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Regular Language is not closed under

Regular language may or may not closed under 

Subset (⊆), Super Set (⊇) , Proper Subset (⊂ ), Proper Superset (⊃), Infinite Union (∞ ∪) and 

Infinite Intersection (∞ ∩).

1. Not closed under Subset (⊆):

Example:

Let language L = a*b* is Regular Language. Now take a subset of a*b*. 

One subset of a*b* is,

 [Here DCFL means Deterministic Context - Free Language and REG means Regular Language]

Another subset of a*b* is:

So, Regular Language may or may not be closed under subset. 

Conclusion:  Regular Language is not closed under subset. 

2. Not closed under Super Set (⊇):

Example:

Let language L = Ф is Regular Language. Now take a superset of  L = Ф.

One superset of  Ф is,

We can take another superset of   Ф : 

So, Regular Language may or may not be closed under superset. 

Conclusion:  Regular Language is not closed under superset. 

3.  Not closed under Proper Subset (⊂ ):

Example

Let language L = a*b* is Regular Language. Now take a Proper Subset (⊂) of a*b*

One proper subset of a*b* is,

Another proper subset of a*b* is,

So, Regular Language may or may not be closed under Proper Subset (⊂).

Conclusion:  Regular Language not closed under Proper Subset (⊂).

4. Not closed under Proper  Superset (⊃) :

Example:

Let language L =Ф is Regular Language. Now take a proper superset of  L = Ф.

One proper superset of Ф is,  

Another proper superset of Ф is,   

So, Regular Language may or may not be closed under Proper Super Set (⊃). 

Conclusion:  Regular Language not closed under Proper  Super Set (⊃).

5. Not closed under Infinite Union (∞ ∪):

Example:

Let one regular language L = a*. Now we can write this regular language by infinite Union:

Now take another example of the infinite union of all regular language: 

Now from the above example, it is clear that the infinite union of all regular language is not regular.

So, Regular Language may or may not be closed under Infinite Union (∞ ∪).

Conclusion:  Regular Language is not closed under Infinite Union (∞ ∪).

6. Not closed under Infinite Intersection (∞ ∩): 

Like the infinite union, it is also not always closed under infinite intersection.

Example:

From the above example, it is clear that the infinite intersection of all regular language is not regular.

So, Regular Language not closed Infinite Intersection (∞ ∩).