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\def\HW{1}
\def\DUE{9/4/2018}
\def\NAME{YOUR NAME HERE}
\title[Homework \HW]{Homework \HW \\
Math 308\\
\small Due: \DUE}
\author{}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
\maketitle %Comment out this line, and uncomment the following for your assignment:
%\begin{flushright}
%\NAME\\
%Math 308\\
%Homework \HW\\
%\DUE
%\end{flushright}


\begin{enumerate}[1.]
\item What is the cardinality of the following sets?
\begin{enumerate}[(a)]
\item $\{1, 2, a, x, \text{apple}\}$
\item $\{1, 2, \{a, x\}, \text{apple}\}$
\item $\{\{1, 2, \{a, x\}\}, \text{apple}\}$
\item $\{\{1, 2, \{a, x\}, \text{apple}\}\}$
\item $\emptyset$
\item $\{\emptyset, \{\emptyset, \{\emptyset\}\}\}$
\end{enumerate}
\item Describe each of the following sets in the format $\Big\{\underbrace{\qquad\qquad}_{\text{ objects }} \Big| \underbrace{\qquad\qquad}_{\text{ conditions }}\Big\}$:
\begin{enumerate}[(a)]
\item $A = \{0,2,4,6,8,10,12,14,16,18,\dots\}$;
\item $B = \{3,5,6,9,10,12,15,18,\dots\}$;
\item $D= \{1,1/2,1/3,1/4,1/5,\dots\}$.
\end{enumerate}
\item  Find $X \cup Y$ and $X \cap Y$ for the following:
\begin{enumerate}[(a)]
 \item $X = \{0,2,5\}$ and $Y = \{1,3,5,7,9\}$;
 \item $X = \{-1,2,17\}$ and $Y = \{1,3,5,7,9\}$;
\item $X=\QQ$ and $Y=\{0,1, \pi, 1/2\}$;
\item $X = \{x \in \RR ~|~ x > 7\}$ and $Y = \{y \in \NN ~|~ x > 5 \}$.
\end{enumerate}
\item Find $\NN \cap \NN$, $\NN \cap \emptyset$, and $\NN \cap \QQ$.
\item Let $X=\{x\in \ZZ ~|~ 0 \le x \le 10 \}$, and $A$ and $B$ be the subsets of $X$ given by $A = \{0, 2, 3, 9\}$ and $B = \{x \in X ~|~ 5 \le x <9\}$. Find $A \cap B$, $A \cup B$, $A - B$, $B - A$, $A \times B$, $A^c$, and $B^c$.
\item In each part give an example of sets $A$, $B$, $C$ that simultaneously satisfy
the stated conditions:
\begin{enumerate}[(a)]
 \item  $A \in B$, $B \in C$, $A \notin C$;
 \item  $A \in B$, $B\in C$, $A\in C$;
 \item $A\notin B$, $B\in C$, $A\in C$.
 \end{enumerate}
\item  Suppose that $A$, $B$ and $C$ are subsets of a set $X$. Use examples to investigate the following sets. In each case, make a hypothesis.
\begin{enumerate}[(a)]
\item $(A \cap B) \cup (A \cap C)$ versus $A \cap (B \cup C)$;
\item $(A \cup B)^c$ versus $A^c \cap B^c$; 
\item $(A \cap B)^c$ versus $A^c \cap B^c$.
\end{enumerate}
\item Let $S$ and $T$ be sets of three elements. How many functions are there
from $S$ to $T$? 
\item Which of the following functions are injective? Which are surjective?
\begin{enumerate}[(a)]
\item The constant function $f: \RR \to \RR$ defined by $f(x) = 3$.
\item The constant function $f: \RR \to \{3\}$ defined by $f(x) = 3$.
\item The identity function $f: \RR \to \RR$ defined by $f(x) = x$.
\item The function $f: \ZZ \to \RR$  defined by $f(x) = x$.
\item The cardinality function, given by  $\displaystyle
	||: \text{Finite sets} \to \ZZ_{\geq 0}$.
\end{enumerate}
\item Analyze how you approached the reading of Chapter 1 of ``How to Think Like a Mathematician''.
\begin{enumerate}[(a)]
\item If you had not met the material in this chapter before, then did you attempt to
understand everything?
\item If you had encountered the material before, did you check to see that the author had not made any
 mistakes? Did you compare how the books definitions agreed with or differed from those your had learned before?
\end{enumerate}



\end{enumerate}


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