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\date{Fall 2012}


\begin{document} 
\begin{center}
{\bf Math 214 -- Foundations of Mathematics\\
Homework 1}\\
Due: noon September 7, 2012.
\end{center}
Solve the following problems. Please remember to use complete sentences
and good grammar. Each problem is 4 points.



\begin{enumerate}
\item Write each of the following sets as specified.
  \begin{enumerate}[(a)]
  \item List the elements in the set $A=\{n\in \N: n^3<100\}$. 
  \item Describe the set $B = \{-3,-2,-1,0,1,2,3\}$ using the notation
    $\{n : p(n)\}$, where $p(n)$ specifies the property of element $n$.
  \end{enumerate}


\item Recall that for a set $A$, $\cP(A)$ denotes the power set of $A$.  
  \begin{enumerate}[(a)]
  \item Find $\cP(\cP(\{1\}))$ and its cardinality.
  \item Give examples of a set $S$ such that $S\subseteq \cP(\N)$ and $|S|=5$.
  \item Give examples of a set $S$ such that $S\in \cP(\N)$ and $|S|=5$.
  \end{enumerate}

\item The following problems involve set operations.
  \begin{enumerate}[(a)]
  \item  Given an example of three sets $A, B$, and $C$ such that $B\not=C$ but $B-A=C-A$.
  \item Let $A=\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}$. Find $\{\emptyset, \{\emptyset\}\}\cap A$. 
  \end{enumerate}


\item For a real number $r$, define $S_r$ to be the interval $[r-1, r+2]$. Let $A=\{1, 3, 4\}$. Determine $\cup_{\alpha\in A} S_{\alpha}$ and $\cap_{\alpha\in A}S_{\alpha}$. 

\item For two sets $A$ and $B$, recall that $A \times B$ is the
  Cartesian product of $A$ and $B$.
  \begin{enumerate}[(a)]
  \item Let $A=\{a, b\}$. Determine $A\times \cP(A)$. 
  \item Let $A = \{0,1\}$ and $B = [0,2] \cap [1,3]$. Depict the set $A \times B$ in $\R^2$.
  \item Let $A = \{0,1\}$, $B = (0,1) \cap A$ and $C = \R$. What is $A\times B \times C$?
  \end{enumerate}
  
\item Determine all different partitions of the set $\{1,2,3\}$. 


\item Let $P$: $15$ is odd, $Q$: $21$ is prime and $R$: $\frac{1}{2} \in \N$.   State each of the following in words, and determine whether they are true or false. 
  \begin{enumerate}[(a)]
  \item $P\vee Q$ 
  \item $P\wedge Q$
  \item $(\sim{P})\vee Q$
  \item $P\wedge (\sim Q \vee R)$
  \end{enumerate}
  

  
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