Doubt: applicability of set theory rules in Grammar

Question

I studied this link. https://gatecse.in/closure-property-of-language-families/ & https://gatecse.in/grammar-decidable-and-undecidable-problems/ .

I have doubt regarding intersection operation.

if we say that Regular ⊂ DCFL ⊂ CFL ⊂ REC ⊂ RE

then by set theory rule intersection of ( Regular Lang & RE ) must give Regular Lang.  And as Regular Lang is subset of Recursive . 

intersection of ( Regular Lang & RE )  must be recursive(which is incorrect according to https://gateoverflow.in/996/gate2006-33)

So is intersection operator does not work as above for grammar?

 

Answer 1

When they write $Regular ⊂ DCFL ⊂ CFL ⊂ REC ⊂ RE$, here it means that $Regular$ is Regular set  i.e. Set of All

Regular  languages --- Not a particular Regular language. Similarly, Other are Set of All DCFL languages, 

Set of All CFL languages, Set of  All REC languages, Set of All RE languages. 

then by set theory rule intersection of ( Regular Lang & RE ) must give Regular Lang. 

Intersection of Regular Set and RE Set is Indeed Regular set. But Intersection of some Regular language and some RE language need not be Regular. 

Remember, Some Regular language need Not be a subset of Some RE language. Regular Set (Set of all Regular languages) is a subset of RE set(Set of all RE languages).

 

Comments

In short, STRICT / PROPER subset ??