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\begin{center}
{
Math 2106-D, Foundations of Mathematical Proof \\
Homework 1 
\\ Due August 31, 2017
}
\end{center}


Do the following problems from Hammack:

\S1.1: 16, 26, 32, 
\quad\quad
\S1.2: 8, 14
\quad\quad
\S1.3: 8, 12, 14
\quad\quad
\S1.4: 8, 18, 20 
\quad\quad
\S1.5: 2
\quad\quad
\S1.6: 2
\quad\quad
\S1.7: 6
\quad\quad
\S2.1: 2, 4, 6, 14
\quad\quad
\S2.2: 8
\quad\quad
\S2.3: 2, 8
\quad\quad
\S2.4: 4
\quad\quad
\S2.5: 6, 8
\quad\quad
\S2.6: 6, 12

\vspace*{1in}

\noindent{\bf Additional exercises} 

\begin{enumerate}[label=A\arabic*]

\item The {\it Inclusion-Exclusion Principle}, which you saw on Worksheet 1, states that for finite sets $A,B$, 
\[
|A\cup B|=|A|+|B|-|A\cap B|
.
\]
Use this principle to determine the number of integers $n\in\{1, 2, \ldots, 2017\}$ which are not divisible by $2$ or $3$. 


\item 
For any $\alpha\in\mathbb R$, let $X_{\alpha}=\{(x,y,\alpha)\, |\, x,y\in\mathbb R,\ x^2+y^2=1-\alpha^2 \}\subseteq\mathbb R^3$. 


What geometric object is 
\[
\cup_{\alpha\in\mathbb R}X_{\alpha}
?
\]
Explain your reasoning and draw a sketch of the situation.
\end{enumerate}
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