Consider the set $\phi$ consisting of the following formulas.
$$ \exists x \exists y \exists z \neg (x = y ~ \lor x = z ~ \lor y = z) $$ $$ \forall x \exists y ~ gimble(x, y) ~ \land \exists x ~ \neg gimble(x, x) $$ $$ \forall x \forall y \forall z ~ (gimble(x, z) ~ \land ~gimble(y, z) \implies x =y) $$
1) How can I describe a model for $\phi?$
Should I use a table? How would I do that?
2) How can I describe an interpretation that is not a model for $\phi?$
3) Is the set $\phi$: (i) valid, (ii) satisfiable but not valid or (iii) unsatisfiable?
I think that $\phi$ is probably satisfiable but not valid (ii) because it holds under some interpretation e.g. the interpretation for 1) but it is not valid because it does not holds under every interpretation i.e the interpretation for 2)