S1 : True ,
S2 : False. Let A be any nonregular language, and let B = C = . Then the pairwise intersections are regular, but the union is nonregular.
S3 : True, Suppose that C ∪ F is context-free and F is finite. We have
The language C ∪F is context-free by assumption. The other language, (F - C)' , is regular by the closure properties because it is the complement of the finite (hence regular!) language F-C. Therefore, C is the intersection of a context-free language and a regular language. Which is context free.
S4 : True , For DPDA acceptance with empty stack and acceptance with Final State are not equivalent. In the case of DPDA, acceptance by the empty stack is weaker, since the language has to satisfy the prefix property.It can also be said that set of languages accepted by a DPDA by empty stack is the set of languages accepted by a DPDA by final state and which has the prefix property.
Note : A DPDA with acceptance by empty stack cannot even accept all regular languages- example a∗.
But for a NPDA Final-state acceptance and empty-stack acceptance are equivalent.
S5 : False. Regular Languages are not closed under infinite union.
Hence Option D