# Math Help - Negating Quantified Statements

1. ## Negating Quantified Statements

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”)

b) There is a student in this class who has chatted with exactly one other student.

I was able to correctly translate the statement, the answer being $\exists x \exists y (Q(x,y) \wedge \forall z(Q(x,z) \rightarrow y = z))$, where Q(x,y) represents, "x has chatted with y."

When I alter the statement by negating it, it becomes $\forall x \forall y (Q(x,y) \rightarrow \exists z(Q(x,y) \vee y \ne z ))$ (this required a few propositional equivalences to get to this state.) The difference between my answer and the true answer is, that the disjunction is actually a conjunction.

What did I do wrong?

2. ## Re: Negating Quantified Statements

Originally Posted by Bashyboy
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”)
b) There is a student in this class who has chatted with exactly one other student.
I was able to correctly translate the statement, the answer being $\exists x \exists y (Q(x,y) \wedge \forall z(Q(x,z) \rightarrow y = z))$, where Q(x,y) represents, "x has chatted with y."
When I alter the statement by negating it, it becomes $\forall x \forall y (Q(x,y) \rightarrow \exists z(Q(x,y) \vee y \ne z ))$ (this required a few propositional equivalences to get to this state.) The difference between my answer and the true answer is, that the disjunction is actually a conjunction.
It is very very basic: $\neg \left[ {A \wedge (B \to C)} \right] \equiv \neg A \vee \left( {B \wedge \neg C} \right)$

3. ## Re: Negating Quantified Statements

Oh, yes. You are very right: it is basic. Thank you very much for the clarification.