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{\bf Math 214 -- Foundations of Mathematics \quad 
Homework 1 \hfill Your name}

\bigskip
Solve the following problems. Six points for each problem. 

\medskip
\begin{enumerate}
\item Write each of the following sets as specified.
  \begin{enumerate}
  \item[(a)] List the elements in the set $A=\{n\in \N: n^3<1000\}$. 
  \item[(b)] Describe the set $B = \{-2,-1,0,1,2,3,4\}$ using the notation
    
    $B = \{n : p(n)\}$, 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}
  \item[(a)] Find $\cP(\cP(\{a\}))$ and its cardinality.
  \item[(b)] Give an example of a set $S$ such that $S\in \cP(\N)$ and $|S|=5$.
  \item[(c)] Give an example of a set $S$ such that $S\subseteq \cP(\N)$ and $|S|=5$.

  \end{enumerate}

\item The following problems involve set operations.
  \begin{enumerate}
  \item[(a)]  Give an example of three two element sets 
  $A, B$, and $C$ such that $B\not=C$ but $B-A=C-A$ is non-empty.
  \item[(b)] Let $A=\{\emptyset, \{\emptyset\}\}$.
  Find $\cP(A) - A$.
  \end{enumerate}

\item
For a real number $r$, define $S_r$ to be the interval $[r-1, r+2)$. Let $A=\{1, 3, 4\}$. 

  \item[(a)]  List the intervals $S_r$ for $r \in A$.
  
  \item[(b)] Determine $\cup_{\alpha\in A} S_{\alpha}$ and $\cap_{\alpha\in A}S_{\alpha}$. 
  
  \item[(c)] (Extra 3 points) Let $B = [0,1]$. Determine $\cap_{r\in B}S_{r}$ and 
  $\cup_{r \in B} S_r$ with explanation.

   
\item For two sets $A$ and $B$, let $A \times B = \{(a,b): a \in A, b \in B\}$ 
be their Cartesian product.
  \begin{enumerate}
  \item[(a)] Let $A=\{a, b\}$. Determine $A\times \cP(A)$. 
  \item[(b)] Let $A = \{0,1\}$ and $B = [0,2] \cap [1,3]$. 
  Describe geometrically the set $A \times B$ in $\R^2 = \R \times \R$.
  (The set is line segment, two line segments, a circular arc, or ???)
  \item[(c)] 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
(Extra credit.)
If a set $A$ has $n$ elements, show that $\cP(A)$ has $2^n$ elements.
\end{enumerate}
\end{document}


