##### Closure Property of Regular language

**Closure Properties of Regular Languages:-**

Recall a closure property is a statement that a certain operation on languages, when applied to languages in a class (e.g., the regular languages), produces a result that is also in that class.

For regular languages, we can use any of its representations to prove a closure property.

**1. **** Closure Under Union
If L and M are regular languages, so is L ∪ M.
Proof:** Let L and M be the languages of regular expressions R and S, respectively.

Then R+S is a regular expression whose language is L

**∪**M.

__2 .Closure Under Concatenation and Kleene Closure__

Proof: Same idea: RS is a regular expression whose language is LM.

R* is a regular expression whose language is L*.

__3. Closure Under Intersection__

If L and M are regular languages, then so is L ∩ M.

**Proof:**Let A and B be DFA’s whose languages are L and M, respectively.

Construct C, the product automaton of A and B.

Make the final states of C be the pairs consisting of final states of both A and B.

**4.Closure Under Difference**

If L and M are regular languages, then so is L – M = strings in L but not M.

Proof: Let A and B be DFA’s whose languages are L and M, respectively.

Construct C, the product automaton of A and B.Make the final states of C be the pairs where A-state is final but B-state is not.

__5.Closure Under Complementation__

The complement of a language L (with respect to an alphabet Σ such that Σ*

contains L) is Σ* – L Since Σ* is surely regular, the complement of a regular language is always regular.

**6. Closure Under Reversal **

Given language L, LR is the set of strings whose reversal is in L.

Example: L = {0, 01, 100};

L^{r} = {0, 10, 001}.

**7. Closure Under Inverse Homomorphism**

** 8**. **Closure Under Homomorphism.**

So Regular language is closed under almost everything except **Infinite UNION.**