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CLOSURE PROPERTIES OF REGULAR LANGUAGE

The set of regular language is closed under the union operation.Finite union is closed for regular language but infinite union is not closed.
The set of regular language is closed under the intersection operation and regular languages are closed under finite intersection but not under infinite intersection.
The set of regular languages is closed under the difference operation , complement operation , concatenation operation , kleene closure operation , positive closure operation and prefix operation as well as suffix operation.
    Init (L) = Prefix (L) = {u | uv is in regular language L }
    suffix (L) = { v | uv is in regular language L }



The set of regular language is closed under the reversal operation and half of a regular language is also regular :Let L is a regular language.Then Half (L) = { u | uv ɛ L  and | u | = | v | }.
One third of a regular language is also regular : Let > is a regular language. Then onethird (L) = { u | uvw ɛ L and | u | = | v | = | w | }. 
Quotient of two regular Language is also regular : L1 / L2 = { x | xy ɛ L1 for some y ɛ L2 }.
Subsequence of a regular language is also regular language : Subsequence (L) = { w|w is obtained by removing symbols from anywhere of a string in L }.
Sub-word of a regular language is also regular : sub-word (L) = { v | for some u and for some w , uvw is in L }. Homomorphism of two language is also regular and inverse homomorphism of two regular language is also regular .Substitution of two regular language is also regular.
Shuffle of two regular language : shuffle ( L1 ,  L2 ) = { x1y1x2y2.......Xkyk | x1x2....Xk is n y1y2....Yk is in L2} is also regular.Symmetric difference of two language is also regular.
If L is regular then under root L is also a regular language : under root L = { w | ww in L }.If L is then max(L) is also regular .If L is then Min(L) is also regular Min (L) = { w | w is also a regular language ɛ L and there is no proper prefix of w is in L }.
Regular language are not closed under subset , super set , infinite union as well as infinite intersection , i,e a subset or super set of a regular language need not be regular,Infinite union and infinite intersection of regular language need not be regular, 




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