Regular set

Regular set --- регулярное множество.

Let [math]\displaystyle{ \Sigma }[/math] be an alphabet. Regular sets over the alphabet [math]\displaystyle{ \Sigma }[/math] are all sets that can be obtained by finitely many applications of the three following rules:

(1) the empty set [math]\displaystyle{ \emptyset }[/math] is a regular set over the alphabet [math]\displaystyle{ \Sigma }[/math],

(2) if [math]\displaystyle{ a }[/math] is in [math]\displaystyle{ \Sigma }[/math], then the singleton set [math]\displaystyle{ \{a\} }[/math] is a regular set over the alphabet [math]\displaystyle{ \Sigma }[/math],

(3) if [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] are regular sets over the alphabet [math]\displaystyle{ \Sigma }[/math], then [math]\displaystyle{ P\cup Q }[/math], [math]\displaystyle{ PQ }[/math] and [math]\displaystyle{ P^* }[/math] are also regular sets over the alphabet [math]\displaystyle{ \Sigma }[/math].

The following property holds for any language [math]\displaystyle{ L }[/math] over [math]\displaystyle{ \Sigma }[/math]:

[math]\displaystyle{ L }[/math] is a regular set iff [math]\displaystyle{ L }[/math] is a regular language.