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\documentclass{math215}

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\doclabel{Math 215: Homework 5}
\docauthor{Your name here}
\docdate{February 24, 2012}

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\begin{document}

\begin{proposition}{2.21} There are no integers $x$ such that
  $0<x<1$.
\end{proposition}

\begin{corollary}{2.22} Let $n\in\Ints$.  There are no integers $x$
  such that $n<x<n+1$.
\end{corollary}

\begin{proposition}{2.23} Let $m,n\in\Nats$.  If $n$ is divisible by $m$,
  then $m\le n$.
\end{proposition}

\begin{proposition}{2.24} For all $k\in\Nats$, $k^2+1>k$.
\end{proposition}

\begin{proposition}{2.27} For all $k\in\Ints$ such that $k\ge 2$, 
  $k^2<k^3$.
\end{proposition}

\begin{proposition}{2.33} Let $A$ be a nonempty subset of $Z$.
  Suppose for some $b\in\Ints$ that $b\le a$ for all $a\in A$.
  Then $A$ has a least element.
\end{proposition}
Hint: Proving the Well-Ordering Principle was hard work.  But proving this
proposition should not be.  Just reformulate it into a form where you
can apply the Well-Ordering Principle.
\begin{project}{2.35} Compute $\gcd(4,6)$, $\gcd(7,13)$, $\gcd(-4,10)$
  and $\gcd(-5,-15)$.  You do {\bf NOT} have to prove that you have
  found the $\gcd$.  But you do have to exhibit the integers $x$ and $y$
  in the definition of the $\gcd$.
\end{project}

\begin{project}{3.1} Express each of the following statements
  using quantifiers.
  \item[(i)] There exists a smallest natural number.
  \item[(ii)] There does not exist a smallest natural number.
  \item[(iii)] Every integer is the product of two integers.
  \item[(iv)] The equation $x^2-2y^2=3$ has an integer solution.
\end{project}

\begin{project}{3.7} Negate each of the following statements
  \item[(i)] Every cubic polynomial has a real root.
  \item[(ii)] $G$ is normal and $H$ is regular.
  \item[(iii)] $\exists ! 0$ such that $\forall x$, $x+0=x$
  \item[(iv)] The newspaper article was neither accurate nor entertaining.
  \item[(v)] If $\gcd(m,n)$ is odd then $m$ or $n$ is odd.
  \item[(vi)] $H/N$ is a normal subgroup of $G/N$ if and only if $H$ is a normal subgroup of $G$
  \item[(vii)] For each $\epsilon>0$, there exists $N\in\Nats$ such that
  for all $n\ge N$, $|a_n-L|<\epsilon$
\end{project}
\end{document}