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Math Help - Negating Quantified Statements

  1. #1
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    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?
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  2. #2
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    Re: Negating Quantified Statements

    Quote Originally Posted by Bashyboy View Post
    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)
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  3. #3
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    Re: Negating Quantified Statements

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