Enderton 2.2.18

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.

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(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.