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Word

Given a set $\Sigma$ , a word (or a string) over $\Sigma$ is a juxtaposition (variously called concatenation or multiplication) of a finite number of elements in $\Sigma$ . The juxtaposition is taken as an associative binary operation on $\Sigma$ . A word with zero number of elements is called an empty word, typically denoted by $\lambda$ or $\epsilon$ . The set of words over $\Sigma$ is denoted $\Sigma^*$ .

Examples.

If $\Sigma = \lbrace a,b,c,\ldots, x,y,z\rbrace$ , the English alphabet written in the lower case, then “good”, “mathematics”, “fasluiwh” are all words (without the double quotes) over $\Sigma$ , where as “PlanetMath” is not, because it contains upper case letters, which are not in $\Sigma$ .
Let $\Sigma=\lbrace 0,1,2,3,4,5,6,7,8,9,+,=\rbrace$ . Then “”, “”, “”, “”, “”, “”, “” are also words over $\Sigma$ .
The notion of words is used extensively in group theory. The juxtaposition here is the group multiplication, as the multiplication is associative. In other words, if are elements in then we can form the word $w=g_1g_2\cdots g_m\in G$ . For example, in the free group $\langle a,b\vert\quad\rangle$ a word could be the commutator $[a,b]=aba^{-1}b^{-1}$ .

Remarks

$\Sigma^*$ is a monoid with juxtaposition as the monoid multiplication, and $\lambda$ , the empty word, as the multiplicative identity.
For any element in $\Sigma$ , define $a^0=\lambda$ , and $a^{n+1}=a^na$ for non-negative integers . Then $a^{n+m}=a^na^m$ since juxtaposition is associative.
A word is called a subword of if , for some words and (may be empty words). If is a subword of , we also say that occurs in , or that contains . For example, “math” is a subword of “mathematics”. Given the equation , we call the triple an occurrence of in . The collection of occurrences of in is denoted . The number of occurrences of in defined as the cardinality of , and written $\vert u\vert _v$ . For example, the number of occurrences of subword in is , since

$\displaystyle O(a^3,a^3ba^5c)=\lbrace (\lambda,a^3,ba^5c),(a^3b,a^3,a^2c), (a^3ba, a^3, ac), (a^3ba^2,a^3,c)\rbrace.$

Generating Words using Rules

Some of the words in the second example above, such as “” and “”, do not make any mathematical sense. The way to define words that make sense is through a process called definition by recursion. First, we declare that certain words over $\Sigma$ are sensible. Then, we have a set of rules or a grammar that dictates how new sensible words can be formed from the old ones. Any word that can be formed from the old words by these rules in a finite number of steps is called sensible.

In the last example, we could declare that all symbols $0,1,\ldots,9$ are sensible words. To form new sensible words, we have the rules:

if do not contain either or , then is a sensible word;
if a two sensible words do not contain the symbol , then and are sensible words;
the only sensible words are the initially declared sensible words and those that can be formed by the previous two rules.

It is not hard to see based on the initially declared sensible words and the rules has one of the forms

$a_1+a_2+\cdots +a_n$
$a_1+a_2+\cdots + a_n = b_1+b_2+\cdots + b_m$ .

where

are words without any occurrence of

and

, over $\Sigma$ . As a result, we see that all words in the previous example are sensible (whether they are right or wrong), except “++231++” and “7=”, since they are not in any one of the forms specified above. Note that the third rule above ensures that “++231++” and “7=” are not sensible. Without it, we would be unable to say for sure if these words are sensible or not.

Generally, any collection of words is called a language. The collection of all sensible words described above is called the language generated by $0,\ldots,9$ under the rules above. In logic, one calls these sensible words well-formed formulas, or formulas or wff for short.