Hi,

This time I've problem with following exercise:

Let $\displaystyle L$ be a first-order language and for each $\displaystyle i \in I$ let $\displaystyle \mathbb{K}_i$ be a class of $\displaystyle L$-structures.

Show that $\displaystyle \mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i) =\bigcap_{i \in I} \mbox{Th}(\mathbb{K}_i)$.

Here are some basic definitions from the book "Model Theory", W. Hodges (which could help):

Asentenceis a formula with no free variables. Atheoryis a set of sentences.

We say that $\displaystyle A$ is amodelof $\displaystyle \Phi$, or that $\displaystyle \Phi$ istruein $\displaystyle A$, when $\displaystyle A \models \Phi$ holds. Given a theory $\displaystyle T$ in $\displaystyle L_{\infty \omega}$, we say that $\displaystyle A$ is amodelof $\displaystyle T$, in symbols $\displaystyle A \models T$, if $\displaystyle A$ is a model of every sentence in $\displaystyle T$.

Let $\displaystyle L$ be a language and $\displaystyle \mathbb{K}$ a class of $\displaystyle L$-structures. We define the $\displaystyle L$-theoryof $\displaystyle \mathbb{K}$, $\displaystyle Th_L(\mathbb{K})$, to be the set (or class) of all sentences $\displaystyle \Phi$ of $\displaystyle L$ such that $\displaystyle A \models \Phi$ for every structure $\displaystyle A$ in $\displaystyle \mathbb{K}$. We omit the subscript $\displaystyle _L$ when $\displaystyle L$ isfirst-order: the theory of $\displaystyle \mathbb{K}$, $\displaystyle \mbox{Th}(\mathbb{K})$, is the set of allfirst-ordersentences which are true in every structure in $\displaystyle \mathbb{K}$.

Thanks for any help.