I am revising predicate logic by answering questions from past papers, but haven't been given the answers. So I was hoping to check to make sure I am understanding this area properly.
The question is:
Let the logical statement K(a,b) stand for "a knows b", where the universe of discourse is the set of people at a party hosted by Liam.
Translate the following into logical notation:
(a) Liam knowns everyone at the party.
(b) Everyone knows each other.
(c) There are people who do not know each other.
(d) For any two people who do not know each other there is someone who know them both.
(e) Translate the following into English:
$\exists x \forall y (y \not = x) \rightarrow (K(y,x) \land \neg K(x,y))$
My answers are:
(a) Vb (K(Liam,b))
(b) Va Vb (K(a,b))
(c) Ea Eb (~K(a,b) & ~K(b,a))
(d) Va Vb (~K(a,b) & ~K(b,a) ==> Ey (K(y,a) & K(y,b)))
(e) There is at least one person NOT at the party who doesn't know anyone at the party, but everyone at the party knows them. (English is not my first language, so I might be completely wrong here).