I'm trying to translate English into Predicate Logic but I am not sure if I did it correctly. So I want to check it before I attempt to prove it.
The sentence is : Everyone who loves Bella also loves either Claire or Daisy, and none of them speak French, but at least one person who loves Daisy speaks German.
L(x, y) = x loves y,
b = Bell, c = Claire, d = Daisy
F(x) = speaks French
G(x) = speaks German
My first translation is
$$\forall x[L(x,b) \to (L(x,c) \lor L(x,d)] \land \neg\exists x[F(x)] \lor \exists x[L(x,d) \land G(x)]$$
For this one, I am unsure whether to put ¬∃x[F(x)] inside or outside the universal quantifier. Would it make a different if I put it out?
My second translation attempt :
$$\forall x\neg\exists y[ [(L(x,b) \to (L(x,c) \lor L(x,c)) \land \neg F(y) \land \neg(x = y)] ] \land \exists x[L(x,d) \land G(x)]$$
Which of these are correct? (If any)
Sentence is : if everyone either speaks French or loves Daisy, and Bella speaks neither French nor German, then somebody must love Daisy.
In this one I translate it as :
$$\forall x[F(x) \lor L(x,d)] \land \neg(F(b) \land G(b)) \to \exists x[L(x,d)]$$
Is the correct? and should I add rule like ¬(x = d) saying that daisy cannot love herself? is it necessary?