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Finished De Bruijn sections

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2024-03-21 12:03:37 -07:00
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6
Advanced/De Bruijn/parts/1 words.tex
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@ -175,12 +175,12 @@ For example, $C_1 = \texttt{0}$, $C_2 = \texttt{011011}$, and $C_3 = \texttt{011
\item If $w$ starts with a \texttt{1}, $w$ must appear in $C_k$ by construction.
\item If $w$ does starts with a \texttt{0} and contains a \texttt{1}, $w$ has the form
$\texttt{0}^x\texttt{1}[..\texttt{y}..]$ \par
$\texttt{0}^x\texttt{1}\overline{\texttt{y}}$ \par
\note{
That is, $x$ copies of \texttt{0} followed by a \texttt{1}, followed by \par
an arbitrary sequence $\texttt{y}$ with length $(k-x-1)$.
an arbitrary sequence $\overline{\texttt{y}}$ with length $(k-x-1)$.
} \par
Now consider the word $\texttt{1}[..\texttt{y}..]\texttt{0}^x\texttt{1}[..\texttt{y}..]\texttt{0}^{(x-1)}\texttt{1}$. \par
Now consider the word $\texttt{1}\overline{\texttt{y}}\texttt{0}^x\texttt{1}\overline{\texttt{y}}\texttt{0}^{(x-1)}\texttt{1}$. \par
This is the concatenation of two consecutive binary numbers with $k$ digits, and thus appears in $C_k$.
$w$ is a subword of this word, and therefore also appears in $C_k$.
\end{itemize}