How do you even start with this problem ?

Question

It's been 2 hours that i am searching everywhere for an idea on how to start on this problem, so i come here to ask for help as i can't grasp the idea behind it.

Considering the Signature Σ = (C {a,b}, F = Ø, P = {R², E², S¹}) and the two formula:

F= ∀x(∃yR(x,y) => (E(x,a) ∨ E(x,b)))

G = ∃x∀y(R(x,y) ∧ S(x))

1. Give a model for G

2. Give an implementation that is not a model for F

3. Give a model for F ∧ G

Answer 1

Let's try to find models with the natural numbers.

1. To model G, we can take R(x,y) to be the relation x < y and S(x) to be the predicate "x is even." Then the natural numbers form a model here, with x = 0.
2. Let's extend our model above by defining the relation E in such a way that F does not hold. Let E(p, q) mean "p – q = 0". Take for instance a = 3 and b = 4. Then the sentence F does not hold for any x not equal to 3 or 4.
3. The model for F and G can be very simple. Take R and S as before; let E(x,y) = R(y,x), and let a = b = 0. Then G holds as before, and F holds because for every x, there does exist a y greater than x and every x is greater than or equal to 0.

There are definitely other more interesting models, but these are the simplest I could come up with.