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\centerline
{
\bf Solutions to Attendance Quiz \# 1 for Dr. Z.'s Number Theory
}
\bigskip
{\bf NAME:} (print!) $\underline
{\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\bigskip
{\bf E-MAIL ADDRESS:} (print!) 
$\underline
{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\bigskip

{\bf 1.} Using {\it unary} (no credit for other methods!),  
compute

$1111 \,\times  \, 111$

{\bf Ans.}
$$
111111111111 
$$

{\bf 2.} Write the integers $0$ through $4$ in 
von-Neumann notation

{\bf Ans.}
$$
0=\{\}
$$
$$
1= \{ \{ \}\}
$$
$$
2=\{\{\},\{ \{ \}\}\}
$$
$$
3=\{\{\},\{ \{ \}\},\{\{\},\{ \{ \}\}\}\}
$$
$$
4=
\{
\{\},
\{ \{ \}\},
\{\{\},\{ \{ \}\}\},
\{\{\},\{ \{ \}\},\{\{\},\{ \{ \}\}\}\}
\}
$$

{\bf 3.} List some {\bf natural} 
members of Frege's class representing three .

{\bf Ans}: $\{Frege, Goedel, Einstein\}$, $\{Kamala, Donald, Tim\}$ $\{Busch, Douglass, Livingston\}$
\end




