Kleene Closure (*)  In TOC

Kleene Closure (*) gives an infinite set of all possible strings of any length, including epsilon (ε), over input values of sigma (Σ). It is also known as Kleene Star and denoted by Σ*. The following is a mathematical notation of Kleene Clouser.

Σ^* = Σ⁰ U Σ¹ U Σ² ………. .  

Where Σ⁰ represents epsilon (ε)  and gives infinite language.

Example: Suppose Σ input value is “(0,1)”, applying Kleene Closure (*) over Σ input which gives the following result

{0,1}* =( 0∪1)* = {ε,0,1,11,00, 01,10 ,0101,1010, …………., n}    // where n is infinity 

Result  (0,1)* will give all possible combinations of “0” and “1”  of any length, including epsilon (ε). So, we can say  Kleene Closure always gives infinite language. 

Examples of Kleene Closure

The following are various examples of Kleene Closure over input values of sigma (Σ).

Example 01: Kleene Closure applied to the empty set

∅* = {ε}

Example 02: Kleene Closure applied to a single character

{a}* = {ε, a, aa, aaa, aaaa, ……………., }     // all combination of “a” with any length 

Example 03: Kleene Closure applied to the set of characters

{x, y, z}* = { ε, x, y, z, xx, xy, xz, yx, yy, yz, za, zb, zz, xxx, xxy, …………}.

Example 04: Kleene Closure applied to a single string

{ab}* = {ε, ab ,abab,ababab, abababab, ………….n}    

Example 05: Kleene Closure applied to the set of strings

{01,0}* = {ε, “01”, “0”, “0101”, “010”, “001”, “00”, “010101”, “01010”, “01001”, “0100”, “00101”, “0010”, “0001”, “000”, …}

Example 06: Kleene Closure applied to some Special case 

{a*bc*} = {a*+b+ c*} = {b, ab, aabcc, ………… n}      // where “b” is compulsory in each string

Positive Closure

Positive Closure is similar to Keleene Closure except epsilon (ε). It also gives an infinite set of all possible strings of any length, excluding epsilon (ε)  over input values of sigma (Σ). Σ+ denotes it. The following is the mathematical notation of Positive Closure.

Σ⁺ = Σ^*– {ε}

Suppose Σ input value is “(0,1)”, applying Positive Closure (⁺) over Σ input which gives the following result

{0,1}⁺ = (0∪1)⁺ = {0,1,11,00, 01,10 ,0101,1010, …………., n}    // where n is infinity 

Result  (0,1)⁺ will give all possible combinations of “0” and “1”  of any length excluding epsilon (ε). So, we can say  Positive Closure always gives infinite language. 

Examples of Positive Closure

All examples of positive Closure are similar to Kleene closure except epsilon (ε). The following are the various examples of Positive Closure over sigma (Σ) input values.

Example 01: Kleene Closure applied to  an empty set, the result is an empty set

∅⁺ = {}  

Example 02: Kleene Closure applied to single character

{a}⁺ = {a, aa, aaa, aaaa, …………….}     // all combination of “a” with any length 

Difference Between Σ^* and Σ⁺

 Σ^* contains the epsilon (ε) sting along with other N strings, but Σ⁺ is just like the Σ^* but does not hold the epsilon (ε) sting.

We can say

Σ^* = Σ⁺ + Σ⁰ . Where Σ⁰ represent epsilon (ε)

So, Σ^* holds the identity(single) element called Absalon, but Σ⁺ does not have the identity element called epsilon (ε)

Tips about Positive and Kleene Clouser

Σ^* is a Universal Set.

Cardinality: The number of elements in a set, basically |Σ ⁿ |,  is called cardinality. So, |Σ³| has  eight cardinality.