A universal ( ) formula is one of the form , where is a quantifier free. An existential ( ) formula is of the dual form . Let be a substructure of , and let .
(a) Show that if and is existential, then . And if and is universal, then .
(b) Conclude that the sentence is not logically equivalent to any universal sentence, nor to any existential sentence.
(a) Since is a substructure of , we have an identity map from into .
Suppose . Then, there are some such that . By homomorphism theorem, we have iff . Therefore, for , i.e., .
Now, to prove,
"If and is universal, then ".
For any , we have
by assumption. If in are also in , then I may use the similar method to the above. But it does not look O.K to assume as well.
Any help will be appreciated.