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).
(a) Yes, you're correct.
$\forall x;K(Liam,x)$
(b) Also correct.
$\forall x,y;K(x,y)$
(c) Also correct.
$\exists x,y;(\neg K(x,y)\land\neg K(y,x))$
(d) Also correct.
$\forall x,y;(\neg K(x,y)\land\neg K(y,x)\implies\exists z;(K(z,z)\land K(z,y)))$
(e) We're given $\exists x;\forall y;(y \neq x\implies K(y,x)\land\neg K(x,y))$
No. There is no one not at the party. It was given at the beginning that "the universe of discourse is the set of people at a party hosted by Liam".
Parsing roughy into english, the statement says "There exists x such that for all y, y does not equal x implies y knows x and x does not know y".
The "x does not equal y" means that they are not the same person.So let's get rid of that part, and replace "x" with "a person", and "y" with an "other".
"There exists a person such that for all others, the other person knows the first person and the first person does not know the other person."
Now we can see that this basically means
"There is at least one person at the party that everyone else knows, but who doesn't know anyone else."