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\centerline{\bf Homework 3, MORALLY Due Feb 19}
\newcommand{\implies}{\Rightarrow}
\newcommand{\goes}{\Rightarrow}
\newcommand{\NSQ}{{\rm NSQ}}
\newcommand{\into}{{\rightarrow}}
\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
\newcommand{\lc}{\left\lceil}
\newcommand{\rc}{\right\rceil}
\newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\Z}{{\sf Z}}
\newcommand{\N}{{\sf N}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\R}{{\sf R}}
\newcommand{\Rpos}{{\sf R}^+}



\begin{enumerate}
\item 
(30 points--6 points each)
In this problem the domain is $\N$. 
\begin{enumerate} 
\item 
Express the following statements using quantifiers. 
There exist an $(x,y,z)$, with $x,y,z\ge 2$ all distinct,  such that $x^2+y^2=z^2$ .



\item 
Express the following statements using quantifies.
There exist an INFINITE NUMBER of $(x,y,z)$, with $x,y,z\ge 2$ and all distinct,  such that $x^2+y^2=z^2$ .
(This happens to be TRUE but you do not need that for this problem.) 


\item
Express the following statements using quantifies.
There is NO $x,y,z,n$ with $x,y,z\ge 2$ and $n\ge 3$  such that $x^n+y^n=z^n$.


\item
The statement in Part d is TRUE.
It is called {\it Fermat's Last Theorem}.
Look it up and write a paragraph about it including
who proved it, when, and how long it was open for. 

\end{enumerate}

\newpage

\item 
(30 points--15 points each)
In this problem the domain is $\R$. 

\begin{enumerate}
\item 
Express the following statements using quantifies.
Every polynomial with real coefficients, of degree 3, has a real solution. 

\item
The statement in Part a is TRUE!
Give a proof (it can be informal, this is NOT Math 410). 

\end{enumerate}

\newpage

\item
(40 points--8 points each)

For this problem we use the following standard terminology:

A domain is {\it dense} if $(\forall x,y)[x<y \implies (\exists z)[x<z<y]]$.

A domain has a {\it min element} if $(\exists x)(\forall y)[x\le y]$. 

A domain has a {\it max element} if $(\exists x)(\forall y)[y\le x]$. 

In this problem we list conditions on a domain. EITHER give a domain that satisfies
the conditions OR state that there is NO such domain (no proof required). 

\begin{enumerate}
\item 
$D$ is finite and dense. 


\item 
$D$ is finite and dense and has $\ge 2$ elements. 


\item
$D$ is infinite and not dense


\item
$D$ is infinite, has a min element, has a max element, and is dense. 



\item
$D$ is infinite, has a min element, has a max element, and is NOT dense. 


\end{enumerate} 

\end{enumerate} 

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