Finding an interpretation that makes a formula true

Question

I understand the basic rules of propositional and predicate logic, and what makes formulas true in a general sense, but I am stuck on how to answer specific questions that ask of "each of the formulas below, state an interpretation that makes it true and an interpretation that makes it false"

For instance, one of first questions asks me to state an interpretation that makes ∃x(F x ∧ ¬∃yGxy) true and one that makes it false. Should my answer simply put:

Domain = a |Fa & ¬Gaa| = F, and |Fa & Gaa| = T? Or would I need to specify, for example, that the interpretation of the predicate '|G|' = \emptyset and the interpretation of '|F|' = a to give an example of an occasion where the formula is true?

Answer 1

No; you have to specify the domain of the interpretation and the "menaing" assigned to the predicates.

E.g.

as domain, the set $\mathbb N$ of natural numbers;

$F(x)$ interpreted as $x=0$;

$G(x,y)$ interpeted as $x > y$.

With this interpretation, the formula :

$∃x(Fx ∧ ¬∃yGxy)$

means :

"there is a number $x$ such that $x=0$ and for all numbers $y : \ x \le y$"

which is true in $\mathbb N$ exactly when $x=0$.