Closure properties Table in TOC

The Below Table shows the Closure Properties of Formal Languages :

REG = Regular Language
DCFL = deterministic context-free languages, 
CFL = context-free languages,
CSL = context-sensitive languages,
RC = Recursive.
RE = Recursive Enumerable

Consider L and M are regular languages :

1. The Kleene star –
    ∑*, is a unary operator on a set of symbols or strings, ∑, that gives the infinite set of all possible strings of all possible lengths over ∑ including λ.
2. Kleen Plus – 
   The set ∑+ is the infinite set of all possible strings of all possible lengths over ∑ excluding λ.
3. Complement – 
   The complement of a language L (with respect to an alphabet E such that E^* contains L) is E*–L. Since E* is surely regular, the complement of a regular language is always regular.
4. Reverse Operator – 
   Given language L, L^R is the set of strings whose reversal is in L.
5. Complement –
   The complement of a language L (with respect to an alphabet E such that E^* contains L) is E*–L. Since E* is surely regular, the complement of a regular language is always regular.
6. Union –
   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 U M).
7. Intersection – 
   Let L and M be the languages of regular expressions R and S, respectively, then it is a regular expression whose language is L intersection M.
8. Set Difference operator – 
   If L and M are regular languages, then so is L – M = strings in L but not M.
9. Homomorphism –
   A homomorphism on an alphabet is a function that gives a string for each symbol in that alphabet.
10. Inverse Homomorphism – 
    Let h be a homomorphism and L a language whose alphabet is the output language of h. h^-1 (L) = {w | h(w) is in L}.
11. Substitution –
    substitution is a letter-to-language mapping, which is likewise extended to a string-to-language mapping. By identifying a singleton language {x} to the tx, morphisms are seen as special cases of substitutions.
12. Left quotient – Left quotient , or quotient of a language L by a word w is The language Lw = {x ∈ Σ* | wx ∈ L}
    The right quotient of L1 with L2 is the set of all strings x where you can pick some  y from L2 and append it to x to get something from L1. That is, x is in the quotient if there is y in L2 for which xy is in L1.)

Operations	REG	DCFL	CFL	CSL	RC	RE
Union	Y	N	Y	Y	Y	Y
Intersection	Y	N	N	Y	Y	Y
Set Difference	Y	N	N	Y	Y	N
Complement	Y	Y	N	Y	Y	N
Intersection with a Regular Language	Y	Y	Y	Y	Y	Y
Union with a Regular Language	Y	Y	Y	Y	Y	Y
Concatenation	Y	N	Y	Y	Y	Y
Kleene Star	Y	N	Y	Y	Y	Y
Kleene Plus	Y	N	Y	Y	Y	Y
Reversal	Y	N	Y	Y	Y	Y
Epsilon-free Homomorphism	Y	N	Y	Y	Y	Y
Homomorphism	Y	N	Y	N	N	Y
Inverse Homomorphism	Y	Y	Y	Y	Y	Y
Epsilon-free Substitution	Y	N	Y	Y	Y	Y
Substitution	Y	N	Y	N	N	Y
Subset	N	N	N	N	N	N
Left Difference with a Regular Language (L-Regular)	Y	Y	Y	Y	Y	Y
Right Difference with a Regular Language (Regular-R)	Y	Y	N	Y	Y	N
Left Quotient with a Regular Language	Y	Y	Y	N	Y	Y
Right Quotient with a Regular Language	Y	Y	Y	N	Y	Y

NOTE : If we do union, intersection or set difference of any language with regular language, language doesn’t change .
Example   

• CFL ∩ Regular is CFL. 
• CFL ∪ Regular is CFL. 

It’s always a good idea to convert the secondary operations into primary operations.
Let L₁ and L₂ be two languages.

NOTE :  For ⊆ , ⊇ , infinite union, infinite intersection, infinite set difference, No language is closed. 
              Under these operations, language may or may not be regular.

Let us consider some cases with concatenation operation :

• Regular . Regular                   ⇒                       Regular
• Regular . Non-Regular           ⇒                       May or may not be regular
• Non-Regular . Non-Regular  ⇒                      May or may not be regular
• If L1 . L2 is regular                 ⇒                       L1 may or may not be regular
• If L1 . L2 is regular                 ⇒                       L2 may or may not be regular
• If L1 . L2 is Non-Regular        ⇒                        Atleast one of them should be Non-regular.