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 |Σ n | is called cardinality. so |Σ3| has 8 cardinality.