Model Theory problem

Question

Good afternoon. I need some help with this little problem. I hope somebody could help me. Thanks a lot

Assume that $A\equiv B$. Then there exists a $C$ such that $A\prec C$ and $B\prec C$.

Answer 1

It's hardly the afternoon for me, but here's a hint: use compactness theorem (if $A\subseteq C$, what does it mean that $A\preceq C$? Try to express it as $C\models T$ for some theory $T$).

As JDH suggested, $A\preceq C$ and $B\preceq C$ in the sense that $A\subseteq C$ and $B\subseteq C$ is not always possible for simple reason that as sets $A\cap B$ may be somewhat different than we would expect from two elementary substructures of $C$ (for example, if $C$ has a constant symbol and $A\cap B=\emptyset$). But there is always a $C$ into which $A,B$ can be elementarily embedded.

Answer 2

The claim is false, if by $A\prec C$ you mean that $A$ is an elementary substructure of $C$. For example, suppose that there is a constant symbol in the language, which is interpreted in $A$ as a different object than in $B$. So they cannot both be elementary substructures of a single structure $C$, since all substructures of $C$ will interpret that constant symbol in the same way.

But if by $A\prec C$ you mean that $A$ embeds elementarily into $C$, that is, that $A$ is isomorphic to an elementary substructure of $C$, then you should follow the hints of tomasz.