Let I(x) be “x has an internet connection” and let
C(x, y) be “x and y have chatted over the internet.” Assume
the universe of discourse consists of all students in your class.
Express the following using quantifiers.There are two students in the class who between them
have chatted with everyone else in the class.The answer of all the websites:∃x∃y(x≠y∧∀z(C(x,z)∨C(y,z))).
doesn't should it be ∃x∃y(x≠y∧(∀z(C(x,z)∨∀z(C(y,z))))?
Because their answers state that there exists two different person x y ,for every people,either x will chat with him or y will.
is it my misunderstanding of the question or the answer of internet is wrong.
Best Answer
You are misunderstanding the expression "between them", I think. This means, e.g. that in the class everyone has chatted with $x$ or with $y$ (but not necessarily with both of them) (or in graph terms these two points form a "vertex cover" of the "chat graph"). Your disjunction means that all $z$ have chatted with $x$, or or all $z$ have chatted with $y$, which is very different. That's why
$$\forall z:(C(z,x) \lor C(z,y))$$ is correct.