regular languages closure properties

Question

What is difference between Kleene closure and infinite union in context of regular languages?

Answer 1

Kleene's closure is nothing but a special case of infinite union of regular language..And the regular language about which Kleens's Closure is about is Σ i.e. if Σ = {a,b} then the regular expression and hence regular language we are focussing on is (a + b)..

So Kleene's Closure for alphabet {a,b} can be written as :

ϵ + (a+b) + (a+b)² + (a+b)³..................................... which is nothing but (a + b)^* which is Kleene's Closure w.r.t alphabet {a,b}..

But this is not necessary that infinite union of any regular language will lead us to regular language as regular languages are not closed under infinite union, subset ,infinite intersection , infinite difference.

But for a language L = (a + b) as described above which is a special case (Kleene's Closure) in that case we are getting regular language (a + b)*..

I hope u have got the difference..