Q1:Prove that Regular Sets are NOT closed under infinite union. (A counterexample suffices).
Ans1: Consider the sets {0}, {01}, {0011}, etc. Each one is regular because it only contains one string. But the infinite union is the set {0i1i | i>=0} which we know is not regular. So the infinite union cannot be closed for regular languages.
Q2: What about infinite intersection?
Ans2: We know that
{0i1i | i>=0} = {0} U {01} U {0011} U ...,
Taking complements and applying DeMorgan's law gives us
{0i1i | i>=0}c = {0}c ^ {01}c ^ {0011}c ^ ...,
Where we are using U to deonte union and ^ to denote intersection. Recall the complement of a regular language is regular, and hence the complement of a not-regular language is not regular. So we can conclude that the left hand side of the equation is not-regular, and each term in the intersection is regular. Therefore infinite intersection does not preserve regularity.
Please explain what is infinite union/ infinite intersection and also explain the answer
This question is from aduni.org