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\headers{Math 303}{Homework 12}{Due: Fri. Apr 19, 2023}

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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.  Submissions must be stapled.  See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item \textit{Proofs involving sets.}  Prove the following.
\begin{enumerate}
	\item $\{6n\st n\in\ZZ\} = \{2n\st n\in\ZZ\} \cap \{3n\st n\in\ZZ\}$.
	\item $\{9^n\st n\in\QQ\} = \{3^n\st n\in\QQ\}$
	\item If $A$ and $B$ be sets, then $A\subseteq B$ if and only if $A\cap B = A$.
	\item $\bigcap_{x\in\RR} [3-x^2, 5+x^2] = [3,5]$.
\end{enumerate}

\item Suppose $B\ne \emptyset$ and $A\times B\subseteq B\times C$.  Prove that $A\subseteq C$.

\item Each of the following is true or false.  Decide which is the case and prove or disprove accordingly, using any method.
\begin{enumerate}
	\item For every natural number $n$, the integer $n^2+17n+17$ is prime.
	\item If $a,b,c\in\NN$ and $ab$, $bc$, and $ac$ all have the same parity, then $a$, $b$, and $c$ all have the same parity.
	\item If $A$ and $B$ are finite sets, then $|A\cup B| = |A| + |B|$.
	\item If $A$ and $B$ are sets, then $\PS(A) \cap \PS(B) = \PS(A\cap B)$.
	\item If $A$ and $B$ are sets, then $\PS(A) \cup \PS(B) = \PS(A\cup B)$.
	\item If $p$ and $q$ are prime numbers for which $p<q$, then $2p+q^2$ is odd.
\end{enumerate}

\item Suppose that $\alpha\in\RR$ and $0 < \alpha < 1$.  A magical cake has icing on one side.  A baker cuts the cake to make a slice with center angle $\alpha \cdot 2\pi$ (radians), flips the slice over (so the piece has icing face-down), and the slice magically reattaches to the rest of the cake.  The baker continues to make wedge slices with center angle $\alpha \cdot 2\pi$, proceeding counter-clockwise around the cake with each subsequent slice starting where the previous slice ended.  For which $\alpha\in\RR$ will this process eventually lead to the cake again having all its icing on the same side (up or down)?  For which $\alpha\in\RR$ will this process lead to the cake having all its icing facing down?

\end{enumerate}
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