## Closer property of Regular Language

1.  Regular Language closed under union.

If  L1 = Regular Language and L2 = Regular Language then L1 U L2 = Regular Language.

Example:

L1 = ${\mathrm{a}}^{\mathrm{n}}$ , L2 = ${\mathrm{a}}^{2\mathrm{n}}$  and L3 = ${\mathrm{a}}^{3\mathrm{n}+5}$  then

L= L1 U  L2 U L3 =    it is also regular language

2.  Regular Language closed under intersection.

If L1= Regular Language and L2 = Regular Language then L1∩ L2 = Regular Language.

Example:

L1 = Mod 2  and L2 = Mod 4 then

L = L1 ∩ L2 = Mod 2 ∩ Mod 4 = Mod 4 = Regular Language

3.  Regular Language closed under concatenation.

If L1 = Regular Language and L2 = Regular Language then L1 . L2 = Regular Language.

Example:

L1 = ${\mathrm{a}}^{\mathrm{n}}$ and L2 = ${\mathrm{a}}^{2\mathrm{n}}$ then L = L1. L2 = ${\mathrm{a}}^{\mathrm{n}}$ . ${\mathrm{a}}^{2\mathrm{n}}$ = Regular Language

4. Regular Language closed under Complement.

If  L = Regular Language then ${\mathrm{L}}^{\mathrm{C}}$ = Regular Language

Example:

L = {At least three a i.e  |a| ≥ 3 }  then,

${\mathrm{L}}^{\mathrm{C}}$ =  = {At most two a i.e. |a| < 3 or |a| ≤ 2}

{At most two a i.e. |a| < 3 or |a| ≤ 2} = Regular Language

5.  Regular Language closed under Kleene star operator.

If L = Regular Language then L*= Regular Language

Example:

L = ${\mathrm{a}}^{\mathrm{n}}$  then L* = ${\left({\mathrm{a}}^{\mathrm{n}}\right)}^{*}$ = Regular Language

6. Regular Language closed under Set Difference.

If L1 = Regular Language and L2 = Regular Language then,

= Regular Language.

Example:

L1 = Mod 2 = {2, 4, 6, 8, 10, 12, 14, 16, 20…} and L2 = Mod 4 = {4, 8, 12, 16, 20….}

Now L = L1 - L2 = {2, 6, 14, ….} = Regular Language

Regular Expression Closed Under:

Union (U), Intersection (), Concatenation (.), Complement, Kleene star operator (*), Set Difference, Symmetric Difference (⊕), NOR, XNOR, Reversal, Homomorphism, Inverse, Homomorphism.