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# 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* = 2N = 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  n |  is called cardinality. so |Σ3| has  8 cardinality.

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