let $A_k = \{kn\st n\in\ZZ\}$.  Find the following sets.
\begin{multicols}{2}
\begin{enumerate}
	\item The examples $A_1$, $A_2$, and $A_3$.
	\item $\bigcup_{k=1}^3 A_k$
	\item $\bigcup_{k=2}^4 A_k$
	\item $\bigcap_{k=1}^3 A_k$
	\item $\bigcap_{k=1}^{\infty} A_k$
	\item $\bigcup_{k\in I} A_k$ where $I=\{3,5,7,9,11,\ldots\}$.
\end{enumerate}
\end{multicols}

\item {} [BP 2.1,evens]  Decide whether or not the following are statements.  In the case of a statement, say if it is true or false, if possible.
\begin{enumerate}
\item Every even integer is a real number.
\item Sets $\ZZ$ and $\NN$.
\item Some sets are finite.
\item $\NN\not\in\PS(\NN)$.
\item $(\RR\times \NN) \cap (\NN\times\RR) = \NN \times \NN$.
\item If the integer $x$ is a multiple of $7$, then it is divisible by $7$.
\item Call me Ishmael.
\item If $x$ is an integer, then $x+y$ is also an integer.
\end{enumerate}

\item {} [BP 2.2] Express each statement or open sentence in one of the forms $P\land Q$, $P \lor Q$, or $\sim P$.  Be sure to state explicitly what $P$ and $Q$ mean in each part.
\begin{enumerate}
	\item The matrix $A$ is not invertible.
	\item $x < y$
	\item The quiz is scheduled for Wednesday or Friday.
	\item $x\in A \cap B$
\end{enumerate}

\item {} An infinite series of nested circles and squares are drawn, all sharing a common center point.  The outermost circle has radius 1.  The space between each circle and the square it circumscribes is shaded.  What is the total area of the shaded regions?
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	\foreach \s in {1.0, 0.70711, 0.5, 0.35355}
	{{
		\begin{scope}[scale=\s]
			\draw[fill=black!25] circle (1) ;
			\draw[fill=white] (-0.70711,-0.70711) rectangle (0.70711,0.70711) ;
		\end{scope}
	}}
	
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\end{center}


\end{enumerate}

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