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\begin{center}\textbf{Problem Set 8 \hspace{.7in} Foundations \hspace{.7in} Due 4/8/15} \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 (From Devlin, $Sets, Functions, and \ Logic$) The Hilbert Hotel, named after the famous German mathematician (and set theory pioneer) David Hilbert, has an inifinte number of rooms.  The rooms are numbered 1, 2, 3, etc.  One eveing --- it was a Super Bowl weekend --- all the rooms were booked.  But late that night, an unexpected VIP turned up and asked for a room.  Fortunately, the desk clerk had been reading Keith Devlin's book $Sets, Functions, and \ Logic$.  With a bit of rearranging, the clerk was able to arrange matters so that the VIP got a room and so did everyone else.  How did he do it?

\item (From Devlin, $Sets, Functions, and \ Logic$) The following evening the rooms were all still booked, but an infinite number of new guests arrived, all without reservations.  The clever desk clerk was again able to arrange matters so that all the new arrivals got rooms and so too did the guests who had advance reservations.  What did the clerk do this second night?

\item Consider the relation $\sim$ on $\mathbb{Z}$ given by $x$ $\sim$ $y$ iff 6 divides $x-y$.  Give 3 elements of $\mathbb{Z}$ that are related to 5 and 3 elements of $\mathbb{Z}$ that are not related to 5.

\item Determine whether the relation from \#3 is reflexive, symmetric, and transitive.

\item Consider the relation $\asymp$ on the set $R(I)$ of integrable functions from [0,1] to $\mathbb{R}$ defined by $f$ $\asymp$ $g$ iff $\int_0^1 f(x) dx = \int_0^1 g(x) dx$.  Give 3 elements of $R(I)$ that are related to 5 and 3 elements of $R(I)$ that are not related to 5.

\item Determine whether the relation from \#5 is reflexive, symmetric, and transitive.

\item Let $S = \{1, 2, 3, 4\}$.  Then $R =\{(1,1), (1,3), (3,1), (2,2), (2,4), (3,3), (4,2), (4,4)\}$ is a relation on $S$.  Write the equivalence classes of $S$ associated with $R$.

\item Let $S = \{1, 2, 3, 4\}$.  Suppose that $\sim$ is an equivalence relation with $1 \sim 2$ and $2 \sim 4$.  What are the possible partitions associated with $\sim$?

\item Do the BinaryGateway on WeBWorK, available via 
\begin{center} http:webwork.coe.edu/webwork2/MTH-215/BinaryGateway/ . \end{center}

\end{enumerate}

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