prove that any two isomorphic structures are elementarily equivalent Imagine we have two L-structures $M$ and $N$.
For each L-sentence $\phi$ , $M$ models $\phi$ iff $N$ models $\phi$. We call $M$ and $N$ two elementary equivalent L-structures.  

We say $M$ and $N$ are isomorphic if there exists a function from $U_M$ to $U_N$ which is 1 to 1 and onto and preserves symbols.  ( i mean an embedding which is also onto ) 

Now i want to prove that if two structures are isomorphic, then they are elementary equivalent.  

Note : When we speak about L-formulas, we prove these kind of questions by induction on the complexity of the formulas. but in this case, we have sentences not formulas.
Choose an isomorphism $f$. Prove the more general statement: for any formula $\phi$, if we consider any assignment of values in $M$ to the variables, and the corresponding (via $f$) assignment in $N$, the formula $\phi$ will be satisfied by the one assignment if and only if it is satisfied by the other.
The structures (2Z, 0,+) and (Z,0,+) are isomorphic in the language of gropus but are not elementary equivalent. E.g. the closed formula "There exists v such that (v+v=2)" is not satisfied in the first but is satisfied in the second.