Express the meaning ofthe following predicate logic statements. Examine whether they are true on infinite or finite world
- $[\forall x R(x,x) \land \forall x \forall y (R(x,y) \land R(y,x) \rightarrow x = y ) \land \forall x \forall y \forall z (R(x,y) \land R(y,z)\rightarrow x = z)] \rightarrow \exists x \forall y (x \neq y \rightarrow \neg R(x,y)) \land \exists x \forall y (x \neq y \rightarrow \neg R(y,x)$
- $[\forall x R(x,x) \land \forall x \forall y (R(x,y) \land R(y,x) \rightarrow x = y ) \land \forall x \forall y \forall z (R(x,y) \land R(y,z)\rightarrow x = z) \land \forall x \forall y (R(x,y) \lor R(y,x))] \rightarrow \exists x \forall y R(x,y) \land \exists x \forall y R(y,x)$
The second one describes that a total order relation has an element x that relates with everyone and and an element x that it is related to everyone, for example if $R(x,y)= x<=y$ and we are in a finite world like [1,100] expresses that there is a maximum and minimum value and hold true in finite worlds.
The first one , the left side express a partial order relation and the consequence I think somehow express that not every - two elements are related ( because a partial order , is a set of values that are not all related) . Can anybody suggest anything else?