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 (∞ ∩).**