# What are the properties of Regular expressions in TOC?

Data Structure AlgorithmsComputer ScienceComputers

A regular expression is basically a shorthand way of showing how a regular language is built from the base set of regular languages.

The symbols are identical which are used to construct the languages, and any given expression that has a language closely associated with it.

For each regular expression E, there is a regular language L(E).

There are some general equalities for the regular expressions.

Properties

All the properties held for any regular expressions R, E, F and can be verified by using properties of languages and sets.

The additive properties of regular expressions are as follows −

R + E = E + R
R + ∅ = ∅ + R = R
R + R = R
(R + E) + F = R + (E + F)

Product (·) properties

The product properties of regular expressions are as follows −

R∅ = ∅R = ∅
R∧ = ∧R = R
(RE)F = R(EF)

Distributive properties

The distributive properties of regular expressions are as follows −

R(E + F) = RE + RF
(R + E)F = RF + EF

Closure properties

The closure properties of regular expressions are as follows −

∅* = ∧ * = ∧
R* = R*R* = (R*)* = R + R*
R* = ∧ + RR* = (∧ + R)R*
RR* = R*R
R(ER)* = (RE)*R
(R + E)* = (R*E*)* = (R* + E*)* = R*(ER*)*

All the properties can be verified by using the properties of languages and sets.

Example 1

Show that

(∅ + a + b)* = a*(ba*)*

Using the properties above:
(∅ + a + b)* = (a + b)* (+ property)
= a*(ba*)* (closure property).

Example 2

Show that

∧ + ab + abab(ab)* = (ab)*

Using the properties above:
∧ + ab + abab(ab)* = ∧ + ab(∧ + ab(ab)*)
= ∧ + ab((ab)*) (using R* = ∧ + RR*)
= ∧ + ab(ab)*= (ab)* (using R* = ∧ + RR* again)