Empty String - Formal Theory

Formal Theory

Formally, a string is a finite sequence of symbols such as letters or digits. The empty string is the extreme case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with λ or sometimes Λ or ε.

The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string.

The empty string has several properties:

• . The string length is zero.
• . The empty string is the identity element of the concatenation operation (which forms a free monoid on the alphabet Σ).
• . Reversal of the empty string produces the empty string.
• The empty string precedes any other string under lexicographical order, because it is the shortest of all strings.