I'm trying to solve this exercise but I'm stuck. I have to consider the following set of formulas H:
∀x∃y(Rx,y)
¬∃x(Rx,x)
∀x∀y∀z(Rx,y∧Ry,z→Rx,z)
I have to show an interpretation that makes H true in the Integer domain.
My biggest issue is that I don't understand how to start the construction of the interpretation. What should I look for? Any useful theoremes/rules?
Well, you are looking for a $2$-place relationship on the integers. The first one that comes to my mind is $<$. And lo and behold, that one works:
$∀x∃y(Rx,y)$ becomes $∀x∃y (x < y)$, i.e. for every integer there is a greater integer. Yes, that's true!
$¬∃x(Rx,x)$ becomes $¬∃x(x<x)$. i.e. there is no integer smaller than itself. Also true!
$∀x∀y∀z(Rx,y∧Ry,z→Rx,z)$ becomes $∀x∀y∀z(x<y∧y<z→x<z)$, i.e. whenever a first integer is smaller than a second, and that second is smaller than a third, then the first is smaller than the third. Again: True!