I need to show that if A and B are isomorphic, then A and B are elementarily equivalent with a proof by induction on formulas.
A and B are first-order structures for a first-order language, L.
A and B are isomorphic if there is a bijection h: domain of A -> domain of B such that
h(c^A)=c^B
h(f^A(a1,...,an))=f^B(a1),...,f^B(an)
(a1,...,an) contained in P^A iff (h(a1),...,h(an)) contained in P^B.
I also am given that
Th(A)={psi l A l= psi}
A is logically equivalent to B iff Th(A) =Th(B)
*where Th(A) means the theory of A
Hi nsarocket17,
Here are the hints:
First prove this lemma:
Let be a language and and be -structures and let be a homomorphism. Then for every term and for all elements of , the following holds: .
Proof is very easy, by induction on .
What you're trying to prove is then immediate consequence of the following proposition:
Let be a language and and be -structures, let be an isomorphism, let be a formula of and let be elements of . Then is satisfied in by the sequence if and only if is satisfied in by sequence .
Proof of the proposition is again very easy, by induction on . If is atomic you just use that isomorphisms preserve relations and the lemma above. Induction steps for logical connectives and for quantification are more or less straightforward verifications.