This time I've problem with following exercise:
Let be a first-order language and for each let be a class of -structures.
Show that .
Here are some basic definitions from the book "Model Theory", W. Hodges (which could help):
A sentence is a formula with no free variables. A theory is a set of sentences.
We say that is a model of , or that is true in , when holds. Given a theory in , we say that is a model of , in symbols , if is a model of every sentence in .
Let be a language and a class of -structures. We define the -theory of , , to be the set (or class) of all sentences of such that for every structure in . We omit the subscript when is first-order: the theory of , , is the set of all first-order sentences which are true in every structure in .
Thanks for any help.