$F = \forall x R(x,x)$
$G = \forall x \forall y \forall z (R(x,y) \land R(y,z) \rightarrow R(x,z))$
a) Find a model for $G$, that isn't a model for $F$.
b) Does a domain $D$ exist, so that $val_{D,I,\beta}(F) = true$ for every Interpretation $(D',I)$ with $D = D'$.
c)Does a domain $D$ exist, so that $val_{D,I,\beta}(G) = true$ for every Interpretation $(D',I)$ with $D = D'$.
d)Is $F$ valid? Is $G$ valid? Explain your answer.
I'm doing exercises from a textbook, but there aren't any answers.
a)$D = \mathbb{N}$
$I(R) = \{(x,y) \in D \times D | x < y\}$
The relation is transitive, but can't be reflexive.
I don't understand what is meant in b) and c).
d)For $F$ or $G$ to be valid, they would have to be true for every Interpretation $I$. Like in a) we found an interpretation for which $F$ isn't valid.
Wouldn't this then answer b), that such a domain doesn't exist for every Interpretation $I$.
Is there a simple example for a non transitive relation?
For b) and c), you need to find a set $D$ such that, however you interpret the relation symbol $R$, $F$ (resp. $G$) will hold. Hint : consider $D := \emptyset$.
For d), Just take $D = \{0,1,2\}$ and interpret $R$ as follow : $\{(0,1),(1,2)\}$.