Model THeories not equivalent

Question

How can we show that Th(Rationals) does not equal Th(Integers) does not equal Th(Naturals)?

Where Th(M) is a the set of all L sentences which are true in a model, M.

I know that L is defined as = {0,+}.

Answer 1

To distinguish between the naturals and the integers, note that for example $\forall x\exists y(x+y=0)$ is true in the integers but not in the naturals.

The same sentence is also true in the rationals, so it serves to distinguish between the two structures.

To distinguish between the rationals and the integers, note for example that the sentence $\forall x\exists y(x=y+y)$ is true in the rationals, but not in the integers.

There are many other sentences that will do the job.