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Section 9.6 Alphabets and words
alphabet
any set can be considered an
alphabet
letters
the elements of an alphabet set
word
a finite-length, ordered list of letters
\(\words{\Sigma} \)
the set of words using
alphabet set
\(\Sigma \)
Example 9.6.2 . English is not a full set of words.
Words
\begin{equation*}
\mathrm{math}, \; \mathrm{qwerty}, \; \mathrm{aabbccddijzuuu}
\end{equation*}
are examples of elements in \(\words{\Sigma} \) for
\begin{equation*}
\Sigma = \EngAlphabet \text{.}
\end{equation*}
So, ignoring punctuation, hyphenation, and capitalization, the English language is a proper subset of \(\words{\Sigma} \text{.}\)
Example 9.6.3 . If digits are letters then numbers are words.
Using alphabet
\(\Sigma = \{ 0,\, 1,\, 2,\, \dotsc,\, 9 \} \text{,}\) then
\(\N \subsetneqq \words{\Sigma} \text{.}\)
Checkpoint 9.6.4 .
In computing science, a certain set of words is of particular importance.
binary word
a word using alphabet
\(\{0,1\} \)
binary string
length (of a word)
given
\(w \in \words{\Sigma} \text{,}\) the
length of \(w \) is the number of elements from
\(\Sigma \) used to form
\(w \text{,}\) counting repetition
\(\length{w} \)
length of the word
\(w \in \words{\Sigma} \)
Example 9.6.6 .
Using alphabet \(\Sigma = \EngAlphabet \text{,}\) we have
\begin{align*}
\length{\mathrm{qwerty}} \amp = 6 \text{,} \amp \length{\mathrm{aabab}} \amp = 5 \text{.}
\end{align*}
The concept of length allows us to identify some special subsets and a special element of \(\words{\Sigma} \text{.}\)
\(\words{\Sigma}_n \)
for
\(n \in \N \text{,}\) the subset of
\(\words{\Sigma} \) consisting of all words of length
\(n \)
empty word
given an alphabet
\(\Sigma \text{,}\) we always consider
\(\words{\Sigma} \) to contain a unique word of length
\(0 \)
\(\emptyword \)
