Cleanup

Advanced/De Bruijn/parts/1 words.tex

@@ -2,24 +2,28 @@
 
 \definition{}
 An \textit{alphabet} is a set of symbols. \par
-For example, $\{\texttt{0}, \texttt{1}\}$ is an alphabet of two symbols, \par
+For example, $\{\texttt{0}, \texttt{1}\}$ is an alphabet of two symbols,
 and $\{\texttt{a}, \texttt{b}, \texttt{c}\}$ is an alphabet of three.
 
 \definition{}
 A \textit{word} over an alphabet $A$ is a sequence of symbols in that alphabet. \par
 For example, $\texttt{00110}$ is a word over the alphabet $\{\texttt{0}, \texttt{1}\}$. \par
-We'll let $\varnothing$ denote the empty word, which exists over every alphabet.
+We'll let $\varnothing$ denote the empty word, which is a valid word over any alphabet.
 
 \definition{}
 Let $v$ and $w$ be words over the same alphabet. \par
 We say $v$ is a \textit{subword} of $w$ if $v$ is contained in $w$. \par
+\note{
+	In other words, $v$ is a subword of $w$ if we can construct $v$ \par
+	by removing a few characters from the start and end of $w$.
+}
 For example, \texttt{11} is a subword of \texttt{011}, but \texttt{00} is not.
 
 \definition{}
-Recall \ref{lockproblem}. From now on, we'll call this the \textit{$n$-subword problem}: \par
+Recall \ref{lockproblem}. Let's generalize this to the \textit{$n$-subword problem}: \par
 Given an alphabet $A$ and a positive integer $n$,
-we want a word over $A$ that contains all possible length-$n$ subwords. \par
-That shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}.
+we want a word over $A$ that contains all possible length-$n$ subwords.
+The shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}.
 
 
 
@@ -55,6 +59,12 @@ Find the following:
 \vfill
 \pagebreak
 
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 \problem{}<sbounds>
 Let $w$ be a word over an alphabet of size $k$. \par
 Prove the following:
@@ -87,6 +97,10 @@ Prove the following:
 
 
 
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 \definition{}
 Let $v$ and $w$ be words over the same alphabet. \par
 The word $vw$ is the word formed by writing $v$ after $w$. \par
@@ -145,6 +159,10 @@ We'll call this the \textit{Fibonacci word} of order $k$.
 
 
 
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 % C_k is called the "Champernowne word" of order k.
 \problem{}<cword>
 Let $C_k$ denote the word over the alphabet $\{\texttt{0}, \texttt{1}\}$ obtained by \par