Suppose we have a string on alphabet . We can then represent the string as , where for all ( ), (this means that each must be a “letter” from the alphabet). Then, the length of is . The length of a string is represented as .

For example, if our alphabet is then the length of the string is , since the string breaks down as follows: , , , . So, our is and therefore . Although you may think that is two separate symbols, our chosen alphabet in fact classifies it as a single symbol.

A “special case” occurs when , i.e. it does not have any symbols in it. This string is called the empty string. Instead of saying , we use to represent the empty string: . This is similar to the practice of using to represent a space, even though a space is really blank.

If your alphabet contains as a symbol, then you must use something else to denote the empty string.

Suppose you also have a string on the same alphabet as . We turn into just as before, and similarly . We say is equal to if and only if both , and for every , .

For example, suppose and , both strings on alphabet . These strings are not equal because the second symbols do not match.