--- a/hws/proof.tex	Tue Oct 31 12:52:36 2023 +0000
+++ b/hws/proof.tex	Sat Nov 04 18:28:09 2023 +0000
@@ -86,11 +86,11 @@
 
 \begin{itemize}
 \item First Case: $P(\ZERO)$ is $L(der\,c\,\ZERO) = Der\,c\,(L(\ZERO))$ (a). We have $der\,c\,\ZERO = \ZERO$ 
-and $L(\ZERO) = \ZERO$. We also have $Der\,c\,\ZERO = \ZERO$. Hence we have $\ZERO = \ZERO$ in (a). 
+and $L(\ZERO) = \emptyset$. We also have $Der\,c\,L(\ZERO) = L(\ZERO$). Hence we have $\emptyset = \emptyset$ in (a). 
 
 \item Second  Case: $P(\ONE)$ is $L(der\,c\,\ONE) = Der\,c\,(L(\ONE))$ (b). We have $der\,c\,\ONE = \ZERO$,
-$L(\ZERO) = \ZERO$ and $L(\ONE) = \{\texttt{""}\}$. We also have $Der\,c\,\{\texttt{""}\} = \ZERO$. Hence we have 
-$\ZERO = \ZERO$ in (b). 
+$L(\ZERO) = \ZERO$ and $L(\ONE) = \{\texttt{""}\}$. We also have $Der\,c\,\{\texttt{""}\} = \emptyset$. Hence we have 
+$\emptyset = \emptyset$ in (b). 
 
 \item Third  Case: $P(d)$ is $L(der\,c\,d) = Der\,c\,(L(d))$ (c). We need to treat the cases $d = c$ and $d \not= c$. 
 
@@ -99,8 +99,8 @@
 $\{\texttt{""}\} = \{\texttt{""}\}$ in (c). 
 
 $d \not=c$: We have $der\,c\,d = \ZERO$.
-We also have $Der\,c\,\{\texttt{"}d\texttt{"}\} = \ZERO$. Hence we have 
-$\ZERO = \ZERO$  in (c). 
+We also have $Der\,c\,\{\texttt{"}d\texttt{"}\} = \emptyset$. Hence we have 
+$\emptyset = \emptyset$  in (c). 
 \end{itemize}
 
 \noindent
@@ -192,7 +192,7 @@
 $Der\,c\,(L(r^*)) = Der\,c\,(L(r)^0 \cup \bigcup_{n \ge 1} L(r)^n) = (Der\,c\,L(r)^0) \cup Der\,c\,(\bigcup_{n \ge 1} L(r)^n)$
 \end{center}
 
-The first union ``disappears'' since $Der\,c\,(L(r)^0) = \ZERO$.
+The first union ``disappears'' since $Der\,c\,(L(r)^0) = \emptyset$.
 
 
 \end{document}