So my problem is the following:
Let $L$ and $L'$ be languages of first order so that every symbol of $L$ is a symbol of $L'$. Furthermore let $S$ be a set of $L$ - formulas. Prove:
$S$ is satisfiable in a model for $L$ $\Leftrightarrow$ $S$ is satisfiable in a model for $L'$
So this is what i got so far:
$\Rightarrow$: If $S$ is satisfiable in a model for $L$ then there exsists an interpretation $b$ such that $S$ is true. We know that every symbol of $L$ is a symbol of $L'$. Therefore $S$ is also a set of $L'$ - formulas and since the symbols are the same we can use the interpretation $b$ so that $S$ is satisfiable for $L'$.
I am not sure how the other implication might go, and it would be appreciated if somebody could tell me, if what i got is correct (we just recently introduced first order languages).
Anyway thanks in advance!
Yes, that half of the proof you did is correct. For the other half:
If $S$ is satisfiable in a model for $L'$, then that means that there is an interpretation for $L'$ that sets $S$ to true. But given the fact that $S$ is a set of $L-formulas$, we can just take that part of the interpretation that deals with the symbols in $L$, and that will be an interpretation for $L$ that still sets $S$ to true. Hence, $S$ is satisfiable in a model for $L$