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\begin{document}


\large{

  \centerline{\bf CMSC 250 Final}

  \begin{enumerate}


    \item This  is  an open-everything exam. You can use anything except ask another person. 
{\bf Caution:} if you copy from
the web or elsewhere mindlessly you will probably get it wrong.

    \item  There are 3 problems which add up to 70 points.  Recall that you already did 30 points of
this midterm take home. 

\item 
The exam is Monday May 17, 8:00PM-10:15PM
unless you have contacted me to
make other arrangements. So the exam is 2 hours and 15 minutes

    \item For each question show all of your work and {\bf use LaTeX or write VERY NEATLY}.
{\bf Clearly indicate} your answers.  No credit for illegible answers.

    \item Please write out the following statement:
          {\it I pledge on my honor that I
          will not give or receive any unauthorized assistance on this
          examination\/}.   
\end{enumerate}

}


\newpage


\begin{enumerate}

\item
(20 points)
Prove that $11^{1/4}$ is NOT rational.
You must state and prove carefully any lemmas you use. 
(Do the problem in the style we did such proofs in class.
That is, do not do a prove that uses Unique Factorization, a technique
we did not do in class, so do not worry if you don't know what that means.) 

You can do this problem on this page and the next page.

\newpage

\hbox{\ } 

\newpage

\hbox{\ }

\item
(20 points- 5 points each) 

We will use the following definitions. 

{\bf Definition}
Let $f$ be a function. 

$f$ is {\it strictly increasing} if, for all $x<y$, $f(x) < f(y)$. 

$f$ is {\it monotone increasing} if, for all $x<y$, $f(x) \le f(y)$. 

$f$ is {\it strictly decreasing} if, for all $x<y$, $f(x) > f(y)$. 

$f$ is {\it monotone decreasing} if, for all $x<y$, $f(x) \ge f(y)$. 

And now FINALLY the problem 

For each of the following sets say whether it is 
(1) finite,
(2) countable, or
(3) uncountable
and PROVE it. 

(A set $X$ is {\bf countable} if there is a BIJECTION to $\N$.) 

\begin{enumerate}
\item
The set of all functions $f$ with 

domain $\N$ and codomain $\N$ that are strictly increasing. 

\item 
The set of all functions $f$ with 

domain $\N$ and codomain $\N$ that are strictly decreasing. 

\item
The set of all functions $f$ with 

domain $\N$ and codomain $\Z$ that are monotone increasing. 

\item 
The set of all functions $f$ with 

domain $\N$ and codomain $\Z$ that are monotone decreasing. 

\end{enumerate}

You can do this problem on this page and the next page.

\newpage

\hbox{\ } 

\newpage

\hbox{\ } 


\item
(30 points-10 points each) 
Let

$$A=\{1,\ldots,100\}$$

and 

$$B=\{1,\ldots,200\}.$$

For each of the following sets say how big they are. 

(You may use the following notations: 

{\bf factorial, e.g., 10!}

{\bf choose, e.g., $\binom{\bm{19}}{\bm{9}}$}

{\bf things like $\bm{5\times 6 \times \cdots \times 120}$.} 

) 



\begin{enumerate}
\item
The set of all functions with 

domain $A$ and co-domain $B$.


\item 
The set of all 1-1 functions with 

domain $A$ and co-domain $B$.


\item 
The set of all 1-1 functions with 

domain $B$ and co-domain $A$.

\newpage

\hbox{\ } 


\end{enumerate}

\end{enumerate}

\end{document}