Question

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.
(a) There is a student in this class who has chatted with exactly one other student.
(b) No student has solved at least one exercise in every section of this book.
(c) Every movie actor has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon.

Answer 1

(a) P(x): a student has chatted with exactly one other students.

$\exists x \in \{class\} \; P(x)$

$\neg \exists x \in \{class\} \; (P(x)) = \forall x \in \{class\} \; ( \neg P(x))$

All students in the class have not chatted with exactly one other student.

(b) Q(x,y) : student has solved at least 1 exercise from y section

$\forall x \in \{class\} \forall y \in \{sections \; of \; book\} (\neg Q(x,y))$

$\neg(\forall x \in \{class\} \forall y \in \{sections \; of \; book\} ( \neg Q(x,y)) = \exists x \in \{class\} \exists y \in \{sections \; of \; book\} ( Q(x,y))$

There is a student in class and there is a section from book from which he/she solved at least one exercise.

(c) P(x): actor has been in a movie with Kevin Bacon

Q(x): actor has been in a movie with someone who has been in a movie with Kevin Bacon.

$\forall x \in \{actors\} (P(x) \lor Q(x))$

$\neg (\forall x \in \{actors\} (P(x) \lor Q(x))) = \exists x \in \{actors\} (\neg P(x) \land \neg Q(x))$

There is an actor that hasn't been in a movie with Kevin Bacon and hasn't been in a movie with someone who has been in a movie with Kevin Bacon.