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\textbf{Name:} \smallskip  \\
\textbf{Collaborators:} \smallskip \\ %%% List anyone with whom you discussed the problems
\textbf{Outside resources:} \smallskip \\ %%% List all resources used OTHER than the textbook, lecture notes, quizzes, worksheets, and previous homework assignments.
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\begin{center}
{
Math 2106, Foundations of Mathematical Proof \\
HW 4 --- Due Feb 27, 2017 (Monday)
}
\end{center}


\noindent Turn in the following problems from Hammack's book: \\
Chapter 10, Problems 8, 10, 16, 18.\\
Section 11.1, Problems 4, 8, 10, 12.\\
Section 11.2, Problems 2, 4, 6, 8, 10, 12.\\
Section 11.3, Problems 2, 4.\\
Section 11.4, Problems 4, 6, 8.\\

\noindent{\bf Additional exercises} (to be turned in)

\begin{enumerate}[label=A\arabic*]
\item Prove that for every real number $x > -1$ and every positive integer $n$, $(1+x)^n \geq 1 + nx$. 
\item Prove that for any integer $n\geq 3$, the sum of the interior angles of an $n$-sided polygon is $(n-2)\cdot 180^\circ$.  (You may assume that the statement is true for $n=3$.)
\item Suppose a plane is divided into some number of regions by a finite number of infinitely long straight lines.  Show that the regions can be colored with two colors so that no two adjacent regions have the same color.
\item Prove that you can cut a square into $n$ smaller squares (which may have different sizes), for all integers $n \geq 6$.
\item Let $R$ be the relation on $(\ZZ \times \ZZ )- \{(0,0)\}$ such that $(a,b) R (c,d)$ if and only if $ad = bc$.
\begin{enumerate}
\item Prove that $R$ is an equivalence relation.
\item What is the equivalence class of $(1,2)$ under this relation?
\item Give a description of all the equivalence classes.
\end{enumerate}
\item Let $H$ be a nonempty subset of $\ZZ$ and let $R = \{(a,b) \in \ZZ \times \ZZ : a - b \in H\}$.  Suppose $R$  is an equivalence relation on $\ZZ$.  Prove the following.
\begin{enumerate}
\item $0 \in H$
\item For any $a \in \ZZ$, if $a \in H$, then $-a \in H$.
\item For any $a,b \in \ZZ$, if $a,b \in H$, then $a+b \in H$.
\end{enumerate}
\end{enumerate}
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