Let $L$ be a first order language with equality such as $F_{0} = \{a,b\}$, $F_{1}=\{g,h,f\}$, $F_{2}=\{f\}$, $R_{1}=\{S,T\}$ and $R_{2}=\{R,Q\}$. For each of the following propositions indicate a structure of $L$ where it is true and one where it is false.
- $\forall y \exists x (x \not= g(y))$
- $\forall x (T(x) \implies \forall y Q(x,y))$
You can always do a lot of different things with the natural numbers (0,1,2,3,...), so that is usually a good domain to play with.
Indeed, for the first one, you can interpret $g$ as the successor function: the statement will be true since for any $y$ you can always pick 0 for $x$.
For the second one, you have a 1-place predicate T (so interpret this as some kind of property of natural numbers ... like 'x is an even number' or 'x is a prime number' or ...) and a 2-place predicate Q (so interpret this as a relation between 2 numbers ... such as 'x is smaller than y' or 'x is divisible by y' or ...). Lots of options here: I am sure you can figure out something that will work!