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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.
Using \(\Sigma = \EngAlphabet\text{,}\) words
\begin{equation*}
\mathrm{math}, \; \mathrm{qwerty}, \; \mathrm{aabbccddijzuuu}
\end{equation*}
are examples of elements in \(\words{\Sigma}\text{.}\) 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
synonym for binary word
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 \(0\)
\(\emptyword\) the empty word
