Kleene Closure (*)  In TOC

Σ*  IS KNOWN AS Kleene Star (Kleene Closure). It gives always infinite language. We can apply Kleene closure on direct values of sigma.

For example: Σ^* = 2^N = N.

N means the STRING OF any LENGTH POSSIBLE  

The Kleene closure of various sigma values are given below

• (a)* = ε, a ,aa,aaa, aaaa,aaaaa…………….n
• (ab)* = ε, ab ,abab,ababab, abababab ………….n
• (a,b)* = (a*+b*)  = (a+b)* = ε, a, b, ab, aa, abb, abbbba….. n (any combination of a,b.)
• (a*+b+ c*) = b, ab, aabcc, ………… n (b is compulsory in string)

Difference Between Σ^* and Σ⁺

 Σ^* contain the Absalom {ε} sting  along with other N strins but Σ⁺ is just like the Σ^* but does not hold Absalom {ε} sting.

We can say

Σ^* = Σ⁺ + Σ⁰

So, Σ^* holds the identity(single) element called Absalom but Σ⁺ does not hold the identity element (Absalom)

Conclusion

Σ^* is a Universal Set.

Σ^* = Σ⁰ U Σ¹ U Σ² ……….  = {ε} U {a, b} U {aa, ab, ba, bb}  ………….   //infinite language.

Cardinality: Number of elements in a set, which is basically |Σ ⁿ |  is called cardinality. so |Σ³| has  8 cardinality.