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\begin{flushright}
Introduction to Abstract Mathematics \\
MATH 310\\
Homework 2\\
Due Friday 10 February
\end{flushright}

\noindent
Please show all your work.
\begin{exer}
Let $U$ be a universal set and let $A$ and $B$ be two subsets of $U$ such that $A \neq B$ and $A\cap B \neq \emptyset$. Draw a Venn diagram for each of the following sets.
\begin{enumerate}[(a)]
\item $\overline{A\cup B}$
\item $\overline{A}\cap \overline{B}$
\item $\overline{A\cap B}$
\item $\overline{A}\cup \overline{B}$.
\end{enumerate}
\end{exer}

\begin{exer}
Suppose sets $A$ and $B$ are in a universal set $U$. Draw  Venn diagrams for $\overline{A\cup B}$ and $\overline{A}\cap\overline{B}$. Based on your drawings, what can you conclude about $\overline{A\cup B}$ and $\overline{A}\cap\overline{B}$?
\end{exer}

\begin{exer}
Let $A =\{x\in \bbr\mid |x-1|\leq 2\}$, $B = \{x\in \bbr \mid |x|\geq 1\}$, and $C = \{x\in \bbr \mid |x+2|\leq 3\}$.
\begin{enumerate}[(a)]
\item Express $A$, $B$, and $C$ using interval notation.
\item Determine each of the following sets using interval notation: $A\cup B$, $A\cap B$, $B\cap  C$, and $B\setminus C$.
\end{enumerate}
\end{exer}

\begin{exer}
For a real number $r$, define $A_r = \{r^2\}$, $B_r$ as the closed interval $[r-1,r+1]$ and $C_r$ as the interval $(r,\infty)$. For $S = \{1,2,4\}$, determine:
\begin{enumerate}[(a)]
\item $\bigcup_{\alpha\in S} A_{\alpha}$ and $\bigcap_{\alpha\in S} A_{\alpha}$
\item $\bigcup_{\alpha\in S} B_{\alpha}$ and $\bigcap_{\alpha\in S} B_{\alpha}$
\item $\bigcup_{\alpha\in S} C_{\alpha}$ and $\bigcap_{\alpha\in S} C_{\alpha}$.
\end{enumerate}
\end{exer}

\begin{exer}
For each of the following, find an indexed collection $\{A_n\}_{n\in \mathbb{N}}$ of distinct sets (that is, no two sets are equal) satisfying the given conditions.
\begin{enumerate}[(a)]
\item $\bigcap_{n=1}^{\infty} A_n = \{0\}$ and $\bigcup_{n=1}^{\infty} A_n = [0,1]$
\item $\bigcap_{n=1}^{\infty} A_n = \{-1,0,1\}$ and $\bigcup_{n=1}^{\infty} A_n = \bbz$.
\end{enumerate}
\end{exer}

\begin{exer}
For $n\in \mathbb{N}$, let $A_n = (-\frac{1}{n},2-\frac{1}{n})$. Determine $\bigcup_{n\in \mathbb{N}} A_n$ and $\bigcap_{n\in \mathbb{N}}A_n$.
\end{exer}

\begin{exer}
Let $A = \{1,2,3,4,5,6\}$. Give an example of a partition $\mathcal{S}$ of $A$ such that $|\mathcal{S}| = 3$.
\end{exer}

\begin{exer}
Give an example of a partition of $\mathbb{C}$ into two subsets.
\end{exer}

\begin{exer}
For $A = \{\emptyset,\{\emptyset\}\}$, determine $A\times \mathcal{P}(A)$ and $|\mathcal{P}(A\times\mathcal{P}(A))|$. 
\end{exer}

\begin{exer}
Let $I$ denote the interval $[0,\infty)$. For each $r\in I$, define
\begin{align*}
	A_r &= \{(x,y)\in \mathbb{R}\times \bbr \mid x^2 + y^2 = r^2\}\\
	B_r &= \{(x,y)\in \bbr \times \bbr \mid x^2+y^2 \leq r^2\}\\
	C_r &= \{(x,y) \in \bbr\times \bbr \mid x^2+y^2 > r^2\}
\end{align*}
\begin{enumerate}[(a)]
\item Determine $\bigcup_{r\in I} A_r$ and $\bigcap_{r\in I} A_r$.
\item Determine $\bigcup_{r\in I} B_r$ and $\bigcap_{r\in I} B_r$.
\item Determine $\bigcup_{r\in I} C_r$ and $\bigcap_{r\in I} C_r$.
\end{enumerate}
\end{exer}

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